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Superspace or One Thousand and One Lessons in Supersymmetry
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Title: Superspace or One Thousand and One Lessons in Supersymmetry
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Creator: Gates, S. James, Grisaru, Marcus T., Rocek, Martin, Siegel, Warren
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Subjects / Keywords: science, high energy physics, Superspace, Supersymmetry, superfields, Supergravity, Covariant derivatives, Constrained superfields, Component expansions, Superintegration, Scalar multiplet, Yang-Mills gauge theories, Superforms, Bianchi identities, DeSitter supersymmetry, Supergraph rules, Covariant Feynman rules, Nonlinear realizations, SuperHiggs mechanism, …
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Abstract: Free college physics textbook on superspace and supersymmetry. Contents: 1) Introduction. 2) A Toy Superspace. 3) Representations of Supersymmetry. 4) Classical, global, simple (N = 1) superfields. 5) Classical N = 1 supergravity. 6) Quantum global superfields. 7) Quantum N = 1 supergravity. 8) Breakdown.
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SUPERSPACEor Onethousandandone lessonsinsupersymmetryS.JamesGates,Jr. MassachusettsInstituteofTechnology,Cambridge,Massachusetts (Present a ddress: UniversityofMaryland,CollegePark,Maryland) gatess@wam. umd.edu MarcusT.Grisaru BrandeisUniversity,Waltham,Massachusetts (Present a ddress: McGillUniversity,M ontreal, Que bec) grisaru@physics.mcgill.ca MartinRo cek StateUniversityofNewYork,StonyBrook,NewYork rocek@insti.physics.sunysb.edu Wa rrenSiegel UniversityofCalifornia,Berkeley,California (Present a ddress: StateUniversityofNewYork) wa rren@wcgall.physics.sunysb.edu

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LibraryofCongressCataloginginPublicationData Mainentryundertitle: Superspace:onethousandandonelessonsinsupersymmetry. (Frontiersinphysics;v.58) Includesindex. 1.Supersymmetry.2.Quantumgravity. 3.Supergravity.I.Ga tes,S.J.II .Series. QC 174.17.S9S971983530.1283-5986 ISBN0-8053-3160-3 ISBN0-8053-3160-1(pbk.)

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Superspaceisthegreatestinventionsincethewheel[1]. Preface Saidto,,and:Letswritear eviewpaper.Saidand:Great idea!Said:Naaa. Butafewdayslaterhadproducedatableofcontentswith1001items. ,,andwrote.Thendidntwrite.Thenwroteagain.Thereviewgrew; andgrew;andgrew.Itbecameanoutlinef orabook;itbecamea“rstdraft;itbecame as econddraft.Itbecameaburden.Itbecameagony.Temperswerelost;andhairs; an da fewpounds(alas,quicklyregained).Theyarguedabout;vs..,about whichvs.that,vs.,  vs .   , +  vs .  .M ad eb ad puns,drewpicturesonthe blackboard,wererudetotheircolleagues,neglectedtheirduties.Bemoaned thepaucityoflettersintheGreekandRomanalphabets,ofhoursintheday,daysin thew eek,weeksinthemonth.,,andwroteandwrote. *** Thismuststop;wewanttogetbacktoresearch,toourfamilies,friendsandstude nt s.Wewanttolookattheskyagain,goforwalks,sleepatnight.Writeasecond volume?Nev er!Well, inaco upleofyears? We be go urreadersindulgence.Wehavetriedtopresentasubjectthatwelike, thatwethinkisimportant.Wehavetriedtopresentourinsights,ourtoolsandour knowledge.Alongtheway,someerrorsandm iscon ceptionshavewithoutdoubtslipped in.Theremustbewrongstatements,mispri nts,mi stakes,awkwardphrases,islandsof incomprehensibility(buttheystartedoutascontinents!).Wecould,probablywe should,improveandimprove.Butwecannolongerwait.Likeclimberswithinsightof thesummitwearerushing,castingasideca ution,reachingtowardsthemomentwhenwe canshoutitsbehindus. Thisisnotapolishedwork.Withoutdoubtsometopicsaretreatedbetterelsewhere.Withoutdoubtwehaveleftouttopicsthatshouldhavebeenincluded.Without doubtwehavetreatedthesubjectfromapersonalpointofview,emphasizingaspects thatwearefamiliarwith,andneglectingsomethatwouldhaverequiredstudyingothers work.Nev ertheless,wehopethisbookwillbeuseful,bothtothosenewtothesubject andtothosewhohelpeddevelopit.Wehavepresentedmanytopicsthatarenotavailableelsewhere,andmanytopicsofinterestalsooutsidesupersymmetry.Wehave [1 ]. A. Oop,Asupersymmetricversionoftheleg,Gondwanalandpredraw(January10,000,000 B.C.), tobediscovered.

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includedtopicsw hosetreatmentisincomplete,andpresentedconclusionsthatarereally onlyconjectures.Insomecases,thisre”ectsthestateofthesubject.Fillinginthe holesandprovingtheconjecturesmaybegoodresearchprojects. Supersymmetryisthecreationofmanyta lentedphysicists.Wewouldliketo thankallourfriendsinthe“el d,wehavemany,fortheircontributionstothesubject, andbegtheirpardonfornotpresentingalistofreferencestotheirpapers. Mostofthew orkonthisbookwasdo newh ilethefourofuswereattheCalifornia InstituteofTechnology,duringthe1982-83academicyear.Wewouldliketothankthe InstituteandthePhysicsDepartmentfortheirhospitalityandtheuseoftheircomputer facilities,theNSF,DOE,theFleischmannF oundationandtheFairchildVisitingScholarsProgramfortheirsupport.SomeoftheworkwasdonewhileM.T.G.andM.R.were visitingtheInstituteforTheoreticalPhysicsatSantaBarbara.Finally,wewouldliketo thankRichardGrisaruforthemanyhourshedevotedtotypingtheequationsinthis book,HyunJeanKimfordrawi ngthedi agrams,andAndersKarlhedeforcarefullyreadinglargepartsofthemanuscriptandforhisu sefulsuggestions;andalltheotherswho helpedus. S.J.G.,M.T .G.,M.R.,W.D.S. Pa sadena,January1983 Au gust2001: Freeversionr eleasedonweb;correctionsandbookmarksadded.

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Contents Preface 1.Introduction 1 2.Atoysuperspace 2.1.Notationandconventions7 2.2.Supersymmetrya ndsuper“elds9 2.3.Scalarmultiplet15 2.4.Vectormultiplet18 2.5.Otherglobalgaugemultiplets28 2.6.Supergravity34 2.7.Quantumsuperspace46 3.Representationsofsupersymmetry 3.1.Notation54 3.2.Thesupersymmetrygroups62 3.3.Representationsofsupersymmetry69 3.4.Covariantderivatives83 3.5.Constraine dsuper “elds89 3.6.Componentexpansions92 3.7.Superintegration97 3.8.Superfunctionaldierentiationandintegration101 3.9.Physical,auxiliary,andgaugecomponents108 3.10.Compensators112 3.11.Projectionoperators120 3.12.On-shellrepresentationsandsuper“elds138 3.13.O-shell“eldstrengthsandprepotentials147 4.Classical,global,simple( N =1)super “elds 4.1.Thescalarmultiplet149 4.2.Yang-Millsgaugetheories159 4.3.Gauge-invariantmodels178 4.4.Superforms181 4.5.Othergaugemultiplets198 4.6. N -extendedmultiplets216 5.Classical N =1superg ravity 5.1.Reviewofgravity232 5.2.Prepotentials244

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5.3.Covariantapproach267 5.4.SolutiontoBianchiidentities292 5.5.Actions299 5.6.Fromsuperspacetocomponents315 5.7.DeSittersupersymmetry335 6.Quantumgloba lsuper “elds 6.1.Introductiontosupergraphs337 6.2.Gauge“xingandghosts340 6.3.Supergraphrules348 6.4.Examples364 6.5.Thebackground“eldmethod373 6.6.Regularization393 6.7.AnomaliesinYang-Millscurrents401 7.Quantum N =1superg ravity 7.1.Introduction408 7. 2. Ba ck gr o und-quantumsplitting410 7.3.Ghosts420 7.4.Quantization431 7.5.Supergravitysupergraphs438 7.6.CovariantFeynmanrules446 7.7.Generalpropertiesoftheeectiveaction452 7.8.Examples460 7.9.Locallysupersymmetricdimensionalregularization469 7.10.Anomalies473 8.Breakdown 8.1.Introduction496 8.2.Explicitbreakingofglobalsupersymmetry500 8.3.Spontaneousbreakingofglobalsupersymmetry506 8.4.Traceformulaefromsuperspace518 8.5.Nonlinearrealizations522 8.6.SuperHiggsmechanism527 8.7.Supergravityandsymmetrybreaking529 Index 542

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1.INTROD UCTION Thereisa“fthdimensionbeyondthatwhichisknowntoman.Itisa dimensionasvastasspaceandast imelessasin“nity.Itisthemiddle groundbetweenlightandshadow,betweenscienceandsuperstition;anditlies be tweenthepitofmansfearsandthesummitofhisknowledge.Thisisthe dimensionofimagination.Itisanareawhichwecall,theTwilightZone. RodSer ling 1001:Asuperspaceodyssey Symmetryprinciples,bothglobalandlocal,areafundamentalfeatureofmodern particlephysics.Attheclassicalandphenomenologicallevel,globalsymmetriesaccount formanyofthe(approximate) regularitiesweobserveinnature,whilelocal(gauge) symmetries explainandunifytheinteractionso fthe basicconstituentsofmatter.At thequantumlevelsymmetries(viaWardidentities)facilitatethestudyoftheultraviolet behavior of“eldtheorymodelsandtheirrenormaliz ation.In particular,theconstructionofmodelswithlocal(internal)Yang-Millssymmetrythatareasymptoticallyfree hasincreasedenormouslyourunderstandingofthequantumbehaviorofmatteratshort distances.Ifthisunderstandingcouldbeextendedtothequantumbehaviorofgravitationalinteractions(quantumgravity)wewouldbeclosetoasatisfactorydescriptionof micronatureintermsofbasicfermioniccons tituentsformingmultipletsofsomeuni“cationgroup,andbosonicgaugeparticlesrespo nsib lefortheirinteractions.Evenmore satisfactorywouldbetheexistenceinnatureofasymmetrywhichuni“esthebosons andthefermions,theconstituentsandtheforces,intoasingleentity. Supersymmetryisthesupremesymmetry:Ituni“esspacetimesymmetrieswith internalsymmetries,fermionswithbosons,an d(localsup ersymmetry)grav itywithmatter.U nderquitegeneralassumptionsitisthelargestpossiblesymmetryoftheSmatrix.Atthequantumlevel,renormalizablegloballysupersymmetricmodelsexhibit improvedultravioletbehavior:Becauseofcancellationsbetweenfermionicandbosonic contributionsquadraticdivergencesareabsen t;somesupersymmetricmodels,inparticularmaximallyextendedsuper-Yang-Millstheory,aretheonlyknownexamplesoffourdimensional“eldtheoriesthatare“nitetoallordersofperturbationtheory.Locally

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21.INTRODUCTIONsupersymmetricgravity(supergravity)m aybetheonlywayinwh ichnaturecanreconc ileEinsteingravityandquantumtheory.Althoughwedonotknowatpresentifitisa “nitetheory,quantumsupergravitydoesexhibitlessdivergentshortdistancebehavior thanordinaryquantumgravi ty.Outsi detherealmofstandardquantum“eldtheory,it isbelievedthattheonlyreasonablestringtheories(i.e.,thosewithfermionsandwithout quantuminconsistencies)aresupersymme tric;thesein cludemodelsthatmaybe“nite (themaximallysupersymmetrictheories). Atthepresent timethereisnodir ectexperimentalevidencethatsupersymmetryis af undamentalsymmetryofnature,butthecurrentlevelofactivityinthe“eldindicates thatmanyphysicistsshareourbeliefthats uchevidencewilleventuallyemerge.Onthe theoreticalside,thesymmetrymakesitpossibletobuildmodelswith(super)natural hierarchies.Onestheticgrounds,theideaofasuperuni“edtheoryisveryappealing. Evenifsupersymmetryandsupergravityar enottheult imatetheory,th eirstudyhas in creasedourunderstandingofclassicalandquantum“eldtheory,andtheymaybean im po rtantstepintheunderstandingofsomeyetunknown,correcttheoryofnature. Wemeanby(P oincar e)supersymmetryanextension ofordinaryspacetimesymmetriesobtainedbyadjoining N spinorialgenerators Q whose an ticommutator yieldsa translationgenerator: { Q Q } = P .Thissy mmetrycanberealizedonordinary“elds (functionsofspacetime)bytransformationsthatmixbosonsandfermions.Suchrealizationssucetostudysupersymmetry(onecanwriteinvariantactions,etc.)butareas cumbersomeandinconvenientasdoingvectorcalculuscomponentbycomponent.A compactalternativetothiscomponent“eldapproachisgivenbythe superspace--super“eld approac h.Superspaceisanextensionofordinaryspacetimetoinclude extra anticomm uting coordinatesintheformof N two-comp onentWeylspinors Super“elds( x )aref unctionsde“nedoverthisspace.Theycanbeexpandedina Taylorserieswi threspecttotheanticommutingcoordinates ;b ecausethesquareofan anticommutingquantityvanishes,this serieshasonlya“nitenumberofterms.The coecientsobtainedinthiswayaretheordinarycomponent“eldsmentionedabove.In superspace,supersymmetryismanifest:Th esupersy mmetryalgebraisrepresentedby translationsandrotationsinvolvingboththespacetimeandtheanticommutingcoordinates.Thetransformationsofthecomponent“eldsfollowfromtheTaylorexpansionof thetranslatedandrotatedsup er“elds.Inparticular,thetransformationsmixingbosons

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1.INTROD UCTION3andfermionsareconstanttranslationsofthe coordinates,andrelatedrotationsof intothespacetimecoordinate x Afurtheradvant ageofsuper“eldsisthattheyautomaticallyinclude,inaddition tothedynamicaldegreesoffreedom,certain unphysical“elds:(1)a uxiliary“elds(“elds withnonderivativekineticterms),neededclassicallyfortheo-shellclosureofthesupersymmetryalgebra,and(2)compensating“elds(“eldsthatconsistentirelyofgauge degreesoffreedom),whichareusedtoenlarg ethe usualgau getransformationstoan entiremultipletoftransformationsforming arepresentationofsup ersymmetry; together withtheauxiliary“elds,theyallowthealgebratobe“eldindependent.Thecompensatorsareparticularlyimportantforquantization,sincetheypermittheuseofsupersymmetricgauges,ghosts,Feynmangraphs ,andsupersy mmetricpower-counting. Unfort unately,ourpresentknowledgeofo-shell extended ( N > 1)supersymmetry is solimitedthatformostextendedtheoriestheseunphysical“elds,andthusalsothe correspondingsuper“elds,areunknown.Onecouldhopeto “nd th e unphysicalcomponentsdirectlyfromsuperspace;theessentiald icultyisthat,ingeneral,asuper“eldisa highlyreduciblerepresentationofthesupersymmetryalgebra,andtheproblembecomes on eo f “nding which representationspermittheconstructionofconsistentlocalactions. Therefore,exceptwhendiscussingthefeatu reswhicharecommontogeneralsuperspace, werest rictourselves inthisvolume toadiscussionof simple ( N =1)sup er“eldsupersymmetry.Wehopetotreatextendedsuperspaceandothertopicsthatneedfurther developmentinasecond(andhopefullylast)volume. Weintro ducesuper“eldsinchapter2forthesimplerworldofthreespacetime dimensions,wheresuper“eldsareverysimilartoordinary“elds.Weskipthediscussion ofno nsuperspacetopics(background“elds,gravity,etc.)whicharecoveredinfollowing ch apters,andconcentrateonapedagogicaltreatmentofsuperspace.Wereturntofour dimensionsinchapter3,wherewedescribehowsupersymmetryisrepresentedonsuper“elds,anddiscussallgeneralpropertiesoffreesuper“elds(andtheirrelationtoordinary “elds).Inchapter4wediscusssimple( N =1)super “eldsinc lassicalgloba lsupersymmetry.Weincludesuchtopicsasgauge-covaria ntderivati ves, supersymmetricmodels, extendedsupersymmetrywithunextendedsuper“elds,andsuperforms.Inchapter5we extendthediscussiontolocalsupersymmetry( supergravity),relyin gheav ilyonthecompensat orapproach.Wediscussprepotentialsandcovariantderivatives,theconstruction

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41.INTRODUCTIONofactions,andshowhowtogofromsuperspacetocomponentresults.Thequantum aspectsofglobalt heoriesisthetopicofchapter6,whichincludesadiscussionofthe background“eldformalism,supersymmetric regularization,anomalies,andmanyexamplesofsupergraphcalculations.Inchapter7wemakethecorrespondinganalysisof quantumsupergravity,includingmanyofthenovelfeaturesofthequantizationprocedure(varioustypes ofghosts).Chapter8describess upersymmetrybre aking,explicit andspontaneous,inc l udingthesuperHiggsmechanismandtheuseofnonlinearrealizations. Wehave notdiscussedcomponentsupersymme tryandsupergravity,realistic superGUTmodelswithorwithoutsupergravity,andsomeofthegeometricalaspectsof classicalsupergravity.Forthe“rsttopicthereadermayconsultmanyoftheexcellent reviewsandlecturenotes.Thesecondisoneofthecurrentareasofactiveresearch.It isourbeliefthatsuperspacemethodseventuallywillprovideaframeworkforstreamliningthe phenomenology,oncewehavebettercon trolofourtools.Thethirdtopicis attractingincreasedattention,buttherearestillmanyissuestobesettled;thereagain, superspacemethodsshouldproveuseful. Wea ssumethereaderhasaknowledgeofstandardquantum“eldtheory(sucient todoFeynmangraphca lculationsinQCD).Wehavetriedtomakethisbookaspedagogicalandencyclopedicaspossible,buthave omittedsomestraightforwardalgebraic detailswhicharelefttothereaderas(necessary!)exercises.

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1.INTROD UCTION5Ahitc hhikersguide Weareh oping,ofcourse,thatthisbookwillbeofinteresttomanypeople,with dierentinterestsandbackgrounds.Thegraduatestudentwhohascompletedacourse inquantum“eldtheoryandwant stost udysuperspaceshould: (1) Read anarticleortworeviewingcomponentglobalsupersymmetryandsupergravity. (2) Read chapter2foraquickand easy(?)introductiontosuperspace.Sections1, 2,and3arestraightforward.Section4introduces,inasimplesetting,theconceptof constrainedcovariantderivatives,andthesolutionoftheconstraintsintermsofprepotentials.Section5couldbes kippedat“rstreading.Sect ion6doesfors upergravity whatsection4didforYang-Mills;super“eld supergravityinthreedimensionsisdeceptivelysimple.Section7introducesquantizationandFeynmanrulesinasimplersituationthaninfour dimensions. (3) Study subsections3.2.a-donsupersymme tryalgebras,andsections3.3.a, 3.3.b.1-b.3,3.4.a,b,3.5and3.6onsuper“e lds,covariantd erivatives,andcomponent expansions. Study section3.10oncompensators;weus ethemextensive lyinsupergravity. (4) Study section4.1aonthescalarmultiplet,andsections4.2and4.3ongauge theori es,theirprepotentials,covariantderivativesandsolutionoftheconstraints.A re ading ofsections4.4.b,4.4.c.1,4.5.aand4.5.emightbepro“table. (5) Takeadee pbreath and slowly studysection5.1,which isourfavoriteapproach togravity,andsections5.2to5.5onsupergravity;thisiswheretheactionis.Foran i nductiveapproachthatstartswiththeprepotentialsandconstructsthecovariant derivati vess ection5 .2issucient,andonecanthengodirectlytosection5.5.Alternatively,onecouldstartwithsection5.3,and ade ductiveapproachbas edonconstrained covariantderivatives,gothroughsection5.4andagainendat5.5. (6) Study sections6.1through6.4onquantizationandsupergraphs.Thetopicsin theses ectionsshou ldbefairlyaccessible. (7) Study sections8.1-8.4. (8)Gobackt othebegi nningand skipnothing thistime.

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61.INTRODUCTIONOu rp articlephysicscolleagueswhoarefamiliarwithglobalsuperspaceshould skim3.1fornotation,3.4-6and4.1,read4.2(no,youdontknowitall),andgetbusy onchapter5. Theexpertsshouldlookforseriousmistakes.Wewouldappreciatehearingabout them. Abriefguidetot heliterature Acomplete listofref erencesisbecomingincreasinglydiculttocompile,andwe havenotattemp tedtodoso.However,thefollowing(incomplete!)listofreviewarticles andproceedingsofvariousschoolsandconfere nces,andthereferencestherein,areuseful andshouldprovideeasyacces st ot hejournalliterature: Forglobals upersymmetry,thestandardreviewarticlesare: P. Fa ye ta nd S. Fe rrara,Supersymmetry,PhysicsReports32C(1977)250. A.SalamandJ.Strathdee,FortschrittederPhysik,26(1978)5. Forcomponent supergravity,thestandardreviewis P. va nN ieuwenhuizen,Supergravity,PhysicsReports68(1981)189. ThefollowingProceedingscontainextensiveandup-to-datelecturesonmany supersymmetryandsupergravitytopics:  RecentDevelopmentsinGravitation(Carges` e 1978),eds.M.LevyandS.Deser, PlenumPress,N.Y.  Sup ergravity(StonyBrook1979),eds.D.Z.FreedmanandP.vanNieuwenhui zen,North-Holland,Amsterdam. TopicsinQuantumF ieldTheoryandGaugeTheories(Salamanca),Phys.77, SpringerVerl ag,Berlin.  Sup erspaceandSupergravity(Cambridge1980),eds.S.W.HawkingandM. Ro cek,CambridgeUniversityPress,Cambridge.  SupersymmetryandSupergravity81(Trieste),eds.S.Ferrara,J.G.Taylorand P.vanNieuwen hui zen,CambridgeUniversityPress,Cambridge.  SupersymmetryandSupergravity82(Trieste),eds.S.Ferrara,J.G.Taylorand P.vanNieuwen hui zen,WorldSc ienti“cPublishingCo.,Singapore.

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Contentsof 2.ATOYSUPERSPACE 2.1.Notationandconventions7 a.Indexcon ventions7 b.Superspace8 2.2.Supersymmetrya ndsuper“elds9 a.Representations9 b.Componentsbyexpansion10 c.Actionsandcomponentsbyprojection11 d.Irreduciblerepresentations13 2.3.Scalarmultiplet15 2.4.Vectormultiplet18 a.Abeliangaugetheory18 a.1.Gaugeconnections18 a.2.Components19 a.3.Constraints20 a.4.Bianchiidentities22 a.5.Mattercouplings23 b.Nonabeliancase24 c.Gaugeinvar iantma sses26 2.5.Otherglobalgaugemultiplets28 a.Superforms:generalcase28 b.Super2-form30 c.Spinorgaugesuper“eld32 2.6.Supergravity34 a.Supercoordinatetransformations34 b.Lorentztransformations35 c.Covariantderivatives35 d.Gaugechoices37 d.1.Asupersy mmetricgauge37 d.2.Wess-Zu minogauge38 e.Fieldstrengths38 f.Bianchiide ntit ies39 g.Actions 42 2.7.Quantumsuperspace46 a.Scalarmultiplet46

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a.1.Generalformalism46 a.2.Examples49 b.Vectormu ltiplet52

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2.ATOYSUPERSPACE 2.1.Notationandconventions Thischapterpresentsaself-containedtreatmentofsupersymmetryinthree spacetimedimensions.Ourmainmotivationfor consideringthiscasei ssimp licity.Irreduciblerepresentationsofsimple( N =1)globals upersymmetryareeasiertoobtain thaninfour dimensions:Scalarsuper“elds(single,realfunctionsofthesuperspacecoordinates)provideonesuchrepresentation,andallothersareobtainedbyappending Lorentzorinternalsymmetryindices.Inaddition,thedescriptionoflocalsupersymmetry(supergravity)iseasier. a.Indexconventions Ourthree-dimensionalnotationisasfollows:Inthree-dimensionalspacetime (withsignature Š ++)theLore ntzgroupis SL (2, R )(insteadof SL (2, C ))andthecorrespondingfundamentalrepresentationactsona real (Majorana)two-componentspinor =( +, Š).IngeneralweusespinornotationforallLorentzrepresentations,denotingspi norindicesbyGreekletters , ... , ... .Thusav ector(thethree-dimensionalrepresentation)willbedescri bedbyasy mmetricsecond-rankspinor V=( V++, V+ Š, VŠŠ)oratra celesssecond-rankspinor V .(Forcompari son,infour dimensionswehavespinors €andavectorisgivenbyahermitianmatrix V€.) Al lo urspinorswillbeanticommuting(Grassmann). Spinorindicesareraisedandloweredbythesecond-rankantisymmetricsymbol C,whichisalso usedtode“nethesquareofaspinor: C= Š C= 0 i Š i 0 = Š C, CC= [ ] Š ; = C, = C, 2=1 2 = i +Š.(2. 1.1) Werepres entsymmetrizat ionandantisymmetrizationof n i ndicesby()and[],respectively(withoutafactorof1 n ).Weoftenmakeuseoftheidentity A[ B ]= Š CAB,(2. 1.2)

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82.ATOYSUPERSPACEwhichfollowsfrom(2.1.1).Weuse C(insteadofthecustomaryreal )tosim p lify therulesforhermiti anconjugatio n.Inparticular,itmakes 2hermitian (r ecall and anticommute)andgivestheconventionalhermiticitypropertiestoderivatives(see below).Notehowev erthatwhereas isreal, isimaginary. b.Superspace Superspaceforsimplesupersymmetryisl abeledbythreespacetimecoordinates xandtwoanticommutin gspi norcoordinates ,denot edcollectivelyby zM=( x, ). Theyhavethehermiticityproperties( zM)= zM.Wede “nederivativesby { , } x [ , x] 1 2 ( ) ,(2. 1.3a) sothat themomentumoperatorshavethehermiticityproperties ( i )= Š ( i ),( i )=+( i ).(2.1 .3b) andthus( i M)= i M.(De “nite)integra tionoverasingleanticommutingvariable is de“nedsothattheintegralistranslat ionallyinvariant(seesec.3.7);henced 1=0,d =aconstantwhichw etaketobe1.Weobservethatafunction f ( )hasaterminatingTaylorseries f ( )= f (0)+ f(0)since { } =0imp lies 2=0.Thusd f ( )= f(0)sotha tintegrationis equivalenttodierentiation.Forourspinorial coordinatesd = andhence d = .(2. 1.4) Thereforethedoubleintegral d22= Š 1,(2.1 .5) andwecan de“nethe -function 2( )= Š 2= Š1 2 *** Weoftenu sethenotation X | toindicatethequantity X evaluatedat =0.

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2.2.Supersymmetryandsuper“elds92.2.Supersymmetryandsuper“elds a.Representations Wede “nefunctionsoversuperspace:...( x )wherethedotss tandforLorentz (s pinor)and/orinternalsymmetryindices.Theytransformintheusualwayunderthe Poincar egroupwith generators P(translations)and M(Lorentzrotations).We grade(ormakesuper)thePoincar ealgebraby introducin ga dditi onal spinor supersymmetrygenerators Q,satisfy ingthe supersymmetryalgebra [ P, P]=0,(2 .2.1a) { Q, Q} =2 P,(2. 2.1b) [ Q, P]=0,(2 .2.1c) aswellastheusualcommutationrelationswith M.Thisalgebra isrealizedon super“elds ...( x )inte rmsofderivativesby: P= i , Q= i ( Š i );(2.2 .2a) ( x, )= exp [ i ( P+ Q)] ( x+ Š i 2 ( ), + ).(2.2 .2b) Thus P+ Qgeneratesasupercoord inatetransformation x = x+ Š i 2 ( ), = + .(2. 2.2c) withreal,constantparameters Thereadercanverifythat(2.2.2)providesarepresentationofthealgebra(2.2.1). Were markinparticularthatiftheanticommutator(2.2.1b)vanished, Qwoulda nnihilateallphysicalstates(seesec.3.3).W ealsonotetha tb ecauseof(2.2.1a,c)and (2.2.2a),notonlyandfunctionsof butalsothespace-timederivatives ca rrya representationofsupersymmetry(aresuper“elds).However,becauseof(2.2.2a),thisis notthecaseforthesp inorialderivatives .Supersymmetricallyinvariantderivatives canbede“nedby DM=( D, D)=( , + i ).(2.2 .3)

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102.ATOYSUPERSPACETheset DM(anti)commut eswiththegenerators:[ DM, P]=[ DM, Q} =0.Weuse [ A B } todenoteagradedcommutator:anticommutatorifboth A and B arefermionic, commutatorotherwise. Thecovariantderivativescanalsobede“nedbytheirgradedcommutationrelations { D, D} =2 iD,[ D, D]=[ D, D]=0;(2 .2.4) or,moreconcisely: [ DM, DN} = TMN PDP; T = i ( ) rest =0.(2. 2.5) Thus,inthelanguageofdierentialgeometry,globalsuperspacehas torsion. The derivati vessatisfythefur theridentities = DD= i + CD2, DDD=0, D2D= Š DD2= i D,( D2)2= .(2. 2.6) TheyalsosatisfytheLeibnitzruleandcanbeintegratedbypartswheninside d3xd2 integrals(sincetheyareacombinationof x and derivati ves).Thefo llowingidentityis useful d3xd2 ( x )= d3x 2( x )= d3x ( D2( x )) | (2.2.7) (whererecallthat | meansevaluationat =0).Theextra space-timederivativesin D(ascomparedto )dropoutafter x -integration. b.Componentsbyexpansion Super“eldscanbeexpandedina(terminating)Taylorseriesin .Forex ample, ...( x )= A ...( x )+ ...( x ) Š 2F ...( x ).(2.2 .8) A B F arethe component “eldsof.Thesupersymmetr ytransfo rmationsofthecomponentscanbederivedfromt hoseofthesuper“eld.Forsimplicityofnotation,weconsiderascalarsuper“eld(noLorentzindices)

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2.2.Supersymmetryandsuper“elds11( x )= A ( x )+ ( x ) Š 2F ( x ),(2.2 .9) Thesupersymmetrytransformation( =0, i n“nitesimal) ( x )= Š ( Š i )( x ) A + Š 2 F ,(2. 2.10) gives,uponequa tingpowersof A = Š ,( 2.2.11a) = Š ( CF + i A ),(2.2 .11b) F = Š i .(2. 2.11c) Itiseasytoverifythatonthecomponent“eldsthesupersymmetryalgebraissatis“ed: Thecommutatoroftwotransformationsgivesatranslation,[ Q( ), Q( )]= Š 2 i etc. c.Actionsandcomponentsbyprojection Theconstructionof(integral)invariantsisfacilitatedbytheobservationthat supersymmetrytransformation sarecoordi natetransformationsinsuperspace.Because wecanign oretotal -derivat ives(d3xd2f=0,whichfo llowsfrom( )3=0)and totalspacetim ederivat ives,we “ndthat any superspaceintegral S = d3xd2 f (, D, ... )(2. 2.12) th at doesnotdependexplicitlyonthecoordinatesisinvariantunderthefullalgebra.If thesuper “eldexpansionintermsofcomponentsissubstitutedintotheintegralandthe -integrationiscarriedout,theresultingcomponentintegralisinvariantunderthe transformationsof(2.2.11)(theintegrandingeneralchangesbyatotalderivative).This alsocanbeseenfromthefactthatthe -integrationpicksoutthe F componentof f whichtransformsasaspacetimederivative(see(2.2.11c)). Wenowdes cribeatechnicaldevicethatcanbeextremelyhelpful.Ingeneral,to obtaincomponentexpressionsbydirect -expansionscanbecumbersome.Amore

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122.ATOYSUPERSPACEecientprocedureistoobservethatthecomponentsin(2.2.9)canbede“nedby projection: A ( x )=( x ) | ( x )= D( x ) | F ( x )= D2( x ) | .(2. 2.13) Thiscanbeused,forexample,in(2.2.12)byrewriting(c.f.(2.2.7)) S = d3xD2f (, D, ... ) | .(2. 2.14) Afterthe derivativesareevaluated(usingtheLeibnitzruleandpayingduerespectto theanticommutativityofthe D s),theresultisdir ectlyexpressibleintermsofthecomponents(2. 2.13).Thereadershouldverifyinafewsimpleexamplesthatthisisamuch moreecientprocedurethandirect -expansionandintegration. Finally,wecanalsoreobtainthecomponenttransformationlawsbythismethod. We“rst note theidentity iQ+ D=2 i .(2. 2.15) Thuswe“nd,forexample A = i Q | = Š ( D Š 2 i ) | = Š .(2. 2.16) Ingeneralwehave iQf | = Š Df | .(2. 2.17) Thisissucienttoobtainallofthecomponent“eldstransformationlawsbyrepeated applicationof(2.2.17),where f is, D, D2andwe use(2. 2.6)and(2.2.13).

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2.2.Supersymmetryandsuper“elds13d.Irreduciblerepresentations Ingeneralatheoryisdescribedby“eldswhichinmomentumspacearede“ned forarbitraryvaluesof p2.For any“xedvalueof p2the“eldsarearepresentationofthe Poincar egro up.Wecallsuch“elds,de“nedfor arbitrary valuesof p2,an o-shell representationoft hePoincar egro up.Similarly,whenasetof“eldsisarepresentationofthe supersy mmetry algebrafor any valueof p2,wecallitano-shellrep resentationofsupersymmetry.Whenthe“eldequationsareimposed,aparticularvalueof p2(i.e., m2)is pi ck ed out.Someofthecomponentsofthe“elds(auxiliarycomponents)arethenconstrainedtovanish;theremaining(physical)componentsformwhatwecallan on-shell representationofthePoincar e(orsupersy mmetry)group. Asuper “eld ...( p )isani rreduciblerepresentationoftheLorentzgroup,with regardtoitsexternalindices,ifitistotallysymmetricintheseindices.Forarepresentationofthe(super)Poincar egroupweca nre duceitfurther.Since inthreedimensions thelittlegroupis SO (2),anditsirreduciblerepresen tationsareonecomponent(complex),thisreductionwillgiveone-componentsuper“elds(withrespecttoexternal i ndices).Suchsuper“eldsareirreduciblerepresentationsofo-shellsupersymmetry, whenarealityconditionisimposedin x -space(butthesuper“eldisthenstillcomplexin p -space,where( p )= ( Š p )). Inanappropriatereferenceframewecanassignhelicity(i.e.,theeigenvalueof the SO (2)gen erator) 1 2 tothespino ri ndices,andtheirreduciblerepresentationswill belabeledbythesupe rhelicity(thehelicityofthesuper“eld):halfthenumberof+ externalindicesminusthenumberof Š s.Inthisframewecanalsoassign 1 2 he licity to .Expa ndingthesuper“eldofsuperhelicity h intocomponents,weseethatthese componentsha vehe licities h h 1 2 h .Forex ample,a scalarmultiplet, consistingof spin s( i.e., SO (2,1)representations)0,1 2 (i.e.,helicities0, 1 2 )isdes cribedbya super“eldofsuperhelicity0:ascalarsuper“eld.A vectormultiplet, consistingofspins1 2 ,1(he licities0,1 2 ,1 2 ,1)isdes cribedbyasuper“eld ofsuperhelicity+1 2 :the+componentofas pinorsuper“eld;the Š compon entbeinggaugedaway(inalight-cone gauge).Ingeneral,thesuperhelicitycontentofasuper“eldisanalyzedbychoosinga gauge(thesupersymmetriclight-conegauge)whereasmanyaspossibleLorentzcomponentsofasuper“eldhavebeengaugedto0:the superhelicitycontentofanyremaining

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142.ATOYSUPERSPACEcomponentissimply1 2 thenumberof+sminus Š s.Unlesso therwise stated ,wew ill automaticallyconsider all thr ee-dimensionalsuper“eldstobe real.

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2.3.Scalarmultiplet152.3.Scalarmultiplet Thesimplestrepresentationofsupersymmetryisthescalarmultipletdescribed bythereals uper“eld( x ),andcontainingthescalars A F andthetwo-component spinor .From(2. 2.1,2)weseethat hasdimension( mass )Š1 2 .Also,thec anonical dimensionsofcomponent“eldsinthreedimensionsare1 2 lessthaninfo urdimensions (b ecauseweused3x insteadofd4x inthekineticterm).There fore,ifthismultiplet istodescribe physical“elds,wemustassigndimension( mass )1 2 tosothat has canonicaldimension( mass )1.(Althou ghitisnotimmediatelyobviouswhichscalar shouldhavecanonicaldimension,thereisonlyonespinor.)Then A willhavedimension ( mass )1 2 andwillbethephysicalscalarpartnerof ,whereas F hastoohighadimensiontodescribeacanonicalphysicalmode. Sincea integralisthesameasa deriva tive,d2 hasdimension( mass )1. Therefore,ondimensionalgroundsweexpectthefollowingexpressiontogivethecorrect (massless)kineticactionforthescalarmultiplet: Skin= Š1 2 d3xd2 ( D)2,(2. 3.1) (recallthatforanyspinor wehave 2=1 2 ).Thisex pressionisreminiscentof thekineticactionforanordinaryscalar“eldwiththesubstitutions d3x d3xd2 and D.Thecompon entexpressioncanbeo btainedb yexp licit -expansionand integration.However,wepr efertousethealternativeprocedure(“rstintegrating Dby parts): Skin=1 2 d3xd2 D2 =1 2 d3xD2[ D2] | =1 2 d3x ( D2 D2+ D DD2+( D2)2) | =1 2 d3x ( F2+ i + A A ),(2.3 .2)

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162.ATOYSUPERSPACEwherewehaveusedtheidentities(2.2.6)andthede“nitions(2.2.13).The A and kinetictermsareconventional,while F isclearlynon-propagating. Theauxiliary“eld F canbeeliminatedfromtheact ionbyusingitsequationof motion F =0(or,inaf unction alintegral, F canbetriviallyintegratedout).The resultingactionisstillinvariantunderthe bo se-fermitransforma tions(2.2.11a,b)with F =0;howev er,thesearenotsupersymmetrytrans formations(notare presentationof thesupersymmetryalgebra)exceptonshell.Thecommutatoroftwosuchtransformationsdoesnotclose(doesnotgiveatranslation)exceptwhen satis“esits“eldequation.Thiso-shellnon-closureofthealgebraistypicaloftransformationsfromwhich auxiliary“eldshavebeeneliminated. Massandinter actiontermscanbeaddedto(2.3.1).Aterm SI= d3xd2 f (),(2 .3.3) leadstoacomp onentaction SI= d3xD2f () | = d3x [ f()( D)2+ f() D2] | = d3x [ f( A ) 2+ f( A ) F ].(2.3 .4) Inarenormalizablemodel f ()canbe atmostquartic.Inparticular, f ()=1 2 m 2+1 6 3givesmassterms,Yukawaandc ubicinterac tionterms.Together withthekineticterm,weobtain d3xd2 [ Š1 2 ( D)2+1 2 m 2+1 6 3] = d3x [1 2 ( A A + i + F2) + m ( 2+ AF )+ ( A 2+1 2 A2F )].(2.3.5) F canagainbeeliminatedusingits(algebraic)equationofmotion,leadingtoa

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2.3.Scalarmultiplet17conventionalmasstermandquarticinteractionsforthescalar“eld A .Moreexotic kineticactionsarepossiblebyusinginsteadof(2.3.1) S kin= d3xd2 ( ,), = D,(2.3 .6) whereissomefunctionsuchthat2 | ,=0= Š1 2 C.Ifweint roducemorethan onemultipletofscalarsuper“elds,then,forexample,wecanobtaingeneralizedsupersymmetricno n linearsigmamodels: S = Š1 2 d3xd2 gij()1 2 ( Di)( Dj)(2. 3.7)

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182.ATOYSUPERSPACE2.4.Vectormultiplet a.Abeliangaugetheory Inaccordancewiththediscussioninsec.2.2,arealspinorgaugesuper“eldwithsuperhelicity h =1 2 ( h = Š1 2 canbegaugedaway)willconsistofcomponentswith he licities0,1 2 ,1 2 ,1.Itcanbeused todescribeamasslessgaugevector“eldandits fermionicpartner.(Inthreedimensions,agaugevectorparticlehasonephysicalcomponentofde“nitehelicity.)Thesuper“eldcan beintrodu cedbyanalogywithscalarQED (thegeneralizationtothenonabeliancaseisstraightforward,andwillbediscussed below).Consideracomp lexscalarsuper“eld(a doubletofrealscalarsuper“elds)transformingundera cons tant phasero tation = eiK, = eŠ iK.(2. 4.1) ThefreeLagrangian | D |2is in va riantunderthesetransformations. a.1.Gaugeconnections Weextend thistoa local phaseinvariancewith K areals calarsuper“elddependingon x and ,bycovaria ntiz ingthespinorderivatives D: D= DŠ+ i ,(2. 4.2) whenactingonor ,respectively.Thespinorgaugepotential(orconnection)transformsintheusualway = DK ,(2. 4.3) toensure = eiKeŠ iK.(2. 4.4) Thisisrequiredby( )= eiK( ),andguaranteesthattheLagrangian | |2is locallygaugeinvariant.(Thecouplingconstantcanberestoredbyrescaling g ).

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2.4.Vectormultiplet19Itisnowstraightforward,byanalogywithQED,to“ndagaugeinvariant“eld strengthandactionforthemultipletdescribedbyandtost udyitscomponentcoup lingstothecomplexscalarmultipletcontainedin | |2.H ow ever,bothtounderstand itsstructureasanirreduciblerepresentati onofsupersymmetry,andasanintroduction tomorecomplicatedgaugesuper“elds(e.g.insupergravity),we“rstgiveageometrical presentation. Althoughtheactionswehave considereddonotcontainthespacetimederivative ,inother contextsweneedthecovariantobject = Š i = K ,(2. 4.5) introducingadistinct(vector)gaugepotentialsuper“eld.Thetransformation of thisconn ectionischosentogive: = eiKeŠ iK.(2. 4.6) (Fromageometricviewpoint,itisnaturaltointroducethevectorconnection;thenandcanberegardedasthecomponentsofasuper1-formA=(,);se es ec. 2.5).However,wewill“ndthatshouldnotbeindependent,andcanbeexpressedin termsof. a.2.Comp onents Togetoriented, weexaminethe componentsofintheTaylorseries -expansion. Theycanbede“neddirectlybyusingthespinorderivatives D: =| B =1 2 D| V=Š i 2 D( )| =1 2 DD| ,(2. 4.7a) and W=| = D| = D( )| T= D2| .(2. 4.7b) Wehavese paratedthecomponentsintoirreduciblerepresentationsoftheLorentzgroup, th atis,traces(orantisymmetrizedpieces,see(2.1.2))andsymmetrizedpieces.Wealso

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202.ATOYSUPERSPACEde“nethecomponentsofthegaugeparameter K : = K | = DK | = D2K | (2.4.8) Thecomponentgaugetransformationsforthecomponentsde“nedin(2.4.7)arefound byrepeatedlydi erentiating(2.4.3-5)withspinorderivatives D.W e “nd: = B = V= =0,(2. 4.9a) and W= = = ( ), T= .(2. 4.9b) Notethat and B suerarbitraryshiftsasaconsequenceofagaugetransformation, and,inparticular,canbegaugedcompletelyaway;thegauge = B =0isca lled WessZumino gauge,and explicitly breakssupersymmetry.However,thisgaugeisusefulsince itrevealsthe physicalcontentofthemult iplet. Examinationofthecomponentsthatremainrevealsseveralpeculiarfeatures: Thereare two componentgaugepotentials Vand Wforonly one gaugesymmetry, andthereisahighdimensionspin3 2 “eld .These problemsw illberesolvedbelow whenweexpressintermsof. Wecanalso “ndsupersymmetricLorentzgaugesby“xing D;such gaugesare usefulforquantization(seesec.2.7).Furthermore,inthreedimensionsitispossibleto chooseasuper symmetriclight-conegauge+=0.(Intheabe liancasethegaugetransformationtakesthesimpleform K = D+( i ++)Š 1+.)Eq.(2.4.14)belowimpliesthatin thisgaugethesuper“eld++alsovanishes.Theremainingcomponentsinthisgaugeare Š, V+ Š, VŠŠ,and Š,with V++=0and + ++Š. a.3.Constraints To understandhowthevectorconnectioncanbeexpressedintermsofthe spinorconnection,r ecallthe(an ti)commutationrelationsfortheordinaryderivatives are:

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2.4.Vectormultiplet21[ DM, DN} = TMN PDP.(2. 4.10) Forthecovarian tderiv atives A=( )thegr adedcommutationrelationscanbe written(from(2.4.2)and(2.4.5)weseethatthetorsion TAB Cisunmodi“ed): [ A, B} = TAB CCŠ iFAB.(2. 4.11) The“eldstrengths FABareinvariant( F AB= FAB) duetothecovarianceofthederivatives A.Observethat the“eldstrengthsareantihermitianmatrices, FAB= Š FBA,so thatthesymmetric“eldstrength Fisimaginarywhiletheantisymmetric“eld strength F is real.Examiningaparticularequationfrom(2.4.11),we“nd: {, } =2 i Š iF=2 i +2Š iF.(2. 4.12) Thesuper“eldwasintrodu cedtocovariantizethespace-timederivative .However,itisclearthatanalternativechoiceis =Ši 2 Fsince Fiscovariant(a “eldstrength).Thenewcovariantspace-time derivati vew illthensatisfy(wedropthe primes) {, } =2 i ,(2. 4.13) withthenewspace-timeconnectionsatisfying(aftersubstitutingin2.4.12theexplicit forms A= DAŠ i A) = Š i 2 D( ).(2. 4.14) Thusthe con ventionalconstraint F=0,(2. 4.15) imposedonthesystem(2.4.11)hasallowedthevectorpotentialtobeexpressedinterms ofthespinorpotential.Thissolvesboththeproblemoftwogauge“elds W, Vand theproblemoftheh ighers pinanddimensioncomponents T:The gauge“elds areidenti“edwitheachother( W= V),andtheextracompone ntsareexpr essedas de rivativesoffamiliarlowerspinanddimension“elds(see2.4.7).Theindependentcomponentst hatremaini nWe ss-Zuminogaugeaftertheconstraintisimposedare Vand .

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222.ATOYSUPERSPACEWestre sstheimportanceoftheconstraint (2.4.15)ontheobjectsde“nedin (2.4.11).Unconstrained“eldstrengthsingeneralleadtoreduciblerepresentationsof supersymmetry(i.e.,thespinorandvectorpo tentials),andtheconstraintsareneededto en su reirreducibility. a.4.Bianchiidentities Inordinary“eldtheories,the“eldstrengthssatisfyBianchiidentitiesbecausethey areexpressedintermsofthepotentials;theyare identi ties andcarrynoinformation. For gaugetheoriesdescribedbycovariantderivatives,theBianchiidentitiesarejust Jacobiidentities: [ [ A,[ B, C )}} =0,(2. 4.16) (where[)isthe graded antisymmetrizationsymbol,identicaltotheusualantisymmetrizationsymbolbutwithanextrafactorof( Š 1)foreachpairofinterchanged fermionicindices).However,onceweimpose constraintssuchas(2.4.13,15)onsomeof the“eldstrengths,theBianch iidentit iesimplyconstraintsonother“eldstrengths.For example,theidentity 0=[ {, } ]+[ {, } ]+[ {, } ] =1 2 [ ( {, )} ](2. 4.17) gives(usingtheconstraint(2.4.13,15)) 0=[ ( )]= Š iF( ).(2. 4.18) Thusthetotallysymmetricpartof F vanishes.Ing eneral,wecandecompose F into irreduciblerepresentati onsoftheLo rentzgroup: F =1 6 F( )Š1 3 C ( |F | )(2.4.19) (where i ndicesbetween| ... |,e.g .,inthiscase ,arenoti ncludedinthesymmetrization).Hencetheonlyremainingpieceis: F = iC ( W ),( 2.4.20a) whereweintroducethesuper“eldstrength W.Wecanco mpute F intermsof

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2.4.Vectormultiplet23and “nd W=1 2 DD.(2. 4.20b) Thesuper“eld Wistheonlyindependentgaugeinvariant“eldstrength,andis constrainedby DW=0,whichfo llowsfromtheBianchiidentity(2.4.16).This impliesthato nlyoneLorentzcomponentof Wisindepe ndent.The“eldstrength describesthephysicaldegreesoffreedom:onehelicity1 2 andonehelicit y1mode.Thus Wisasuita bleobjectforconstructinganaction.Indeed,ifwestartwith S =1 g2 d3xd2 W2=1 g2 d3xd2 (1 2 DD)2,(2. 4.21) wecancom putethecomponentaction S =1 g2 d3xD2W2=1 g2 d3x [ WD2WŠ1 2 ( DW)( DW)] | =1 g2 d3x i Š1 2 ff .(2. 4.22) Here(cf.2.4.7) W| while f= DW| = DW| isthespi norformoftheusual “eldstrength F | =( Š ) | =1 2 ( ( f ) )= Š i1 2 [ D( )Š D( )] | .(2. 4.23) Toderivethea bovecompon entactionwehaveusedtheBianchiidentity DW=0,and itsconse quen ce D2W= i W. a.5.Matterc ouplings Wenowexami nethecomponent Lagrangiandescribingthecouplingtoacomplex scalarmultip let.Wecouldstartwith S = Š1 2 d3xd2 ( )( )

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242.ATOYSUPERSPACE= Š1 2 d3xD2[( D+ i ) ][( DŠ i )],(2 .4.24) andworkouttheLagrangianintermsofcomponentsde“nedbyprojection.However,a moreecientprocedure,whichleadstophysicallyequivalentresults,istode“ne covariantcomponents ofby covariant projection A =( x ) | = ( x ) | F = 2( x ) | .(2. 4.25) Thesecomponentsarenotequaltotheordinaryonesbutcanbeobtainedbya(gauge“elddependent)“eldrede“nitionandprovideanequallyvaliddescriptionofthetheory. Wecanalsouse d3xd2 = d3xD2| = d3x 2| ,(2. 4.26) whenactingonaninvariantandhence S = d3x 2[ 2] | = d3x [ 2 2+ 2+ ( 2)2] | = d3x [ FF + ( i + V ) +( i A + h c .)+ A ( Š iV)2A ].(2.4.27) Wehave usedthecommu tationrelationsofthecovariantderivativesandinparticular 2= i + iW, 2= Š i Š 2 iW,( 2)2= Š iW,where is the covariant dAlembertian(covariantizedwith). b.Nonabeliancase Wenowbrie”ycon siderthenonabeliancase:Foramu ltipletofscalarsuper“elds transformingas= eiK,where K = KiTiand TiaregeneratorsoftheLiealgebra, weintro ducecovariantspinorderivatives preciselyasfortheab eliancase(2.4.2). Wede “ne= iTisothat

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2.4.Vectormultiplet25= DŠ i = DŠ i iTi.(2. 4.28) Thespinorconnectionnowtransformsas = K = DK Š i [, K ],(2.4 .29) leaving(2.4.4)unmodi“ed.Thevectorconnectionisagainconstrainedbyrequiring F=0;inotherwo rds,wehave =Š i 2 {, } ,( 2.4.30a) = Š i1 2 [ D( )Š i { ,} ].(2.4 .30b) Theformoftheaction(2.4.21)isunmodi“ed(exceptthatwemustalsotakeatraceover groupindices).Theconstraint(2.4.30)impliesthattheBianchiidentitieshavenontrivialconsequences,andallowsustosolve(2.4.17)forthenonabeliancaseasin (2.4.18,19,20a).Thus,weobtain [ ]= C ( W )(2.4.31a) intermsofthenonabelianformofthe covariant “eldstrength W : W=1 2 DDŠi 2 [, D] Š1 6 [, { ,} ].(2.4 .31b) The“eldstrengthtransformscovariantly: W = eiKWeŠ iK.There mainingBianchi identityis [ {, } ] Š{( ,[ ), ] } =0 .( 2.4.32a) Contractingindiceswe“nd[ {, } ]= {( ,[ ), ] } .Howev er, [ {, } ]=2 i [ ]= 0a nd he n ce,using(2.4.31a), 0= {( ,[ ), ] } = Š 6 {, W} .(2. 4.32b) ThefullimplicationoftheBianchiidentitiesisthus: {, } =2 i (2.4.33a) [ ]= C ( W ), {, W} =0(2.4 .33b) [ ]= Š1 2 i ( ( f ) ), f1 2 {( W )} .(2. 4.33c)

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262.ATOYSUPERSPACEThecomponentsofthemultipletcanbede“nedinanalogyto(2.4.7)by projections of: =| V=| B =1 2 D| = W| (2.4.34) c.Gaugeinvar iantmasses Acurio usfeaturewhichthistheoryhas,andwhichmakesitratherdierentfrom fo ur di mensionalYang-Millstheory,istheexistenceofagauge-invariantmassterm:In theabeliancasetheBianchiidentity DW=0canbeuse dtoprovetheinvarianceof Sm=1 g2 d3xd2 1 2 m W .(2. 4.35) Incomponentsthisactioncontainstheusualgaugeinvariantmasstermforthree-dimensionalelectrodynamics: m d3xVV = m d3xVf,(2. 4.36) whichisgaugeinvariantasaconsequenceoftheusualcomponentBianchiidentity f=0. Thesuper“eldequationswhichresultfrom(2.4.21,35)are: i W+ mW=0,(2. 4.37) whichdescribesanirreduciblemultipletofmass m .TheBianch iidentity DW=0 impliesthato nlyoneLorentzcomponentof W isindepe ndent. Forthe nonabeliancase,themas stermissom ewhatmorecomplicatedbecausethe “eldstrength W iscovariantratherthaninvariant: Sm= tr1 g2 d3xd21 2 m (W+i 6 { ,} D+1 12 { ,}{ ,} )

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2.4.Vectormultiplet27= tr1 g2 d3xd21 2 m ( WŠ1 6 [,]).(2. 4.38) The“eldequations,however,arethecovariantizationsof(2.4.37): i W+ mW=0.(2. 4.39)

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282.ATOYSUPERSPACE2.5.Otherglobalgaugemultiplets a.Superforms:generalcase Thegaugemultipletsdiscussedinthelastsectionmaybedescribedcompletelyin termsofgeometricquantities.ThegaugepotentialsA (,)whichcova riantize thederivatives DAwithrespecttolocalphaserotationsofthemattersuper“eldsconstituteasuper1-fo rm.Wede“nesuper p -formsastensorswith p covariantsupervector indices(i.e.,supervectors ubscripts)thathavetotal graded antisymmetrywithrespectto theseindices(i.e.,aresymmetricinanypairofspinorindices,antisymmetricinavector pairor inamixedpair).Forexample,the“eldstrength FAB ( F , F , F )constitutesasuper2-form. IntermsofsupervectornotationthegaugetransformationforA(from(2.4.3)and (2.4.5))takestheform A= DAK .(2. 5.1) The“eldstrengthde“nedin(2.3.6)whenexpressedintermsofthegaugepotentialcan bewri ttenas FAB= D[ AB )Š TAB CC.(2. 5.2) Thegaugetransformationlawcertainlytakesthefamiliarform,butevenintheabelian case,the“eldstrengthhasanunfamiliarn onderivativeterm.Onewaytounderstand howthistermarisesistomak eachan geofbasisforthecomponentsofasupervector. Wecanexpand DAintermsofpartialderivativesbyintroducingamatrix, EA M,such that DA= EA MM, M ( , ), EA M= 01 2 i ( )1 2 ( ) .(2. 5.3) Thismatrixisthe ”atvielbein; itsinverseis

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2.5.Otherglobalgaugemultiplets29EM A= 0 Š1 2 i ( )1 2 ( ) .(2. 5.4) Ifwede“neMbyA EA MM,then M= MK .(2. 5.5) Similarly,ifwede“ne FMNby FAB ( Š )A ( B + N )EB NEA MFMN,(2. 5.6a) then FMN= [ MN ).(2. 5.6b) (IntheGrassmannparityfactor( Š )A ( B + N )thesuper scripts A B ,and N areequalto onewhent heseindicesrefertospinorialindicesandzerootherwise.)Wethusseethat thenonderivativeterminthe“eldstrengthisabsentwhenthecomponentsofthis supertensorarereferredtoadierentcoordi natebasis.Furthermore,inthisbasisthe Bianchiidentitiestakethesimpleform [ MFNP )=0.(2. 5.7) Thegeneralizationtohigher-rankgradedantisymmetrictensors(superforms)is nowevident.Thereisabasis inwhichthegaugetransformation,“eldstrength,and Bianchiidentitiestaketheforms M1... Mp=1 ( p Š 1)! [ M1KM2... Mp), FM1... Mp +1=1 p [ M1M2... Mp +1), 0= [ M1FM2... Mp +2).(2. 5.8) Wesimplymu ltiplythesebysuitablepowersofthe ”atvielbeinandappropriateGrassmannparityfactorstoobtain A1... Ap=1 ( p Š 1)! D[ A1KA2... Ap)Š1 2( p Š 2)! T[ A1A2| BKB | A3... Ap),

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302.ATOYSUPERSPACEFA1... Ap +1=1 p D[ A1A2... Ap +1)Š1 2( p Š 1)! T[ A1A2| BB | A3... Ap +1), 0=1 ( p +1)! D[ A1FA2... Ap +2)Š1 2 p T[ A1A2| BFB | A3... Ap +2).(2. 5.9) (The | sindicatethatalloft heindicesaregradedantisymmetricexceptthe B s.) b.Super2-form Wenowdiscuss indetailthecaseofasuper2-formgaugesuper“eldABwith gaugetransformation = D( K )Š 2 iK, = DKŠ K, = KŠ K.(2. 5.10) The“eldstrengthforABisasuper3-form: F , =1 2 ( D( )+2 i ( )), F , = D( ), + Š 2 i , F , = D + Š , F , = + + .(2. 5.11) Alloftheseeq uationsarecontainedintheconcisesupervectornotationin(2.5.9). Thegaugesuper“eldAwassubj ecttoconstraintsthatallowedonepart( )to beexpr essedasafunctionoftheremainingpart.Thisisageneralfeatureofsupersymmetricgaugetheories;constraintsareneededtoensureirreducibility.Forthetensor gaugemultipletweimposetheconstraints F , =0, F , = i ( ) G = T G ,(2. 5.12) which,asweshowbelow,allowustoexpressallcovariantquantitiesintermsofthesinglerealscalarsuper“eld G .Theseco nstraintsc anbesolvedasfollows:we“rstobserve thatinthe“eldstrengths alwaysappearsinthecombination D( )+2 i ( ).

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2.5.Otherglobalgaugemultiplets31Therefore,withoutchangingthe“eldstrengthswecanrede“ne byabsorbing D( )intoit.Thus disappearsfromthe“eldstrengthswhichmeansitcouldbe settozerofromthebeginning(equivalently,wecanmakeitzerobyagaugetransformation).The“rstconstraintnowimpliesthatthetotallysymmetricpartof is zero andhencewecanwrite = iC ( )intermsofas pinorsuper“eld.The remainingequationsandconstraintscanbeusednowtoexpress andtheother “eldstrengthsintermsof.We “ndasolution =0, = iC ( ), =1 4 ( ( [ D ) )+ D ) )], G = Š D.(2. 5.13) ThustheconstraintsallowABtobeexpr essedintermsofaspinorsuper“eld.(The generalsolutionoftheconstraintsisagaugetransform(2.5.10)of(2.5.13).) Thequantity G isbyde “nitiona“eldstrength;hencethegaugevariationofmustleave G invariant.Thisimpliesthatthegaugevariationofmustbe(see (2.2.6)) =1 2 DD,(2. 5.14) whereisanarbitraryspinorgaugeparameter.Thisgaugetransformationisofcourse consistentwithwhatremainsof(2.5.10)afterthegaugechoice(2.5.13). Weexp ectthephysicaldegreesoffreedomtoappearinthe(onlyindependent) “eldstrength G .Sinceth isisascalarsuper“eld,itmustdescribeascalarandaspinor, and(orAB)providesa variantrepresentation ofthesupersymmetryalgebranormallydescribedbythescalarsuper“eld.Infactcontainscomponentswithhelicities0,1 2 ,1 2 ,1just likethevectormultiplet,butnowthe1 2 ,1 co mponentsareauxiliary “elds.(= + A + vŠ 2).Forwithcanonicaldimension( mass )1 2 ,on dimensionalgroundsthegaugeinvariantactionmustbegivenby S = Š1 2 d3xd2 ( DG )2.(2. 5.15) Wri tteninthisformweseethatintermsofthecomponentsof G ,theacti onha sthe

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322.ATOYSUPERSPACEsameformasin(2.3 .2).Theonlydier encesarisebecause G isexpr essedintermsof .W e “nd thatonlytheauxiliary“eld F ismodi “ed;itisreplacedbya“eld F.An exp licitcomputationofthisquantityyields F= Š D2D| = i D| V| V1 2 iD( ).(2. 5.16) Inplaceof F thedivergenceofavectorappears.Toseethatthisvector“eldreallyisa gau ge “eld,wecomputeitsvariationunderthegaugetransformation(2.5.14): V=1 4 ( [ D )+ D )].(2.5 .17) Thisisnotthetransformationofanordinarygaugevector(see(2.4.9)),butratherthat ofasecond-ra nkantisymmetrictensor(inthreedimensionsasecond-rankantisymmetric tensoristhesameLorentzrepresentationasavector).Thisisthecomponentgauge “eldthat appearsatlowestorderin in ineq.(2.5.13).A“eldofthistypehasno dynamicsinthreedimensions. c.Spinorgaugesuper“eld Superformsarenottheonly gaugemultipletsonecanstudy,butthepatternfor othercasesissimilar.Ingeneral,(nonvari ant)supersymmetricgaugemultipletscanbe describedbyspinorsuper“eldscarryingadditionalinternal-symmetrygroupindices.(In aparticularc ase,theadditionalindexcanbeaspi norindex:seebelow.)Suchsuper“e ldscontaincomponentgauge“eldsand,asintheYang-Millscase,theirgaugetransformationsaredeterminedbythe =0partofth esuper “eld gaugeparameter(cf. (2.4.9)).Thegaugesuper“eldthustakesth eformoftheco mponent“eldwithavector i ndexreplacedbyaspinorindex,andthetransformationlawtakestheformofthecomponenttransf ormationlawwiththevectorderivati verepla cedbyaspinorderivative. Forexample,todes cribeamu ltipletcontainingaspin3 2 componentgauge“eld,we introduceaspinorgaugesuper“eldwithanadditionalspinorgroupindex: = DK.(2. 5.18) The“eldstrengthhasthesameformasthevectormultiplet“eldstrengthbutwitha spinorgroupindex:

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2.5.Otherglobalgaugemultiplets33W =1 2 DD .(2. 5.19) (Wecan,ofcourse,introduceasupervectorpotentialM inexactanalogywiththe abelianvectormultiplet.The“eldstrengthheresimplyhasanadditionalspinorindex. Theconstraintsareexactlythesameasforthevectormultiplet,i.e., F =0.) Inthreedimensionsmassless“eldsofspingreaterthan1havenodynamical de greesoffreedom.Thekinetictermforthismultipletisanalogoustothe massterm forthevector mult iplet: S d3xd2 W.(2. 5.20) Thisactiondescribescomponent“eldswhichareallauxiliary:aspin3 2 gauge“eld ( ) ,av ector,andascalar,ascanbeveri“edbyexpandingincomponents.The invarianceoftheactionin(2.5.20)isnotmanifest:ItdependsontheBianchiidentity DW=0.Theex p licitappearanceofthesuper“eldisagen eralfeatureofsupersymmetricgaugetheories;itis not alwayspossibletowritethesuperspaceactionfora gaugetheoryintermsof“eldstrengthsalone.

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342.ATOYSUPERSPACE2.6.Supergravity a.Supercoordinatetransformations Supergravity,thesupersymmetricgenera lizationofgravity,isthegaugetheoryof thesupertranslations.Theglobaltransformationswithconstantparameters , generatedby Pand Qarereplacedbylocalonesparametrizedbythesupervector KM( x )=( K, K).Forascalarsuper“eld( x )wede “nethetransformation ( z ) ( z )= eiK( z )= eiK( z ) eŠ iK,(2. 6.1) where K = KMiDM= Ki + KiD.(2. 6.2) (Toexhibittheglobalsupersymmetry,itisconvenienttowrite K intermsof Drather than Q(or ).Thisamountstoarede“nitionof K).Thesecondformofthe transformationofcanbeshowntobeequivalenttothe“rstbycomparingtermsina powerseriesexpan sionofthetwoformsandnotingthat iK =[ iK ,].It iseasy tosee that(2.6.1)isageneralcoordin atetransformationinsuperspace: eiK( z ) eŠ iK=( eiKzeŠ iK);de “ning z eŠ iKzeiK,(2. 6.1)becomes( z)=( z ). We mayexpect,byanalogytotheYang-Millscase,tointroduceagaugesuper“eld H Mwith(linearized)transformationlaws H M= DKM,(2. 6.3) (weint roduce H Maswell,butaconstraintwillrelateitto H M)and de“necovariant derivati vesbyana logyto(2.4.28): EA= DA+ HA MDM= EA MDM.(2. 6.4) EA Misthe vielbein. Thepotentials H , H havealar genumberofcomponents amongwhichweidentify,accord ingtothe discussionfollowingequation(2.5.17),aseco nd-ranktensor(thedreibein,minusits”at-spacepart)describingthegravitonanda spin3 2 “elddescribingthegravitino,whosegaugeparametersarethe =0part softhe v ectorandspinorgaugesuperparameters KM| .Other componentswilldescribegauge de greesoffreedomandauxiliary“elds.

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2.6.Supergravity35b.Lorent ztransfo rmations Thelocalsupertranslationsintroducedso farincludeLorentztransformationsofa scalarsuper“eld,actingonthecoordinates zM=( x, ).Tode “netheiractionon spinorsuper“eldsitisnecessarytointroducetheconceptoftangentspaceandlocal framesattachedateachpoint zMandlocalLorentztransformationsactingonthe i ndicesofsuchsuper“elds ...( zM).(Inchapter5wediscuss thereasonsfo rthisprocedure.)Theenlargedfulllocalgroupisde“nedby ...( x ) ...( x )= eiK ...( x ) eŠ iK,(2. 6.5) wherenow K = KMiDM+ K iM .(2. 6.6) Herethes uper“eld K parametrizesthelocalLorentzt ransformationsandtheLorentz generators M actoneachtangentspaceindexasindicatedby [ X M ,]= X ,(2. 6.7) forarbitrary X Missymmetric,i.e., M istra celess(whichmakesitequivalentto av ectorinthreedimensions).Thus, X isanel ementoftheLorentzalgebra SL (2, R ) (i.e., SO (2,1)).Therefore,th epar ametermatrix K isalsotr aceless. Fromnowonwem ustdistinguishtangentspaceandworldindices;todothis,we denotetheformerbyle ttersfromthebeginningofthealphabet,andthelatterbyletters fromthemiddleofthealphabet.Byde“nition,theformertransformwith K whereas thelattertransformwith KM. c.Covariantderivatives Havingintro ducedlocalLorentztransformationsactingonspinorindices,wenow de“necovariantspinorderivativesby = E MDM+ M ,(2. 6.8) aswellasvectorderivatives .H ow ev er ,j us ta si nt he Ya ng -M illscase,weimposea conventionalconstraintthatde“nes = Š i1 2 {, } ,(2. 6.9)

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362.ATOYSUPERSPACEThe co nnectioncoecients A ,whichap pearin A= EA MDM+A M ,(2. 6.10) andactasgauge“eldsfortheLorentzgroup,willbedeterminedintermsof H Mby imposingfurt hersuitableconstraints.Thecova riantderivativestransformby AA = eiKAeŠ iK.( 2.6.11a) All “elds ...(asopposedtotheoperator )trans formas ...= eiK...eŠ iK= eiK...(2.6.11b) whenallindicesare”at(tangentspace);wealwayschoosetouse”atindices.Wecan usethevielbein EA M(anditsinverse EM A)toconvertbe tw eenworldandtangentspace i ndices.Forexample,ifMisaworldsupervector,A= EA MMisatangentspace supervector. Thesuperderivative EA= EA MDMis to be understoodasatangentspacesuperv ector.Ontheotherhand, DMtr an sformsunderthelocaltranslations(supercoordinate transformations),andthisinducestransformationsof EA Mwithrespecttoitsworld index(inthiscase, M ).Wecanexhibitthis,andverifythat(2.6.6)describesthefamilia rl oc al Lorentzandgeneralcoordinatetransformations,byconsideringthein“nitesimal versionof( 2.6.11): A=[ iK A],(2.6 .12) whichimplies EA M= EA NDNKMŠ KNDNEA MŠ EA NKPTPN MŠ KA BEB M, A = EAK Š KMDMA Š KA BB Š K A + K A = AK Š KMDMA Š KA BB ,(2. 6.13) where TMN Pisthetorsionof ”at,global superspace(2.4.10),and K 1 2 K( ( ) ). The“rstthreetermsinthetransformationlawof EA Mcorrespondtotheusualformof thegeneralcoordinatetransformationofaworldsupervector(labeledbyM),whilethe lasttermisalocalLorentztransformationonthetangentspaceindex A .Therelation betw een K and K impliestheusualreducibilityoftheLorentztransformationson

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2.6.Supergravity37thetangentspace,correspondingtothede “nitionofvectorsassecond-ranksymmetric spinors. d.Gaugechoices d.1.Asupersymmetricgauge Aswehavementio nedabove,thegauge“e lds(orthevielbein EA M)containa largenumberofgaugedegreesoffreedom,andsomeofthemcanbegaugedawayusing the K transformations.Forsimplicitywediscussthisonlyatthelinearizedlevel(where wen eednotdistinguishworldandtangentspaceindices);wewillreturnlatertoamore complete treatment.From(2.6.13)thelinearizedtransformationlawsare E = DKŠ K , E = DKŠ i ( K ).(2. 6.14) Thus K canbeusedtogaugeawayallof E exceptitstrace(recallthat K is tra celess)and Kcangaugeawaypartof E .Intheco rrespondinggaugewecan write E = , E =0;(2. 6.15) this globallysupersymmetric gaugeismaintainedbyfurthert ransformationsrestricted by K =1 2 D( K ) DKŠ1 2 DK, K= Ši 3 DK.(2. 6.16) U ndertheserestrictedtr ansformationswehave =1 6 K, E( )= D( K ).(2. 6.17)

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382.ATOYSUPERSPACEInthisgaugethetracelesspart h( )oftheordinarydreibein(thephysicalgraviton “eld)appearsin E( ).Thetrace h = h iscontaine din(the =0partof )and hasanidentical(linearized)transformationlaw.(Insuperconformaltheoriesthevielbe in al soundergoesasuperscaletransformationwhosescalarparametercanbeusedto gaugeto1,stillinagloballysupersymmetricway.Thus E( )containstheconformalpartofthesupergravitymultiplet,whereascontainsthetraces.) d. 2.Wess-Zuminogauge Theabovegaugeisconvenientforcalculationswherewewishtomaintainmanifest globalsupersymmetry.HoweverjustasinsuperYang-Millstheory,wecan“ndanonsupersymmetricWess-Zuminogaugethatexhibitsthecomponent“eldcontentofsupergravitymostdirectly.Insuchagauge = h + Š 2a E( )= h( )Š 2( ),(2. 6.18) where h and h( )aretheremainingpartsofthedreibein, and ( )ofthegravitino,and a isascalarauxiliary“eld.Theresidualgaugeinvariance(whichmaintains theaboveform)isparametrizedby K= + ( ),(2. 6.19) where ( x )par ametrizesgeneralspacetimecoo rdinatetransformationsand ( x ) parametrizeslocal(component) supersymmetrytransformations. e.Fieldstrengths Wenowret urntoastudyofthegeometricalobjectsofthetheory.The“eld strengthsforsupergravityaresupertorsions TAB Candsupercurvatures RAB ,de “nedby [ A, B} TAB CC+ RAB M .(2. 6.20) Ourdetermi nati onof intermsof (see(2.6.9)),isequivalenttotheconstraints T = i ( ) T = R =0.(2. 6.21) Wen eedonefurtherconstraint torelatetheconnection (thegauge“eldforthe

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2.6.Supergravity39localLorentztransformations)tothegaugepotential H M(orvielbein E M).Itturns outtha tsucha constraintis T =0.(2. 6.22) Tosolvethisc onstraint,andactually“ndintermsof E Mitisconvenienttomake someadditionalde“nitions: E E, EŠi 2 { E, E} [ EA, EB} CAB C EC.(2. 6.23) Theconstraint(2.6.22)isthensolvedfor asfollows:First,express[ ]in termsof andthecheckobjectsof(2.6.23)using(2.6.9).Then,“ndthecoecientof Einthisexpression .Theco rrespondingcoecientoftheright-handsideof (2.6.20)is T .Thisgiv esustheequation T = C Š1 2 ( ) ( )+1 2 ( ( ) )= C Š1 2 C ( ( ) )=0.(2. 6.24) (FromtheJaco biidentity[ E( { E, E )} ]=0,wehave,i ndependentof(2 .6.21,22), C( ) =0.)Wethen solvefor :Wemulti ply(2. 6.24)by Candusetheidentity =1 2 (( ) Š C( ) ).We “nd =1 3 ( C , Š C ( ) ),(2.6 .25) the C sbeingcalculablefrom(2.6.23)asderivativesof E M. f.Bianchiidentities Thetorsionsandcurvaturesarecovariantandmustbeexpressibleonlyinterms ofthephysicalgaugeinvariantcomponent“eldstrengthsforthegravitonandgravitino andauxiliary“elds.Weproceedintwosteps:First,weexpressallthe T sand R sin (2.6.20)intermsofasmallnumberofindepe ndent“eldstrengths;then,weanalyzethe contentofthesesuper“elds.

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402.ATOYSUPERSPACETheJacobiidentitiesforthecovariantderivativesexplicitlytaketheform: [[ [ A, B} C )} =0.(2. 6.26) Thepresenceoftheconstraintsin(2.6.21,22)allowsustoexpressallofthenontrivial torsionandcurvaturetensorscomp letelyintermsoftwosuper“elds R and G(where Gistotallysymmetric),andtheirspinorialderivatives.Thisisaccomplishedbyalgebraica llysolvingtheconstraintsplusJacobiidentities(whicharetheBianchiidentities fo rt he torsionsandcurvatures).WeeitherrepeatthecalculationsoftheYang-Mills case,orwemakeuseoftheresultsthere,asfollows: Weobservetha ttheco nstraint (2.6.21) {, } =2 i isidenticaltotheYangM illsconstraint(2.4.13,30a).TheJacobiidentity[ ( {, )} ]=0hasthes amesolutionasin(2.4.17-20a,31a): [ ]= C ( W ),(2. 6.27) where Wisexpa ndedoverthesupergravitygenerators i and iM (thefactor i is introducedtomakethegeneratorshermitian): W= W i + W i + W iM .(2. 6.28) ThesolutiontotheBianchiidentitiesisthus(2.4.33),withtheidenti“cation(2.6.28). Theconstraint(2.6.22)implies W =0,andwecansolve {, W} =0(see (2.4.33b))explicitly: W= Š CR W= G+1 3 C ( )R G= Š2 3 i R ,(2. 6.29) wherewehaveintroducedascalar R andatotallysymmetricspinor G.Thefull so lutionoftheBianchiidentitiesisthustheYang-Millssolution(2.4.33)withthesubstitutions Š iW= Š R +2 3 ( R ) M + G M G= Š2 3 i R Š if= Š1 3 ( ( R ) )+ G Š 2 Ri +2 3 ( 2R ) M

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2.6.Supergravity41+1 2 ( i ( R ) M ) + W M (2.6.30) where W1 4! ( G ).Wehaveused = i Š C2to“nd(2.6. 30).Individualtorsion sandcurvat urescanbereaddirectlyfromtheseequationsbycomparing withthede“nition(2.6.20).Thus,forexample,wehave R , =1 2 ( ( r ) ) r W Š1 3 ( ) 2R +1 4 ( ( i ) )R .(2. 6.31) The -independentpartof r istheRiccitensorinaspacetimegeometrywith( -indepe ndent)tor sion. Insec.2.4.a.3wediscussedcovariantshiftsofthegaugepotential.In any gauge theorysuch shiftsdonotchangethetransformationpropertiesofthecovariantderivati ve sa ndthusareperfectlyacceptable;theshiftedgauge“eldsprovideanequallygood descriptionofthetheory.Insec.2.4.a.3weusedtherede“nitionstoeliminatea“eld strength.Herewerede“netheconnection toeliminate T by = Š iRM.(2. 6.32) (Thiscorrespo ndstoshifting a b cbyaterm a b cR toca ncel T a b c;wetempora rilymake useofvectorindices a torepres enttracelessbispinorssincethismakesitclearthatthe shift(2.6.32)ispossibleonlyinthreedimensions.)Theshifted r ,dro ppingprimes, is r = W Š1 4 ( ) r r 4 3 2R +2 R2.(2. 6.33) Thisrede“nitionof isequivalenttoreplacingtheconstraint(2.6.9)with {, } =2 i Š 2 RM.(2. 6.34) Wew ill“ndthatthea nalogofthenewtermappearsintheconstraintsforfour dimensionalsupergravity(seechapter5).Thisisbecausewecanobtainthethree di mensionaltheoryfromthefourdimensionalone,andthereisnoshiftanalogousto (2.6.32)possibleinfourdimensions. Thesuper“elds R and Garethevariationsofthesupergravityaction(see below) withrespecttothetwounconstrainedsuper“eldsand E( )of(2.6.15-17).

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422.ATOYSUPERSPACEThe“eldequationsare R = G=0;theseares olvedonlyby”atspace(justasfor ordi narygravityinthree-dimensionalspacetime),sothree-dimensionalsupergravityhas no dy na mics(all“eldsareauxiliary). g.Actions Wenowt urntotheconstructionofactionsa ndtheirexpansi onintermsofcomponent “elds.Asweremarkedearlier,in”atsuperspacetheintegralof any (scalar) super“eldexpressionwiththe d3xd2 measureis globally supersymmetric.Thisisno longertrueforlocallysupersymmetrictheo ries.(Thenewfeaturesthatarisearenot speci“callylimitedtolocalsupersymmetry,butareageneralconsequenceoflocalcoordinateinvariance). Wer ecallthatinourfo rmalismanarbitrary matter super“eldtransforms accordingtotherule = eiK eŠ iK= eŠ iK eiK, K= KMiD M+ K iM ,(2. 6.35) where D Mmeansthatweletthedierentialoperatoractoneverythingtoitsleft.(The variousf ormsofthetransformationlawcanbes eentobeequivalentafterpowerseries expansionoftheexponentials,orbymultiplyingbyatestfunctionandintegratingby parts).L agrangiansarescalarsuper“elds,andsinceanyLagrangian IL isconstr ucted fromsuper“ eldsand operators,aLagrangiantransformsinthesameway. IL= eiKILeŠ iK= eŠ iKILeiK.(2. 6.36) Thereforetheintegral d3xd2 IL is not invariantwithrespecttoourgaugegroup.To “ndinvariants,weconsiderthevielbeinasasquaresupermatrixinitsindicesandcomputeitssuperdeterminant E .Thefo llowingresultwillbederi vedinour discussionof four-dimension s(sees ec.5.1): ( EŠ 1)= eiKEŠ 1eŠ iK(1 eiK) = EŠ 1eiK.(2. 6.37)

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2.6.Supergravity43Thereforetheproduct EŠ 1IL transformsinexactlythesamewayas EŠ 1: ( EŠ 1IL )= EŠ 1ILeiK.(2. 6.38) Sinceeverytermbutthe“rstoneinthepowerseriesexpansionofthe eiKisatotal derivative,weconcludethatuptosurfaceterms S = d3xd2 EŠ 1IL ,(2. 6.39) isinvariant.Wethereforehaveasimpleprescriptionforturninganygloballysupersymmetricactionintoalocallysupersymmetricone: [ IL ( DA,)]glob al EŠ 1IL ( A,),( 2.6.40) inanalogytoordinarygravity.Thus,theactionforthescalarmultipletdescribedbyeq. (2.3.5)takesthecovariantizedform S= d3xd2 EŠ 1[ Š1 2 ( )2+1 2 m 2+ 3! 3].(2.6 .41) Forv ectorgaugemultipletsthes impleprescriptionofreplacing”atderivatives DAbygravit ationallycovariantones Aissucienttoconvertglobalactionsintolocal actions,ifweincludetheYang-Millsgeneratorsinthecovariantderivatives,sothatthey arecovariantwithrespecttobothsupergravityandsuper-Yang-Millsinvariances.However,suchaprocedureisnotsucientformoregeneralgaugemultiplets,andinparticularthesuperformsofsec.2.5.Ontheo therha nd,itispossibl etoformulate all gauge theorieswithinthesuperform framew ork,atleastattheabelianlevel(whichisallthat isrelevantfor p -formsfor p > 1).Additionaltermsduetothegeometryofthespace willautomaticallyappearinthede“nitionsof“eldstrengths.Speci“cally,thecurvedspaceformulationofsuperformsisobtainedasfollows:Thede“nitions(2.5.8)holdin arbitrarysuperspaces,independentofanymetricstructure.Converting(2.5.8)toatangent-space basiswiththecurvedspace EA M,weobtaine quationsthatdierfrom(2.5.9) onlybythereplacementofthe”at-spacecovariantderivatives DAwiththecurved-space ones A. To illustratethis,letusreturntotheabelianvectormultiplet,nowinthepresence ofsupergravity.The“eldstrengthforthevectormultipletisa2-form:

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442.ATOYSUPERSPACEF= + Š 2 i F = ŠŠ T , F = ŠŠ T EE.(2. 6.42) We againimposetheconstraint F=0,which implies F = iC ( W ), W=1 2 + R ;(2. 6.43) wherewehaveused(2.6.30)substitutedinto(2.4.33).Comparingthistotheglobal“eld strengthde“nedin(2.4.20),weseethatanewtermproportionalto R appears.The extratermin Wisnecessaryforgaugeinvarianceduetotheidentity = i2 3 [ ].Inthegloballimitthecommutatorvanishes,butinthelocal caseitgivesacontributionthatispreciselycanceledbythecontributionofthe R term. Theseresultscanalsobeobtainedbyuseofderivativesthatarecovariantwithrespect tobothsupergravityandsuper-Yang-Mills. Weturnnowtot heacti onforthegauge“eldsoflocalsupersymmetry.Weexpect toconstructitoutofthe“eldstrengths Gand R .Bydimensi onalanalysis(noting that hasdimen sions( mass )Š1 2 inthreedimensions),wededuceforthePoincar esupergravityactionthesupersymmetricgeneralizationoftheEinstein-Hilbertaction: SSG= Š 2 2 d3xd2 EŠ 1R .(2. 6.44) Wecancheckthat (2.6.44)leadstothe correctcomponentactionasfollows:d2 EŠ 1R 2R 3 4 r (see(2.6.33)),andthusthegravitationalpartoftheactionis correct.Wecanalsoaddasupers ymmetriccosm ologicalterm Scosmo= 2 d3xd2 EŠ 1,(2. 6.45) whichleadstoanequationsofmotion R = G=0.Theonly solution tothisequation(inthreedimensions)isemptyanti-deSitterspace:From(2.6.33), r =2 2, W=0. Higher-derivativeactionsarepo ssiblebyusingotherfunctionsof Gand R .For ex ample,theanalogofthegauge-invariantmasstermfortheYang-Millsmultipletexists

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2.6.Supergravity45hereandisobtainedbythereplacementsin(2.4.38)(alongwith,ofcourse, d3xd2 d3xd2 EŠ 1): A iTi A iM W iTi G iM +2 3 ( R ) iM .(2. 6.46) Thisgives ILmass= d3xd2 EŠ 1 ( G +2 3 R Š1 6 ( ) ).(2.6 .47)

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462.ATOYSUPERSPACE2.7.Quantumsuperspace a.Scalarmultiplet InthissectionwediscussthederivationoftheFeynmanrulesforthree-dimensionalsuper“eldperturbationtheory.Sincethestartingpoint,thesuper“eldaction,is somuchlikeacomponent(ordinary“eldtheory)action,itispossibletoreadothe rulesfordoingFeynmansupergraphsalmostbyinspection.However,asanintroduction tothefour-dimensionalcaseweusethefullmachineryofthefunctionalintegral.After derivingtherulesweapplythemtosomeone-loopgraphs.Themanipulationsthatwe pe rformonthegraphsaretypicalandillustratethemannerinwhichsuper“eldshandle thecancellationsandothersimpli“cations duetosupersymmetry.Formoredetails,we referthereadertothefour-dime nsionaldiscussi oninchapter6. a.1.Genera lforma lism TheFeynmanrulesforthescalarsuper“eldcanbereaddirectlyfromthe Lagrangian:Thepropagatorisde“nedbythequadraticterms,andtheverticesbythe interactions.Thepropagatorisanoperatorinboth x and space,andatthevertices weintegr ateoverboth x and .ByFouriertr ansformationwechangethe x integration toloop-momentumintegration,butweleavethe integrationalone.( canalsobe Fouriertra nsformed,butthiscauseslittlechangeintherules:seesec.6.3.)Wenow derivetherulesfromth ef unction alintegral. Webeginbyc onsideringthegeneratingfunctionalforthemassivescalarsuper“eld witharb itraryself-interaction: Z ( J )= ID exp d3xd2 [1 2 D2+1 2 m 2+ f ()+ J ] = ID exp [ S0()+ SINT()+ J ] = exp [ SINT( J )] ID exp [ 1 2 ( D2+ m )+ J ].(2.7.1) Intheusualfashionwecompletethesquare,dothe(functional)Gaussianintegralover ,andobtain

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2.7.Quantumsuperspace47Z ( J )= exp [ SINT( J )] exp [ Š d3xd21 2 J 1 D2+ m J ].(2.7 .2) Usingeq.(2.2.6)wecanwrite 1 D2+ m = D2Š m Š m2 .(2. 7.3) (Note D2behavesj ustas /inconventional“eldtheory.)Weobtain,inmomentum space,thefollowingFeynmanrules: Propagator: J ( k ) J ( Š k ) d3k (2 )3 d21 2 J ( k ) D2Š m k2+ m2 J ( Š k ) = D2Š m k2+ m2 2( Š ).(2.7 .4) Verti ces:Aninteractionterm,e.g.d3xd2 D D ... ,gives avertexwith linesleavingit,withtheappropriateoperators D, D,etc.actingont hecorresponding lines,andanintegralover d2 .Theoper ators Dwhichappearinthepropagators,or arecomingfromavertexandactonaspeci“cpropagatorwithmomentum kleaving thatvertex,d ependonthatmomentum: D= + k.(2. 7.5) Inadditionwehaveloop-momentumintegralstoperform. Ingeneralwe“nditconvenienttocalculatetheeectiveaction.Itisobtainedin standardfashionbyaLegendretransformationonthegeneratingfunctionalforconnectedsupergraphs W ( J )anditco nsistsofasumofone-particle-irreduciblecontributionsobtainedbyamputatingexternallinepropagators,replacingthembyexternal“eld factors( pi, i),andintegratingover pi, i.Therefo re,itwillhavetheform ()=n1 n d3p1... d3pn(2 )3 n d21... d2n( p1, 1) ... ( pn, n) (2 )3 ( pi)loops d3k (2 )3 internal vertices d2 propagators vertices (2.7.6)

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482.ATOYSUPERSPACEAswehavealreadymentioned,allofthiscanbereaddirectlyfromtheaction,byanalogywiththederivationoft heusualFeynmanrules. Theintegrandintheeectiveactionis apriori anonloca lf unction ofthe x s(nonpolynomi alinthe p s)andofthe 1, ... n.Howev er,wecanmanipulatethe -integrationssoastoexhibititexplicitlyasafunctionalofthesallevaluatedatasinglecommon asfollows:Ageneralmultiloopintegralconsistsofverticeslabeled i i +1,conn ectedbypropagatorswhichcontainfactors ( iŠ i +1)withope rators Dactingon them.Consideraparticularloopinthediagramandexamineonelineofthatloop. Thefactorsof D canbecombinedbyusingtheresult(transferrule): D( i, k ) ( iŠ i +1)= Š D( i +1, Š k ) ( iŠ i +1),(2.7 .7) aswellastherulesofeq.(2.2.6),afterwhichwehaveatmosttwofactorsof D actingat oneendoftheline.Atthevertexwherethisendisattachedthese D scanbeint egrated bypartsontotheot herlines(orexternal“elds)usingtheLeibnitzrule(andsomecare withminussignssincethe D santicommut e).Thentheparticular -functionnolonger hasanyderivativesactingonitandcanbeusedtodothe iintegration,th useectively shrinking the( i, i +1) linetoapointin -space.Wecanrepeatthisprocedureon eachlineoftheloop,integratingbypartso neatatimeandshrink ing.Thiswillgenerateasumofterms,fromtheintegrationb yparts.The procedurestopswhenineach termweareleftwithexactlytwolines,onewith ( 1Š m)whichisfree ofanyderivatives,andonewith ( mŠ 1)whichmayca rryzero,one,ortwoderivatives.Wenow usetherules(whichfollowfromthede“nition 2( )= Š 2), 2( 1Š m) 2( mŠ 1)=0, 2( 1Š m) D2( mŠ 1)=0, 2( 1Š m) D22( mŠ 1)= 2( 1Š m).(2.7 .8) Thus,inthosetermswhereweareleftwithno D orone D weget zero,while inthe termsinwhichwehavea D2actingononeofthe -functions, mult ipliedbytheother -function,weusetheaboveresult.Weareleftwiththesingle -function,whichwecan usetod oone more integration,thus “nallyreducingthe -spacelooptoapoint.

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2.7.Quantumsuperspace49Theprocedurecanberepeatedloopbyloop,untilthewholemultiloopdiagram hasbeenreducedto one pointin -space,givingacontributiontotheeectiveaction ()= d3p1... d3pn(2 )3 n d2 G ( p1, ... pn)( p1, ) ... D( pi, ) ... D2( pj, ) ... ,(2. 7.9) where G isobtainedbydoingordinaryloop-mo mentumintegrals,withsomemomentum factorsinthenumeratorscomi ngfrom anticommutatorsof D sarisingintheprevious manipulation. a.2.Examples Wegivenowtwoex amples,inamasslessmodelwith3interactions,toshowhow the manipulationworks.The“rstoneisthecalculationofaself-energycorrection representedbythegraphinFig.2.7.1 k k + p ( Š p ) ( p ) Fig.2.7.1 2= d3p (2 )3 d2 d2( Š p )( p ) d3k (2 )3 D2 ( Š ) k2 D2 ( Š ) ( k + p )2 .(2. 7.10) Thetermsinvolving canbemanipulatedasfollows,usingintegrationbyparts: D2 ( Š ) D2 ( Š )( p ) = Š1 2 D ( Š )[ DD2 ( Š )( p )+ D2 ( Š ) D( p )]

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502.ATOYSUPERSPACE= ( Š )[( D2)2 ( Š )( p )+ DD2 ( Š ) D( p ) + D2 ( Š ) D2( p )].(2.7.11) However,using( D2)2= Š k2and DD2= kDweseethata ccordingtotherulesin eq.(2.7.8)onlythelasttermcontributes.We“nd 2= d3p (2 )3 d2 ( Š p ) D2( p ) d3k (2 )3 1 k2( k + p )2 .(2. 7.12) Doingtheintegrationbypartsexplicitlycanbecomerathertediousanditis preferabletoperformitbyindicating D sandmovingthemdirectlyonthegraphs.We showthisinFig.2.7.2: D2D2D2D2D2D2D2DDFig.2.7.2 Onlythelastdiagramgivesacontribution.Onefurtherruleisusefulinthisprocedure: Ingeneral,afterintegrationbyparts,various D -factorsendupindierentplacesinthe “nalexpressionandonehastoworryaboutminussignsintroducedinmovingthempast eachother.Theoverallsigncanbe“xedattheendbyrealizingthatwestartwitha particularorderingofthe D sandwecanexaminewhathappenedtothisorderingat theendofthecalculation.Forexample,wemaystartwithanexpressionsuchas D2... D2... D2... =1 2 DD...1 2 DD...1 2 DD... andend upwith D... D... D... D... D... D... wherethevarious D sactondier ent“elds.The overallsignc anobviouslybedeterminedbyjustcountingthenumberoftranspositions. Forexample,inthecaseabov ewewouldendu pwithaplussign.N otethatt hisrule alsoappliesiffactorssuchas karise,providedonepaysa ttentiontot hemannerin whichtheywereproduced(e.g.,atwhichendofthelinewerethe D sacting?Didit comefrom DDorfrom DD?). Oursecondexampleisthethree-pointd iagrambelow,whichwemanipulateas showninthesequenceofFig.2.7.3:

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2.7.Quantumsuperspace51 D2 DDD2D2D2D2D2D2D2D2D2D2D2D2D2 DDDDFig.2.7.3 Atthe“rsts tagewehaveintegratedbypartsthe D2othebottomlineandimmediatelyreplaced( D2)2by = Š k2.Atthes econdstagewehaveintegratedbypartsthe D2otherightside,butkeptonlythosetermsthatarenotzero:Thebottomlinehas alreadybeenshrunktoapointbythecorresponding -function(butweneednotindicate thisexplicitly;anylinethathasno D sonitcanbeconsideredashavingbeen shrunk)andintheendwekeeponlytermswith exactly twofac torsof D intheloop.For themiddlediagramthismeansusing DD2D= DkD= Š kD2+aterm withno D swhichmaybedropped.Theintegrandintheeectiveactioncanbewrittenthenas d3k (2 )3 1 k2( k + p1)2( k Š p3)2 ( p3, )[ Š ( p1, )( p2, ) k2Š D( p1, ) D( p2, ) k+ D2( p1, ) D2( p2, )],(2.7.13) andonl ythe k -momentumintegralremainstobedone. Ingeneral,theloop-momentumintegralsmayhavetoberegularized.Theprocedurew euse, whichisguaranteedtopreservesupersymmetry, istodo allthe D -manipulations“rst,untilwere ducetheeectiveactiontoanintegraloverasingle ofan expressionthatisaproductofsuper“elds,andthereforemanifestlysupersymmetric. Theremainingloop-momentumintegralsmaythenberegularizedinanymanner

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522.ATOYSUPERSPACEwhatsoever,e.g.,byusingdimensionalregularization.Weshalldiscusstheissues involvedinthiskindofregularizationinsec.6.6.Analternativeprocedure,somewhat cumbersomeinitsapplicationbutbetterunde rstood,issupersymmetricPauli-Villars regularization.Inthreedimensionsthisisapplicableeventogaugetheories,sincegauge invariantmasstermsexist. b.Vectormultiplet Nothingnewisen counteredinthederivationorapplicationoftheFeynman rules.However,thederivationmustbeprecededbyquantization,i.e.,introductionof gauge-“xingtermsandFaddeev-Popovghosts,whichwenowdiscuss. Webeginwit hthecla ssicalaction SC=1 g2 tr d3xd2 W2.(2. 7.14) Thegaugeinvarianceis = K and,bydirectanalogywiththeordinaryYang-Mills case,wecanchoosethegauge-“xingfunction F =1 2 D.Weuseanaver aging procedurewhichleadstoa gauge-“xingtermwithoutdimensionalparameters, FD2F ,and obtain,forthequadraticaction, S2=1 g2 tr d3xd2 [1 2 (1 2 DD)(1 2 DD) Š1 (1 2 D) D2(1 2 D)] =1 2 1 g2 tr d3xd2 [1 2 (1+1 ) +1 2 (1 Š1 )i D2].(2.7 .15) Variouschoi cesofthegaugeparameter arepossible: Thechoice = Š 1givesthe kineticterm1 2 i D2,wh ilethechoice =1gives1 2 ,whichresu ltsinthe simplestpropagator. TheFaddeev-Popovactionissimply SFP=1 g2 tr d3xd2 c( x )1 2 Dc ( x ),(2.7 .16) withtwoscalarmultipletghosts.(Notethatinabackground-“eldformulationofthe

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2.7.Quantumsuperspace53theory,similartotheonewediscussinsec.6.5,onewouldreplacetheoperator D2in th e gau ge “xingtermbythebackground-covariantoperator 2,andthis wouldgiv erise toa third, Nielsen-Kallosh,ghostaswell.) TheFeynmanrulesarenowstraightforwardtoobtain.Theghostpropagatoris conventional,followingfromthequadraticghostkineticterm cD2c ,wh ilethegauge “e ldpropagatoris k2 2( Š ).(2.7 .17) Verti cescanbereadofromtheinteractionterms.Thegauge-“eldself-interactions(in thenonabeliancase)are g2LINT= Ši 4 DD[, D] Š1 12 DD[, { ,} ] Š1 8 [, D][, D]+i 12 [, D][, { ,} ] +1 72 [, { ,} ][, { ,} ],(2.7 .18) thoseoftheghostsare g2LINT= Š i1 2 cD[, c ],(2.7 .19) whilethoseofacomplexscalar“eldare g2LINT= ( 2Š D2)= [ Š i DŠ i1 2 ( D) Š 2].(2. 7.20)

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Contentsof 3.REPRESENTATIONSOFSUPERSYMMETRY 3.1.Notation 54 a.Indexconventions54 b.Superspace56 c.Symmetriz ationandantisymmetrization56 d.Conjugation57 e.Levi-Civitatensorsandindexcontractions58 3.2.Thesupersymmetrygroups62 a.Liealgebras62 b.Super-Lie algebras63 c.Super-Poincar ealgebra63 d.Positivityof theenergy64 e.Superconformalalgebra65 f.Super-deSitteralgebra67 3.3.Representationsofsupersymmetry69 a.Particlerepresentations69 a.1.Masslessrepresentations69 a.2.Massiverepresentationsandcentralcharges71 a.3.Casimiroperators72 b.Representationsonsuper“elds74 b.1.Superspace74 b.2.Acti onofgeneratorsonsuperspace74 b.3.Acti onofgeneratorsonsuper“elds75 b.4.Extendedsupersymmetry76 b.5.CPTinsuperspace77 b.6.Chir alrepresentationsofsupersymmetry78 b.7.Superconformalrepresentations80 b.8.Super-deSitterrepresentations82 3.4.Covariantderivatives83 a.Construction83 b.Algebraicr elations84 c.Geometryof”atsuperspace86 d.Casimiroperators87 3.5.Constraine dsuper “elds89 3.6.Componentexpansions92

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a. -expansions92 b.Projection94 c.Thetransformationsuper“eld96 3.7.Superintegration97 a.Berezinintegral97 b.Dimensions99 c.Superdeterminants99 3.8.Superfunctionaldierentiationandintegration101 a.Dierentiation101 b.Integr ation103 3.9.Physical,auxiliary,andgaugecomponents108 3.10.Compensators112 a.Stueckelbergformalism112 b.CP(1)m odel 113 c.Cosetspaces117 3.11.Projectionoperators120 a.General120 a.1.Poincar eproj ectors121 a.2.Super-Poincar eproj ectors122 b.Exam ples 128 b.1.N= 0 128 b.2.N= 1 130 b.3.N= 2 132 b.4.N= 4 135 3.12.On-shellrepresentationsandsuper“elds138 a.Fieldstrengths138 b.Light-cone formalism142 3.13.O-shell“eldstrengthsandprepotentials147

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3.REPRESENTATIONSOFSUPERSYMMETRY 3.1.Notation Anifor ani,anda2fora2. Wenowt urntofourdimensi ons.Ourtreatmentwillbeentirelyself-contained;it willnotassumefamiliaritywithourthree-dimensionaltoy.Althoughsupersymmetryis morecomplicatedinfourdimensionsthaninthree,becausewegiveamoredetaileddiscussion,somegeneralaspectsofthetheorymaybeeasiertounderstand.Webeginby givingthenotationandconventionsw eusethro ughouttherestofthework. a.Indexconventions Ourindexconventionsareasfollows:Thesimplestnontrivialrepresentationof theLorentzgroup,thetwo-componentco mplex(Weyl)spinorrepresentation(1 2 ,0)of SL (2, C ),islabeledbyatwo-valued(+or-)lower-caseGreekindex(e.g., =( +, Š)), andthecomplex-conjugaterepresentation(0,1 2 )islabel edbyado ttedindex ( €=( €+, €Š)). Af ou r-componentDiracspinoristhecombinationofanundotted spinorwithad ottedone(1 2 ,0)+(0,1 2 ),andaMajoranaspinorisaDiracspinorwhere th ed o tte ds pinoristhecomplexconjugateoftheundottedone.Anarbitraryirreduciblerepresentation( A B )isthenconve nientlyrepresentedbyaquantitywith2 A undottedindicesand2 B dottedindices,totallysymmetricinitsundottedindicesandin itsdottedindices.Anexampleistheself-dualsecond-rankantisymmetrictensor(1,0), whichisrepresentedbyasecond-ranksymmetricspinor f.(Thechoiceo fselfdualvs. anti-self-dualfollowsfromWickrotationfromEuclideanspace,wherethesignisunambigu ous.) Anotherexampleisthevector(1 2 ,1 2 ), la be le dw it ho neundottedandonedotted i ndex,e.g., V€.Arealv ectorsatis“esthehermiticitycondition V€= V€= V€.As as horthandnotation,weoftenuseanunderlinedlower-caseRomanindextoindicatea v ectorindexwhichisacompositeofthecorrespondingundottedanddottedspinor indices:e.g., V a V€.Weconsiders uchanindexmerelyasanabbreviation:Itmay appearononesideofanequationwhiletheexplicitpairofspinorindicesappearsonthe

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3.1.Notation55other,oritmaybecontractedwithanexplicitpairofspinorindices.Whendiscussing Lorentznoncovariantquantit ies(as,e.g.,inlight-coneformalisms),wesometimeslabel thevaluesofavector indexasfollows: V a=( V+€+, V+€Š, VŠ€+, VŠ€Š) ( V+, VT, VT, Š VŠ),(3.1 .1) where VTisthecomplex conjugateof VT,and VarerealinMinkowskispace(but V+isthecomplex conjugateof VŠinWick-rotatedEuclideans pace).Moregenerally,we canrelateavectorlabel ainan arbitrary basis,where a = € ,tothe € basisbyaset ofClebsch-Gordancoecients,thePaulimatrices:Wede“ne forfields : V€=1 2 b €Vb Vb =1 2 b €V€; forderivatives : €= b €b b =1 2 b €€; forcoordinates : x€=1 2 b €xb xb = b €x€.(3. 1.2a) ThePaulimatricessatisfy b €c €=2 b c b €b €=2 €€.(3. 1.2b) TheseconventionsleadtoanunusualnormalizationoftheYang-Millsgaugecoupling constant g ,since € €Š igV€= b €b Š ig1 2 b €Vb b €( b Š i gVb ) andhenceour g is 2timestheu sualone g .(Weuset hesummationconvention:Any indexofanytypeappearingtwiceinthesameterm,oncecontravariant(asasuperscript)andoncecovariant(asasubscript),issummedover.) NexttoLorentzi ndices,thetypeofindiceswemostfrequentlyuseare isospin i ndices:internalsymmetryindices,usuallyforthegroup SU ( N )or U ( N ).Theseare representedbylower-caseRomanletters,withoutunderlining.Weusean underlined i ndexonlytoindicateacompositeindex,anabbreviationforapairofindices.Inadditiontothevectorindexde“nedabove,wede “neacompositespinor-isospinorindexby an underlinedlower-caseGreekindex(undottedordotted): a € a€,

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563.REPRESENTATIONSOFSUPERSYMMETRY a € a€. b.Superspace Wede “ne N -extendedsuperspacetobeaspacewithboththeusualrealcommutingspa cetimecoordinates x€= x a= x a,andantico mmutingcoordinates a = (andtheircompl exconjugates € =( ))whichtran sformasaspinorandan N -componentisos pinor.Todenotethesecoordinatescollectivelyweintroduce supervector i ndices,usingupper-caseRomanletters: zA=( x a, € ),(3.1 .3a) andthecorresp o ndingpartialderivatives A=( a, € ), AzB A B,(3. 1.3b) wherethenonvanishingpartsof A Bare a b, a b ,and € € b a€€.The derivativesare de“nedtosatisfya graded Leibnitzrule,givenbyexpressingdierentiationas gradedcommutation: ( AXY ) [ A, XY } =[ A, X } Y +( Š )XAX [ A, Y } ,(3. 1.4a) where( Š )XAis Š whenboth X and Aareanticommuting,and +otherwise ,andthe graded commutator[ A B } AB Š ( Š )ABBA istheanticommutator { A B } when A and B arebothoperatorswithfermistatistics,andthecommutator[ A B ]otherwise. Eq.(3.1.4a)followsfromwritingeach(anti)commutatorasadierence(sum)oftwo terms.Thepartialderivativesalsosatisfygradedcommutationrelations: [ A, B} =0.(3. 1.4b) c.Symmetrizationand antisymmetrization Ournot ationforsymmetrizingandantisymmetrizingindicesisasfollows:Symmetrizationisindicatedbyparentheses(),whileantisymmetrizationisindicatedby brackets[].Bysymme trizationwemeansimplythesumoverallpermutationsof indices,withoutadditionalfactors(andsimilarlyforantisymmetrization,withtheappropriatepermutationsigns).Allindicesbetweenparentheses(brackets)aretobe (anti)symmetrized exceptthosebetweenverticallines || .Forex ample, A( | B | )=

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3.1.Notation57AB+ AB.Ina ddition,justasitisconvenienttode“nethegradedcommutator [ A B } ,wede “negr adedantisymmetrization[)tobeasumofpermutationswithaplus signforanytranspositionoftwospinorindices,andaminussignforanyotherkindof pair. d.Conjugation Whenworkingwithoperatorswithfermist atistics,theonlytypeofcomplexconjugationthatisusuallyde“nedishermitianconjugation.Itisde“nedsothatthehermitianconjugateofapr o ductistheproductofthehermitianconjugatesofthefactorsin reverseorder.Foranticommutingc-numbershermitianconjugationagainisthemost naturalformofcomplexconjugation.Wedenotetheoperationofhermitianconjugation by ad agger,andi ndicatethehermitianconjugateofagivenspinorbyabar: ( ) €,or( €) .Inparticular, thisappliestot hecoordinates and introducedabove.Hermitianconjugationofanobjectwithmany(upper)spinorindicesis de“nedasforaproductofspinors: ( 1 1... j j 1€1... k€k)= k k... 1 1 j€j... 1€1=( Š 1)1 2 [ j ( j Š 1)+ k ( k Š 1)]1 1... k k 1€1... j€j,(3. 1.5a) andhence ( 1... j€1...€k) ( Š 1)1 2 [ j ( j Š 1)+ k ( k Š 1)] 1... k€1...€j.(3. 1.5b) Inaddition,isospinindicesfor SU ( N )gofrom uppertolower,orviceversa,uponhermitianconjugation.Hermitianconjugationofpartialderivativesfollowsfromthereality of A B=( AzB)=[ A, zB} : ( A)= Š ( Š )AA,(3. 1.6a) where( Š )Ais Š 1forspin orindicesand+1otherwise: ( a)= Š a,( )=+ € .(3. 1.6b) Hermitianconjugationasappliedtogeneraloperatorsisde“nedby O ( O ) ,(3. 1.7)

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583.REPRESENTATIONSOFSUPERSYMMETRYwheretheintegrationisovertheappropriatespace(aswillbedescribedinsec.3.7)and isthehermitianconj ugateofthefunction ,asde “nedabove. Si nc ei nt eg rationde“nesnotonlyasesquilinear(hermitian)metric onthe sp aceoffunctions,asusedtode“neaHilbertspace,butalsoabilinearmetric ,we canalsode“nethetransposeofanoperator: O ( Ot ) ,(3. 1.8) where is Š for O and anticommuting,+otherwise.Whentheoperatorisexpressed asamatrix,thehermitianconjugateandtransposetaketheirfamiliarforms.Wecan alsode“necomplexconjugationofanoperator: O O ,(3. 1.9) with asin(3.1.8).Forc-numberswehave t= and *= .Forpartia lderivatives,integrationbypartsimplies( A)t= Š A.Ingen eral,wealsohav etherelation O *t= O,andtheorderin grelations( O1O2)t= O2 tO1 tand( O1O2)*= O1* O2*,as wellasth eusual( O1O2)= O2 O1 . e.Levi-Civitat enso rsandindexcontractions Thereisonlyonenontrivialinvariantmatrixin SL (2, C ),theantisymmetricsymbol C(anditscom plexconjugateandtheirinverses),duetothevolume-preserving natureofthegroup(unitdeterminant).Similarly,for SU ( N )wehavethe antisymmetricsymbol Ca1... aN(anditscom plexconjugate).Inadditionwe“nditusefultointroduce theantisymmetricsymbolof SL (2 N C ) SU ( N ) SL (2, C ), C 1... 2 N.B ecauseofanticommutativity,itappearsintheantisymmetricproductofthe2 N sof N -extended supersymmetry.Theseobjectssat isfythefollowingrelations: C= C€€, CC= [ ] ;( 3.1.10a) Ca1... aN= Ca1... aN, Ca1... aNCb1... bN= [ a1b1... aN] bN;(3. 1.10b) C 1... 2 N= C€ 2 N...€ 1, C 1... 2 NC 1... 2 N= [ 1 1... 2 N] 2 N.(3. 1.10c) The SL (2 N C )symbolcan beexpr essedintermsoftheothers:

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3.1.Notation59C 1... 2 N= 1 N !( N +1)! ( Ca1... aNCaN +1... a2 NC1N +1... CN2 N permutationsof i).(3.1 .11) Themagnitudesofthe C sare“xedbythe conventions C= C€€, C 1... 2 N= C€ 2 N...€ 1,(3. 1.12) whichsettheabsolutevaluesoftheircomponentsto0or1. Wehavethefo llowingrelationfort heproductofallthe s(b ecause { } =0, thesquareofanyonecomponentof vanishes): 1... 2 N= C 2 N... 1(1 (2 N )! C 2 N... 1 1... 2 N) C 2 N... 12 N,(3. 1.13) andasimilarrelationfor ,where,uptoa phasefactor, 2 Nissimplytheproductofall the s.Ourconventionsforcomplexconjugationofthe C simply 2 N = 2 N.Altho ugh seldomneeded(exceptforexpressingthe SL (2, C ) SU ( N )covariantsint ermsofcovariantsofasubgro up,as,e.g.,whenperformingdimensionalreductionorusingalight-cone formalism),wecan“xthephases(uptosigns)inthede“nitionofthe C sbythefollowingconventions: C= C€€, C 1... 2 N= C€ 1...€ 2 N Ca1... aN= Ca1... aN.(3. 1.14) Inparticular,wetake C= 0 i Š i 0 (3.1.15) For N =1wehave 2=1 2 C= i +Š. Cisthusthe SL (2, C )metri c,andcanbe usedforraisingandloweringspinorindices: = C, = C,( 3.1.16a) = 21 2 C=1 2 =1 2 = i +Š;(3. 1.16b) €= €C€€, €= C€€ €,(3. 1.16c)

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603.REPRESENTATIONSOFSUPERSYMMETRY € €= 21 2 C€€ € €=1 2 € €=1 2 = i €+ €Š;(3. 1.16d) V a= V bCC€€ V b b a, V a= CC€€V b a bV b,(3. 1.16e) V W V aW a= W V V21 2 b aV aV b=1 2 V aV a=1 2 V V = V+VŠ+ VTVT= Š detV€.(3. 1.16f) (Asindicatedbytheseequations,wecontractindiceswiththecontravariantindex“rst.) Ourunusualde“nitionofthesquareofavectorisusefulforspinoralgebra,butwecautionthereadernottoconfus eitwiththest a ndardde“nition.Inparticular,wede“ne 1 2 a a.(However,wh enwetransform(witha nonunimodular transformation)toa cartesianbasis,thenwehavetheusual = a a .Forthecoordin ates,wehave x2=1 4 xa xa .Ourconv entionsareconvenientforsuper“eldcalculations,butmayleadto afewun usualcomponentnormalizations.) De“ning € a b€, € b a€;(3. 1.17) wehavethei dentit ies €€= € € = .(3. 1.18) Fr om(3.1.10a)weobtainthefrequentlyusedrelation [ ]= C( C)= C( ),(3.1 .19) whichistheWeyl-spinorformoftheFierzidentities.Similarrelationsfollowfrom (3.1.10b,c). Thecomplexconjugationpropertiesof Cimplythatthecomplexconjugatesof covariant (lowerindex)spinors,includingspinorpartialderivatives(cf.(3.1.6)),havean additionalminussign: ( )= Š €.(3. 1.20) From(3.1 .11)and(3.1.18),ordirectlyfromthef acttha tantisy mmetricsymbolsde“ne

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3.1.Notation61determinants( detV € =( detV€)N=( Š V2)N),wehavethefollowingidentity: C€ 2 N...€ 1 1€ 1... 2 N€ 2 N= C 1... 2 N( Š )N.(3. 1.21) Finally,wede“nethe SO (3,1)Levi-Civitatensoras a b c d= i ( CCC€€C€€Š CCC€€C€€), a b c d e f g h= Š [ a e b f c g d ] h.(3. 1.22)

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623.REPRESENTATIONSOFSUPERSYMMETRY3.2.Thesupersymmetrygroups LiealgebrasandLiegroupsplayanimportantrolein“eldtheory;groupssuchas thePoincar egroup ISO (3,1),theLorentzgroup SO (3,1), SU (3)and SU (2) U (1)are fa miliar.ThenewfeatureneededforsupersymmetryisageneralizationofLiealgebras tosuper-Liealgebras(alsocalledgradedLie algebras;however,thistermissometimes usedinadierentway). a.Liealgebras ALieal gebraconsistsofasetofgenerators { A} ,A=1,. .., M .Theseob jects closeunderanantisymmetricbinaryoperationcalledaLiebracket;wewriteitasa commutator: [A,B]=ABŠ BA.(3. 2.1) TheLiealgebraisde“nedbyitsstructureconstants fAB C: [A,B]= ifAB CC.(3. 2.2) Thestructureconstantsarerest rictedbytheJacobiidentities fAB DfDC E+ fBC DfDA E+ fCA DfDB E=0(3.2 .3) whichfollowfrom [[A,B],C]+ [[B,C],A]+ [[C,A],B]=0.(3 .2.4) Thegeneratorsformabasisforvectorsoftheform K = AA,wherethe AarecoordinatesintheLiealgebrawhichareusually takentocommutewiththegeneratorsA.In mostphysicsapplicationstheyaretakentobereal,complex,orquaternionicnumbers. B ecausethestructureconstantssatisfytheJaco biidentities,itisalwayspossibletorepresentthegeneratorsasmatrices.WecanthenexponentiatetheLiealgebraintoaLie groupwithelements g = eiK;ingen eral,dierentrepresentationsoftheLiealgebrawill giverisetoLiegroupswithdierenttopologicalstructures.Ifasetof“elds( x )transformslinearlyundertheactionoftheLiegroup,wesay( x )isinorc arriesarepresentationofthegroup.Abstractly,wewrite ( x )= eiK( x ) eŠ iK;(3. 2.5)

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3.2.Thesupersymmetrygroups63togivethismeaning,wemustspecifytheactionofthegeneratorson,i.e.,[A,].For example,if K isamatrixrepresentationandisaco lumnv ector,theexpressionabove istobeinterpretedas= eiK. b.Super-Liealgebras Forsupersy mmetrywegeneralizeandconsidersuper-Liealgebras.Theessential newfeatureisthatnowtheLiebracketofso megeneratorsissymmetric.Thosegeneratorswhosebracketissymmetricarecalledfe rmionic;therestarebosonic.Wewritethe bracketasa gradedcommutator [A,B} =ABŠ ( Š )ABBA [AB).(3. 2.6) Thestructureconstantsofthesuper-Liealgebraobeysuper-Jacobiidentitiesthatfollow from: 0=1 2 ( Š )AC[[[A,B} ,C)} ( Š )AC[[A,B} ,C} +( Š )AB[[B,C} ,A} +( Š )BC[[C,A} ,B} .(3. 2.7) Again,wecande“neavectorspacewiththegeneratorsAactingasabasis;however, inthiscasethecoordinates Aassociatedwiththefermionicgeneratorsare antico mmuting numbersorGra ssmannparametersthatanticommutewitheachotherandwiththe fermionicgenerators.Gra ssmannparameterscommutewithordinarynumbersand bosonicgen erators;thesepropertiesensurethat K = AAisbosonic.Formally,we obtainsuper-LiegroupelementsbyexponentiationofthealgebraaswedoforLie groups. c.Super-Poincar ealgebra Fieldtheoriesinordinaryspacetimeareusuallysymmetricundertheactionofa spacetimesymmetrygroup:thePoincar egroupform assivetheoriesin”atspace,the conformalgroupformasslesstheories,andthedeSittergroupfortheoriesinspacesof constantcurvature.Forsupersymmetry,weconsiderextensionsofthesegroupsto supergroups.ThesewereinvestigatedbyHaag,/Lopusza nski,andSohnius,whoclassi“ed themostgeneralsymmetriespossible(actu ally,theycon sideredsymmetriesoftheSmatrixandgeneralizedtheColeman-Mandulatheoremonuni“edinternalandspacetime

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643.REPRESENTATIONSOFSUPERSYMMETRYsymmetriestoincludesuper-Liealgebras).Theyprovedthatthemostgeneral super-Poincar ealgebra contains,inadditionto { J, J€€, P€} (thegen eratorsofthe Poincar egro up), N fermionicsp inorialgenerators Qa (andtheirhermi tianconjugates Š Qa€),where a =1,. .., N isanisospinindex,andatmost1 2 N ( N Š 1)complexcentral char ges(calledcentralbecausetheycommutewithallgeneratorsinthetheory) Zab= Š Zba.Theal gebrais: { Qa Qb€} = a bP€,(3. 2.8a) { Qa Qb } = CZab,(3. 2.8b) [ Qa P€]=[ P€, P€]=[ J€€, Qc ]=0,(3 .2.8c) [ J, Qc ]=1 2 iC ( Qc ),(3. 2.8d) [ J, P€]=1 2 iC ( P )€,(3. 2.8e) [ J, J]= Š1 2 i ( ( J ) ),(3. 2.8f) [ J, J€€]=[ Zab, Zcd]=[ Zab, Zcd]=0.(3 .2.8g) TheessentialingredientsintheproofaretheColeman-Mandulatheorem(whichrestricts thebosonicpartsofthealgebra),andthesuper-Jacobiidentities.The N =1caseis calledsimplesupersymmetry,whereasthe N > 1caseisc alledextendeds upersymmetry. Centralchargescanariseonlyinthecaseofextended( N > 1)supersymmetry.The supersymmetrygenerators Q actassquarerootsofthemomentumgenerators P d.Positivityoftheenergy Adir ectconsequenceofthealgebraisthepositivityoftheenergyinsupersymmetrictheories.Thesimplestwaytounderstandthisresultistonotethatthetotal energycanbewri ttenas=1 2 ( P+Š PŠ)=1 2 €P€= Š1 2 €P€.(3. 2.9)

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3.2.Thesupersymmetrygroups65Since P€canbeobtainedfromtheanticommutatorofspinorcharges,wehave= Š1 2 N €{ Qa Qa€} =1 2 N { Qa ,( Qa )} (3.2.10) (weuse Qa€= Š ( Qa )).Therightha ndsideofeq.(3.2.10)ismanifestlynon-negative: Foranyop erator A andanystate | > < |{ A A}| > =n ( < | A | n >< n | A| > + < | A| n >< n | A | > ) =n (|< n | A| >|2+|< n | A | >|2).(3.2 .11) Hen ce,isalsononnegative.Further,ifsupersymmetryisunbroken, Q musta nnihilate thevacuum;inthiscase,(3.2.10)leadstotheconclusionthatthevacuumenergyvanishes.Althoughthisargumentisformal,itcanbemademoreprecise;indeed,itispossibletocharacterizesupersymmetrictheoriesbytheconditionthatthevacuumenergy vanish. e.Superconformalalgebra Fo rm asslesstheories,Haag,/Lopusza nski,andSohniusshowedwhatformextensionsoftheconformalgroupcantake:Th egen eratorsofthesuperconformalgroups consistofthegeneratorsoftheconformalgroup( P€, J, J€€, K€,) (thesearethe generatorsofthePoincar ealgebra,thesp ecialconformalboostgenerators,andthedilationgenerator),2 N spinorgenerators( Qa Sa )(andtheir hermitianconjugates Š Qa€, Š Sa€withatotalof8 N components),and N2furtherb osoniccharges( A Ta b) where Ta a=0.Theal gebrahasstructureconstantsde “nedbythefollowing(anti)commuta tors: { Qa Qb€} = a bP€, { Sa Sb€} = b aK€,( 3.2.12a) { Qa Sb } = Š i a b( J +1 2 ) Š1 2 a b(1 Š4 N ) A +2 Ta b(3.2.12b) [ Ta b, Sc ]=1 2 ( a cSb Š1 N a bSc ),(3.2 .12c)

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663.REPRESENTATIONSOFSUPERSYMMETRY[ A Sc ]=1 2 Sc ,[, Sc ]= Š i1 2 Sc ,(3. 2.12d) [ J Sc ]= Š1 2 i ( | |Sc ),[ P€, Sc ]= Š Qc€,(3. 2.12e) [ Ta b, Qc ]= Š1 2 ( c bQa Š1 N a bQc ),(3.2 .12f) [ A Qc ]= Š1 2 Qc ,[, Qc ]= i1 2 Qc ,( 3.2.12g) [ J Qc ]=1 2 i ( Qc ),[ K€, Qc ]= Sc€,(3. 2.12h) [ Ta b, Tc d]=1 2 ( a dTc bŠ c bTa d),(3.2 .12i) [, K€]= Š iK€,[, P€]= iP€,(3. 2.12j) [ J K€]= Š1 2 i ( | |K )€,[ J P€]=1 2 i ( P )€,(3. 2.12k) [ J, J]= Š1 2 i ( ( J ) ),(3. 2.12l) [ P€, K€]= i ( €€J + J€€+ €€)= i ( J a b+ a b).(3.2.12m) Allother(anti)commutatorsvanisho rarefo undbyhermitianconjugation. Thesuperconformalalgebracontainsthesuper-Poincar ealgebraasas ubalgebra; however,inthesuperconformalcase,thereare no centralcharges(thisisadirectconsequenceoftheJacobiidentities).Inthesamewaythatthesupersymmetrygenerators Q actassquarerootsofthetranslationgenerators P ,the S -supersymmetrygenerators S actassquarerootsofthespecialconformalgenerators K .The newbos oniccharges A and Ta bgeneratephaserotationsofthespinors(axialor 5rotations)and SU ( N )transformationsrespectively(allbutthe SO ( N )s ubgroupofthe SU ( N )isaxial).For N =4, theaxialcharge A dropsoutofthe { Q S } anticommutatorwhereasthe[ Q A ]and [ S A ]commuta torsare N i ndependent.Thenormalizationof A ischosen suchthat Ta b+1 N a bA generates U ( N )(e.g.,[ Ta b+1 N a bA Qc ]= Š1 2 c bQa ).

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3.2.Thesupersymmetrygroups67f.Super-deSitteralgebra Finally,weturntothesupersymmetricextensionofthedeSitteralgebra.The generatorsofthisalgebraarethegeneratorsofthedeSitteralgebra( P, J, J ),spinorial generators( Q, Q ),and1 2 N ( N Š 1)bosonic SO ( N )charges Tab= Š Tba.Theycanbe constructedoutofthesuperconformalalgebra(justasthesuper-Poincar ealgeb raisa subalgebraofthesuperconformalalgebra,soisthesuper-deSitteralgebra).Wecan de“nethegeneratorsofthesuper-deSitteralgebraasthefollowinglinearcombinations ofthesuperconformalgenerators: P€= P€+ | |2K€, Qa = Qa + abSb J= J, Tab= c [ bTa ] c,(3. 2.13) where,sincewebreak SU ( N )to SO ( N ),wehavelo weredtheis ospini ndicesofthe superconformalgeneratorswithakronecke rdelta.(We couldalsofo rmallymaintain SU ( N )invariancebyu singinstead absatisfying ab= baand ac bc a b,with ab= abinanappropriate SU ( N )fra me.)Thuswe“ndthefollowingalgebra: { Qa Qb } =2 ( Š i abJ+ CTab), (3.2.14a) { Qa Qb€} = a bP€,(3. 2.14b) [ Qa P€]= Š Cab Qb€,(3. 2.14c) [ J, Qc ]=1 2 iC ( Qc ),(3. 2.14d) [ J, P€]=1 2 iC ( P )€,(3. 2.14e) [ P€, P€]= Š i 2 | |2( C€€J+ C J€€),(3.2 .14f) [ J, J]= Š1 2 i ( ( J ) ).( 3.2.14g) [ Tab, Qc ]=1 2 c [ aQb ] ,(3. 2.14h)

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683.REPRESENTATIONSOFSUPERSYMMETRY[ Tab, Tcd]=1 2 ( b [ cTd ] aŠ a [ cTd ] b)(3. 2.14i) Thisalgebra,incontrasttothesuperconformalandsuper-Poincar ecases ,dep endsona dimensionalconstant .Physi ca lly, | |2isthecurvatureofthedeSitterspace.(Actually,thesignissuchthattherelevantspaceisthespaceofconstant negative curvature, oranti-deSitterspace.Thisisaconsequen ceofsupersymmetry:Thealgebradeterminestherelativesigninthecombination P + | |2K above.)

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3.3.Representationsofsupersymmetry693.3.Representationsofsupersymmetry a.Particlerepresentations Beforedisc ussing“eldrepresentationsofsupersymmetry,westudytheparticle contentofPoincar esupersy mmetrictheories.WeanalyzerepresentationsofthesupersymmetrygroupintermsofrepresentationsofitsPoincar es ubgroup.Because P2isa Casimiroperatorofsupersymmetry(itcommut eswithallthegenera tors),allelements ofagivenirreduciblerepresentationwillhavethesamemass. a.1.Masslessrepresentations We“rstco nsidermasslessrepresentations.WethencanchooseaLorentzframe wheretheonlynonvanishingcomponentofthemomentum p ais p+.Inthisf ramethe anticommutationrelationsofthesupersymmetrygeneratorsare { Qa+, Qb+} =0, { Qa+, Qb€+} = p+a b, { QaŠ, QbŠ} =0, { QaŠ, Qb€Š} =0, { Qa+, QbŠ} =0, { Qa+, Qb€Š} =0.(3. 3.1) Sincetheanticommutatorof Qa Šwithitshermitianconjugatevanishes, Qa Šmustvanishidenticallyonallphysicalstates:From(3.2.11)wehavetheresultthat 0= < |{ A A}| > =n (|< n | A| >|2+|< n | A | >|2) < n | A | > = < n | A| > =0.(3. 3.2) Ontheotherhand, Qa+anditshermitianconjugatesatisfythestandardanticommutationrelationsforannihilationandcreationo perato rs,uptonormalizationfactors(with theexceptionofthecase p+=0,whichinth isframemeans p a=0andd escribesthe physicalvacuum).Wecanthusconsiderastate,the Cliordvacuum | C > ,whichis annihilatedbyalltheannihilationoperators Qa+(orconstructsuchastatefromagiven statebyoperatingonitwithasucientnumberofannihilationoperators)andgenerate allotherstatesbyactionofthecreationoperators Qa€+.Sin ce,asusual,anannihilation operatoractingonanystateproducesanotherwithonelesscreationoperatoractingon

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703.REPRESENTATIONSOFSUPERSYMMETRYth eC li or dv ac uum,thissetofstatesisclosedundertheactionofthesupersymmetry generators,andthusformsarepresentatio nofthesupers ymmetryalgebra.Furthermore,iftheCliordvacuumisanirreduc iblerepresentationofthePoincar egro up,this setofsta tesisanirreduciblerepresentation ofthesupersymmetr ygro up,sinceany attempttoreducetherepresentationbyimposingaconstraintonastate(oralinear combinat ionofstates)wouldalsoconstrain theCliordvacuum(a fterapplyingan appropriatenumberofannihilationoperators;seealsosec.3.8.a).TheCliordvacuum mayalsocarryrepresentationsofisosp inandotherinternalsymmetrygroups. TheCliordvacuum,beinganirreduciblerepresentationofthePoincar egro up,is alsoaneigenstateofhelicity.Inthisframe, Qa€+hashe licity Š1 2 ,thusdet erminingthe he licitiesoftheotherstatesintermsofthat oftheCliordvacuum.(Ingeneralframes, thehelicity Š1 2 componentof Q€ isthecreationoperator,andthehelicity+1 2 component,whichisthelinearlyindependentLorentzcomponentof P€ Qa€,vanishes: { P€ Qa€, P€Qb } = b aP2P€=0,since p2=0inthema sslesscase.)Therepresentati on so ft hestatesunderisospinarealsodeterminedfromthetransformationproperties oftheCliordvacuumandthe Q s:Wetakethete nsorproductoftheCliordvacuumsrepresentationwi ththatofthecreationoperators(namely,thatformedbymultiplyingtherepresentationsoftheindividualoperatorsandantisymmetrizing). Asexampl es,weconsiderthecasesofthemasslessscalarmultiplet( N =1,2), su pe r-Yang-Mills( N =1,. ..,4),andsupe rgravity( N =1,. ..,8),de“ne dbyC liord vacuawhicharei soscalarsandhavehelicity+1 2 ,+1, and+2,respectively.(Inthe sc alarandYang-Millscases,thestatesmaycarryarepresentationofaseparateinternal symmetrygroup.)ThestatesarelistedinTab le3.3.1.Eachstateistotallyantisymmetricintheisospinindices,andthus,foragiven N ,sta teswithmorethan N isospin indicesvanish.Thescalarmultipletcontainshelicities(1 2 ,...,1 2 ŠN 2 ),superYang-Mills containshe licities(1,...,1 ŠN 2 ),andsupergravitycontainshelicities(2,...,2 ŠN 2 ).In addition,anyrepresentationofaninternalsymmetrygroupthatcommuteswithsupersy mmetry(suchasthegaugegroupofsuperYang-Mills)carriedbytheCliordvacuum iscarriedbyallstates(soinsuperYang-Millsallstatesareintheadjointrepresentation ofthegaugegroup).Thusthetotalnumberofstatesinamasslessrepresentationis 2Nk ,where k isthenumberofstatesi ntheC liordvacuum.

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3.3.Representationsofsupersymmetry71 he licityscalarmultipletsuper-Yang-Millssupergravity +2 = | C > + 3/2 a+1 = | C >ab+ 1/2 = | C >aabc0 aaba bcd-1/2 ababca bcde-1 a bcda bcdef-3/2 a bcdefg-2 a bcdefgh Table3.3 .1.Statesintheoriesofphysicalinterest TheCPTconjugateofastatetransformsasthecomplexconjugaterepresentation. JustasforrepresentationsofthePoincar egro up,onemayidentifyasupersymmetry representationwithitsconjugateifithasthesamequantumnumbers:i.e.,ifitisareal representation.(Intermsofclassical“elds,or“eldsinafunctionalintegral,thisselfconjugacyconditionrelates“eldstotheircom plexconjugates:see(3 .12.4c)or(3.12.11). Thus,inafunctionalintegralformalism,se lf-conjugacyiswithrespecttoatypeof char geconjugation:Achargeconjugationiscomplexconjugationtimesamatrix(see sec.3.3.b.5).)Fortheaboveexamples,thisself-conjugacyoccursfor N =4supe rYan gM illsand N =8superg ravity.(Thisisnottrueforthe N =2scal armultiplet,sincean SU (2)isospinorcannotbeidenti“edwithitsc omplexconj ugate,unlessanextraisospin i ndexoftheinternal SU (2)symmetry,i ndependentofthesupersymmetry SU (2),is a dded.Theself-conjugacythensimplycancelsthedoublingintroducedbytheextra i ndex.) a.2.Massiverepresentationsandcentralcharges Themassivecaseistreatedsimilarly,exceptthatwecannolongerchoosethe Lorentzframeabove;instead,wechoosetherestframe, p€= Š m €: { Q Q } =0, { Q Q€ } = Š m € .(3. 3.3) Nowwehavetwiceasmanycreationandannihilationoperators,the QŠsaswellasthe

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723.REPRESENTATIONSOFSUPERSYMMETRYQ+s.Thereforethenumberofstatesinamassiverepresentationis22 Nk .(Forexample, an N =1ma ssivevectormult iplethashelicitycontent(1,1 2 ,1 2 ,0).) Thecasewithcentralchargescanbeanalyzedbysimilarmethods,butitissimpler to understandifwerealizethatsupersymmetryalgebraswithcentralchargescanbe obtainedfromsupersymmetryalgebraswitho utcentralchargesinhigher-dimensional spacetimesbyinterpretingsomeoftheextracomponentsofthemomentumasthecentralchargegenerators(theywillcommutewithallthefour-dimensionalgenerators). Theanalysisofthestatecontentisthenthesameasforthecaseswithoutcentral char ges,sincebothcasesareobtainedfromthesamehigher-dimensionalsetofstates (ex ceptthatwedonotkeepthefullhigher-dimensionalLorentzgroup).However,the twodisting uishingcas esarenow,intermsof P2 high er Š dimens ional= P2+ Z2= 1 2 ( P aP a+ ZabZab):(1) P2+ Z2=0,which hasthesamesetofstatesasthemassless Z =0case (thoughthestatesarenowmassive,haveasmallerinternalsymmetrygroup, an dt ransformsomewhatdierentlyundersupersymmetry),and(2) P2+ Z2< 0,which hasthesamesetofstatesasthemassive Z =0case.Bythissamea nalysis,weseethat P2+ Z2> 0isnot allowed(ju stasfor Z =0weneverhave P2> 0). a.3.Casimiroperators Wecanconst ructotherCasimiroperatorsthan P2.We“rstd e“nethesupersymmetricgeneralizationofthePauli-Lubanskivector W€= i ( P€JŠ P€ J€€) Š1 2 [ Qa Qa€],(3.3 .4) wherethelasttermisabsentinthenonsupersymmetriccase.Thisvectorisnotinvarian t undersupersymmetrytransformations,butsatis“es [ W a, Q ]= Š1 2 P aQ ,[ W a, Q€ ]=1 2 P a Q€ .(3. 3.5) Asaresult, P[ aW b ]commuteswith Q ,andth usitssquare P2W2Š1 4 ( P W )2commuteswitha llthegen eratorsofthesuper-Poincar ealgebraandisaCa simiroperator. InthemassivecasethisCasimiroperatorde“nesaquantumnumber s ,the superspin. Thegeneralizationofthenonsupersymmetricrelation W2= m2s ( s +1)is

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3.3.Representationsofsupersymmetry73P2W2Š1 4 ( P W )2= Š m4s ( s +1).(3 .3.6) Inthemasslesscase,notonly P2=0, butalso P€Qa = P€ Qa€=0,and hence P W = P[ aW b ]=0.Howev er,usingthegenerator A ofthesuperco nformalgroup (3.2.12),wecanconstructa nobj ectthatcommuteswith Q and Q : W aŠ AP a.Thus wecand e“neaquantumnumber ,the superhelicity, thatgeneralizeshelicity 0(de “nedby W a= 0P a): W aŠ AP a= P a.(3. 3.7) Wealsocanc onstructsupersymmetryinvariantge nera lizationsoftheaxialgenerator A andofthe SU ( N )gen erators: W5 P2A +1 4 P€[ Qa Qa€], Wa b P2Ta b+1 4 P€([ Qa Qb€] Š1 N a b[ Qc Qc€]).(3.3.8) Inthemassivecase,the superchiralcharge andthe superisospin quantumnumberscan thenbede “nedastheusualCasimiroperatorsofthemodi“edgroupgenerators Š mŠ 2W5, Š mŠ 2Wa b.Inthema sslesscase,wede“netheoperators W5 € P€A +1 4 [ Qa Qa€], Wa b € P€Ta b+1 4 ([ Qa Qb€] Š1 N a b[ Qc Qc€]).(3.3.9) Thesecommutewith Q and Q whenthecondition P€Qa =0holds, whichisprecisely thema sslesscase.Since P€W5 €= P€Wa b €=0,wecan “ndmatrixrepr esentations g5, ga bsuchthat W5 c= g5P c, Wa b c= ga bP c.(3. 3.10) Thesuperchiralchargeis g5,andsuperi sospinquantumnumberscanbede“nedfromthe tra celessmatrices ga b.Allsupersy mmetricallyinvariantoperatorsthatwehaveconstructedcanbereexpressedintermsofcovariantderivativesde“nedinsec.3.4.a;seesec 3.4.d.

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743.REPRESENTATIONSOFSUPERSYMMETRYb.Representationsonsuper“elds Weturnnowto“eld(o -shell)representationsoft hesupersymmetryalgebras. Thesecanbedescribedinsuperspace,whichisanextensionofspacetimetoinclude extraanticommutingcoordinates.Todiscovertheactionofsupersymmetrytransformationsonsupersp ace,weuset hemethodofinducedrepresentations.Wediscussonly simple N =1supersy mmetryforthemoment. b.1.Superspace Ordinaryspacetimecanbede“nedasthecosetspace(Poincar egro up)/(Lorentz group).Similarly, global”atsuperspace canbede“nedasthecosetspace (super-P oincar egro up)/(Lorentzgroup):ItspointsaretheorbitswhichtheLorentz groupsweepsoutinthesuper-Poincar egro up.Relative tosomeorigin,thiscosetspace canbeparametrizedas: h ( x , )= ei ( x€ P€+ Q+ € Q€)(3.3.11) where x , arethecoor dinatesofsuperspace: x isthecoordinateofspacetime,and arenewfermionicspinorcoordinates.Thehaton P and Q i ndicatesthatthey areabstractgroupgenerators, not tobeconfusedwiththed ierentialoperators P and Q usedtorepresentthembelow.Thestatisticsof aredetermine dbythoseof Q Q : { } = { } = { Q } =[ x ]=[ P ]=0,(3 .3.12) etc.,thatis, areGrassmannparameters. b.2.Acti onofgeneratorsonsuperspace Wede “netheactionofthesuper-Poincar egrouponsupe rspacebyleftmultiplication: h ( x, )= gŠ 1 h ( x , ) modSO (3,1)(3.3.13) where g isagroup element,and modSO (3,1)meansthatanytermsinvolvingLorentz generatorsaretobe pushedthroughtotherightandthendropped.To“ndtheaction ofthegenerators( J P Q )onsuperspa ce,weconsider

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3.3.Representationsofsupersymmetry75 g = eŠ i ( J + €€ J€€), eŠ i ( € P€), eŠ i ( Q+ € Q€) ,(3. 3.14) respectively.UsingtheBaker-Hausdortheorem( eAeB= eA + B + 1 2 [ A B ]if [ A ,[ A B ]]=[ B ,[ A B ]] =0 )t or earrangetheexponents,we“nd: J & J : x €=[ e] [ e ]€€x€, =[ e] €=[ e ]€€ €, P : x a= x a+ a, = €= €, Q & Q : x a= x aŠ i1 2 ( €+ €), = + €= €+ €.(3. 3.15) Thusthegeneratorsarerealizedascoordinatetransformationsinsuperspace.The Lorentzgroupacts reduci bly: Underitsactionthe x sand sdo not transformintoeach other. b.3.Acti onofgeneratorsonsuper“elds Togetrepresent ationsofsupersymmetryonphysical“elds,weconsider super“elds ...( x , ):(generalized)multispinorfunction soversupersp ace.Unde rthesupersymmetryalgebratheyarede“nedtotransformascoordinatescalarsandLorentzmultispinors.Theymayalsobeina matrixrepresentationofaninternalsymmetrygroup. Thesimplestcaseisascalarsuper“eld,whichtransformsas:( x, )=( x , )or, i n“nitesimally, ( z ) Š ( z )= Š zMM( z ).Using(3.3.13),wewritethetransformati onas = Š i [( Q+ € Q€),]= i [( Q+ € Q€),],etc.Hen ce,justasin theordinaryPoincar ecase,theg enerators Q ,etc.,arerepre sented bydierentialoperators Q ,etc.: J= Š i1 2 ( x( € )€+ ( )) Š iM, P€= i €, Q= i ( Š1 2 €i €), Q€= i ( €Š1 2 i €);(3.3 .16) where Mgenera testhe matrix Lorentztransformationsofthesuper“eld:

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763.REPRESENTATIONSOFSUPERSYMMETRY[ M, ...]=1 2 C ( ) ...+ ... Forfut ureuse,w ewrite Q and Q as Q= e1 2 Ui eŠ1 2 U, Q€= eŠ1 2 Ui €e1 2 U,( 3.3.17a) where U = €i €.(3. 3.17b) Finally,fromtherelation { Q Q } = P ,weconcl udethat thedimensionof and is ( m )Š1 2 b.4.Extendedsupersymmetry Wenowgen eralizetoextendedPoincar esupersy mmetry.Inprinciple,theresults wepres entcouldbederivedbymethodssimilartotheabove,orbyusingasystematic dierentialgeometryprocedure.Inpracticethesimplestprocedureistostartwiththe N =1Poincar eresultsan dgen eralizethembydimensionalanalysisand U ( N )sy mmetry. Forgen eral N ,superspac ehasco ordi nates zA=( x€, a a€) ( x a, € ). Super“elds ... ab ...( x , )transfo rmasmultispinorsandisospinors,andascoordinate scalars.Includingcentralcharges,thesuper-Poincar egen eratorsactonsuper“eldsas thefollowingdierentialoperators: Qa = i ( a Š1 2 a€i €Š1 2 b Zba), (3.3.18a) Qa€= i ( a€Š1 2 a i €Š1 2 b€ Zba),(3.3 .18b) J= Š i1 2 ( x( € )€+ a ( a )) Š iM,(3. 3.18c) J€€= Š i1 2 ( x (€€ )+ a (€ a€ )) Š i M€€,(3. 3.18d) P€= i €.(3. 3.18e) Centralchargesarediscussedinsection4.6.

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3.3.Representationsofsupersymmetry77b.5.CPTinsuperspace Poincar esupersy mmetryiscompatiblewiththediscreteinvariancesCP(charge conjugation parity)andT(timereversal).WebeginbyreviewingC,P,andTin ordinaryspacetime.We describethetransformationsasactingon c -number“el ds, i.e., weuset hefunctionalinte gralformalism,ratherthanactingon q -number“eldsor Hilbertspa cestates. U ndera re”ection withrespecttoanarbitrary(butnotlightlike)axis u a,( u = u u2= 1)thecoordinatestransformas x a= R ( u ) x a= Š uŠ 2u€u€x€= x aŠ uŠ 2u au x R2= I (3.3.19) ( u x chan gessign,whilethecomponentsof x orthogonalto u are unchanged.)Tthen actsonthecoordinatesas R ( a 0)wh ileaspacere” ectioncanberepresentedby R ( a 1) R ( a 2) R ( a 3)(inte rmsofatimelikevector a 0,a nd threeorthogonalspacelike v ectors a i, i =1,2,3). Wede “netheactionofthediscretesymmetriesonarealscalar“eldby ( x)= ( x ).TheactiononaWeylspinoris ( x)= iu€ €( x ), €( x)= iu€( x ); ( x )= u2( x ).(3.3 .20) Sincethistransformationinvolves complexconjugatio n,weinterpret R asgivingCPand T.Indeed,sinceundercomplexconjugation eŠ ipx e+ ipx,wehave p a= Š ( p aŠ uŠ 2u au p ).Ther efore p0chan gessignforspacelike u ,andth isisconsistentwithourinterpretation.Thecom binedtransformationCPTissimply x Š x and the“eldstransformwithoutanyfactors(exceptforirrelevantphases).ThetransformationofanarbitraryLorentzrepresentationisobtainedbytreatingeachspinorindexas in(3.3.20). Thede“nitionofC,andthusPandCT,requirestheexistenceofanadditional, internal,discretesymmetry,e.g.,asymmetr yinvolving onlysignchanges:Forthephoton“eldC A a= Š A a;forap airofrealscalars,C 1=+ 1,C 2= Š 2gives

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783.REPRESENTATIONSOFSUPERSYMMETRYC( 1+ i 2)=( 1+ i 2).Forap airofspi nors,C 1 = 2 ,C 2 = 1 gives,forthe Diracspinor( 1 2€),thetransformationC( 1 2€)=( 2€, 1 ), i.e.,complexconjugationtimesamatrix.Therefore,Cgenera llyinvolvescomplexconjugationofa“eld, asdoCPandT,whereasPand CTdonot.(However,notethatthede“nitionofcomplexconjugationdependsonthede“nitionofthe“elds,e.g.,combining 1and 2as 1+ i 2.) Thegeneralizationtosuperspaceisstraightforward:Inadditiontothetransformation R ( u ) x givenabove,wehave(asforanyspinor) a = iu€ a€, a€= iu€a .(3. 3.21) Areals calarsuper“eldandaWeylspinorsuper“eldthustransformasthecorresponding component“elds,butnowwithallsuperspacecoordinatestransformingunder R ( u ).To preservethechiralityofasuperspaceorsuper“eld(seebelow),wede“ne R ( u )to always complexconjugatethesuper“elds.Wethushave,e.g., ( z)= ( z ), ( z)= iu€ ( z )= iu€ €( z ).(3.3 .22) Asforcomponents,Ccanbe de“nedasanadditional(internal)discretesymmetry whichcanbeexpressedasamatrix timeshermitianconjugation. Were markthat R ( u )transfo rmsthesupersymmetrygeneratorscovariantlyonly for u2=+1.For u2= Š 1thereisarel ativesignchangebetween and € i € .This isb ecauseCPchangesthesignof p0,whichis neededtomain tainthepositivityofthe energy(see (3.2.10)). b.6.Chiralrepresentationsofsupersymmetry Asinthe N =1case(s ee(3.3 .17)), Q Q canbewrittencompactlyforhigher N eveninthepresenceofcentralcharges: Q = e1 2 Ui ( Š1 2 b Zba) eŠ1 2 U, Q€ = eŠ1 2 Ui ( € Š1 2 b€ Zba) e1 2 U,( 3.3.23a) U €i €, €= c d€.(3. 3.23b)

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3.3.Representationsofsupersymmetry79Thisallowsusto“ndotherrepresentationsofthesuper-Poincar ealgebra inwhich Q (or Q )takeaverysi mpleform.Weperformnonunitarysi milaritytransformationson all generatorsA: A ( )= eŠ+1 2 UAe1 2 U,(3. 3.24) whichleadsto: Q (+)= i ( Š1 2 b Zba), Q€ (+)= eŠ Ui ( € Š1 2 b€ Zba) eU,(3. 3.25) or Q ( Š )= eUi ( Š1 2 b Zba) eŠ U, Q€ ( Š )= i ( € Š1 2 b€ Zba). (3.3.26a) Thegeneratorsactontransformedsuper“elds ( )( z )= eŠ+1 2 U( z ) e1 2 U(3.3.26b) Theserepresentationsarecalled chiral or antichiral representations,whereastheoriginal oneiscalledthe vector representation.Theycanalsobefounddirectlybythemethod ofinducedrepresentationsbyusingaslightly dierentparametrizationofthecosetspace manifold(superspace)(cf.(3.3.11)): h(+)= ei Qeix(+) Pei Q, h( Š )= ei Qeix( Š ) Pei Q,( 3.3.27a) where x( )= x i1 2 = e1 2 UxeŠ+1 2 U(3.3.27b) arecomplex(nonhermitian)coordinates.The correspondingsuperspacesarecalledchiralandantichiral,respectively.Thesimilaritytransformations(3.3.26b)canberegarded ascomplexcoordinatetransformations: ( z )= e1 2 U( )( z ) eŠ+1 2 U=( )( z( )),

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803.REPRESENTATIONSOFSUPERSYMMETRYz( )= e1 2 UzeŠ+1 2 U=( x( ), ).(3.3 .28) Hermitianconjugationtakesusfromachiralrepresentationtoanantichiralone: ( V(+))= V( Š ).Consequently,ah ermitianquantity V = V inthev ectorrepresentation satis“es V = eŠ U VeU(3.3.29) inthechiralrepresentation. b.7.Superconformalrepresentations Themethodofinducedrepresentationscanbeusedto“ndrepresentationsforthe superconformalgroup.However,weuseadierentprocedure.Therepresentationsof Q P ,and J areasinthesuper-Poincar ecase.Ther epresentationsoftheremaining generatorsarefoundasfollows:Inordinaryspacetime,theconformalboostgenerators K canbeconstructedby“rstperforming aninversion,thenatranslation( P transformation),and“nallyperforminganotherinversion;asimilarsequenceofoperationscanbe usedinsuperspacetoconstruct K from P and S from Q Wede “netheinversion operationasthefollowingmapbetweenchiralandantichiralsuperspace: x ( ) €=( x(Š+))Š 2x(Š+) €= ( x( ))Š 2x( ) €, a = i ( x( Š ))Š 2x( Š ) € a€= Š i ( x(+))Š 2x(+) €a a€= i ( x(+))Š 2x(+) €a = Š i ( x( Š ))Š 2x( Š ) € a€;(3. 3.30) wehave z= z .Thee ssentialpropertyofthismappingisthatitscalesasupersymetricallyinvariantextensionof thelineelement.Wewrite ds2=1 2 s€s€,where s€= dx€+i 2 ( a d a€+ a€d a ),(3.3 .31) isasupersymmetricallyinvariant1-form(invariancefollowsatoncefrom(3.3.15)). Underinversions(3.3.30),we“nd s €= Š ( x(+))Š 2( x( Š ))Š 2x(+) €x( Š ) €s€,

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3.3.Representationsofsupersymmetry81ds 2=( x(+))2( x( Š ))2ds2.(3. 3.32) Super“eldstransformas II ...€ ...( z )=( x(+))Š 2 d+( x( Š ))Š 2 dŠf€... f€ ...€ ...( z), (3.3.33a) f€ i ( x(+))Š 1x(+) €, f€ i ( x( Š ))Š 1x( Š ) €.(3. 3.33b) Here d d++ dŠisthecanonicaldimension(Weylweight)of,and dŠŠ d+isproportionalt othechiral U (1)weight w .Notethat chiral super“elds(“eldsdependingonly onand x(+)and ,not ;sees ec.3.5)with dŠ=0 an do nlyundottedindicesremainchiralafteraninversion. Wecancal culate S asdescribedabove:Weusetheinversionoperator II and compute Sa = II Q€ II and Sa€= IIQ II .Usingthes uperconformalcommutatoralgebra wethen compute K A T ,and. We “nd A =1 2 ( Š € € ) Š Y ,( 3.3.34a) Ta b=1 2 ( b a Š a€ b€Š1 N a b( Š € € ))+ ta b,(3. 3.34b) Sa = i ( x€Š i1 2 b b€) Qa€+ a b i ( b + i1 2 b€€) Š 2 i b [ ( tb a+1 4 b a(1 Š4 N ) Y ) Š1 2 b a( M +1 2 d d d d )],(3.3.34c) Sa€= i ( x€+ i1 2 b b€) Qa + a€ b€i ( b€+ i1 2 b €) Š 2 i b€[ Š €€( ta b+1 4 a b(1 Š4 N ) Y ) Š1 2 a b( M€€+1 2 €€d d d d )],(3.3.34d) = Š i1 2 ( { x€, €} +1 2 ([ ]+[ € € ])) Š id d d d ,(3. 3.34e) K€= Š i ( x€x€€+ x€ a€ a€+ x€a a Š1 4 a a€b b€€) +1 2 ( a a€b b Š a a€ b€ b€)

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823.REPRESENTATIONSOFSUPERSYMMETRYŠ i ( x€+ i1 2 a a€) M Š i ( x€Š i1 2 a a€) M€€Š x€id d d d Š 2 a b€( ta b+1 4 a b(1 Š4 N ) Y ).(3.3 .34f) Here d d d d isthematrixpieceofthegenerator;it seigenva lueistheca nonicaldimension d .Sim ilarly, Y ta barethematrixpiecesoftheaxialgenerator A andthe SU ( N )gen erators Ta b;theeige nval ueof Y is1 2 w .Thete rmsin S S prop ortionalto Y and ta bdo notfollowfromtheinversion(3.3.33),butaredeterminedbythecommutationrelations and(3.3.34a,b). b.8.Super-deSitte rrepre sentations Toconstr uctthegeneratorsofthesuper-deSi tteralgebra,weusetheexpressions fortheconformalgeneratorsandtakethelin earcombinationsprescribedin(3.2.13). Tosu mmarize,forgeneral N ,ineac hofthecaseswehavecon sideredthegeneratorsactasdierentialoperators.Inadditionthesuper“eldsmaycarryanontrivial matrixrepresentationofallthegeneratorsexceptfor P and Q inthePoincar eand deSittercases,and P Q K ,and S inthesuper conformalcase.Theymayalsocarrya representationofsomearbitraryinternalsymmetrygroup.

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3.4.Covariantderivatives833.4.Covariantderivatives Inordinary”atspacetime,theusualcoordinatederivative aistranslation invariant:thetranslationgenerator P a,whichis representedby €,commuteswith itself.Insupersymmetrictheories ,thesupert ranslationgenerator Qhasanontrivial an ti commutator,andhenceisnotinvariantundersupertranslations;asimplecomputationrevealsthatthefermion iccoordi natederivatives €arenotinvarianteither. Thereis,however,asimplewaytoconstructderivativesthatareinvariantundersupersymmetrytransformationsgeneratedby Q, Q€(a ndarecovariantunderLorentz,chiral, andisospinrotationsgeneratedby J, J€€, A ,and Ta b). a.Construction Intheprecedingsectionweusedthemethodofinducedrepresentationsto“nd theactionofthesuper-Poincar egen eratorsinsuperspace.Thesamemethodcanbe usedto“ndcovariantderivatives.Wede“netheoperators Dand D€bytheequation ( e D + D)( ei ( x P + Q + Q )) ( ei ( x P + Q + Q ))( ei ( Q + Q )).(3.4 .1) Theanticommutatorof Q with D canbeexamine dasfo llows: ( eŠ i ( Q + Q ))( e D + D)( ei ( Q + Q ))( ei ( x P + Q + Q )) =( eŠ i ( Q + Q )) ( ei ( Q + Q ))( ei ( x P + Q + Q ))( ei ( Q + Q )) =( ei ( x P + Q + Q ))( ei ( Q + Q )) =( e D + D)( ei ( x P + Q + Q )).(3.4 .2) Thusthe D s ar ei nv ar ia nt undersupertranslations(andalsounderordinarytranslations): { Q D } = { Q D } =[ P D ]=0.(3 .4.3) WecanusetheBak er-Hausdortheorem,(3.4.1),and(3.3.11,13)tocomputethe exp licitformsofthe D sfromthe Q s.W e “nd D= Š iQ+ €P€, D€= Š i Q€+ P€.(3. 4.4)

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843.REPRESENTATIONSOFSUPERSYMMETRYFor N =1,whenactingo nsuper “elds,theyhavetheform D= +1 2 €i €, D€= €+1 2 i €,(3. 4.5) andarecovariantgeneralizationsoftheordinaryspinorderivative €.For general N ,with centralcharges,thecovariantderivativeshavetheform: D = Da = +1 2 € i € +1 2 b Zba, D€ = Da€= € +1 2 i € +1 2 b€ Zba.(3. 4.6) Theycanberewrittenusing eUas: D = eŠ1 2 U( +1 2 b Zba) e1 2 U, D€ = e1 2 U( € +1 2 b€ Zba) eŠ1 2 U.(3. 4.7) Consequently,justasthegenerators Q simp lifyinthechiral(antichiral)representation, thecovariantderivativeshavethesimplebutasymmetricform: D (+)= eŠ U( +1 2 b Zab) eU, D€ (+)= € +1 2 b€ Zab, D ( Š )= +1 2 b Zab, D€ ( Š )= eU( € +1 2 b€ Zab) eŠ U.(3. 4.8) Inanyrepresentation,theyhavethefollowing(anti)commutationrelations: { D D } = CZab, { D D€ } = i € .(3. 4.9) ItisalsopossibletoderivedeSittercovariantderivativesbythesemethods.However,thereisaneasier,moreuseful,andmorephysicalwaytoderivethemwithinthe frameworkofsupergravity,sincedeSitterspaceissimplyacurvedspacewithconstant curvature.Thiswillbedescribedinsec.5.7. b.Algebraicrelations Thecovariantderivativessatisfyanumberofusefulalgebraicrelations.For N =1,theonlypo ssiblepowerof D is D2=1 2 DD.(B ecauseofantic ommutativity higherpowersvanish:( D )3=0.)Fromthea nticommuta tionrelationswealsohave

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3.4.Covariantderivatives85[ D, D2]= i € D€, D2 D2D2= D2, DD= D2, D22= Š 1.(3.4 .10) For N > 1wehavesim ilarrelations;forvanishingcentralcharges: Dn 1... n D 1... D n, Dn n +1... 2 N1 n C 2 N... 1Dn 1... n, Dn 1... n=1 (2 N Š n )! C 2 N... 1Dn n +1... 2 N, D2 N Š n 1... nDn 1... n= [ 1 1... n] nD2 N, ( Dn 1... n)= Dn €n... €1, ( D2 N Š n 1... n)=( Š 1)n D2 N Š n €n... €1, D2 N2 N=( Š 1)N, D2 ND2 N D2 N= N D2 N.(3. 4.11) Itisoftennecessarytoreducetheproductof D sor D swithrespectto SU ( N ),as wellaswith respectto SL (2, C ).Foreach,thereduction isdonebysymmetrizingand antisymmetrizingtheindices.Speci“cally,w e “ndtheirreduciblerepresentationsasfollows:Apr o duct D D .... D istotallyantisymmetricinitscombinedindicessincethe D santicommute;however,antisymmetryin impliesoppositesymmetriesbetween a b and ,(onep airsymmetric,theotherantisymmetric),andhenceaYoung tableauforthe SU ( N )i ndicesispairedwiththesameYoungtableau re”ected aboutthe diagonal forthe SL (2, C )i ndices.Thelatterisactuallyan SU (2)tableausinceifwe haveonly D s th enonlyundottedindicesappear,andhasatmosttworows.(Actually, for SU (2)acolumnof2isequivalen ttoacolumnof0 ,and hencethe SL (2 C )tab leau canbereducedtoasinglerow.)Therefore,theonly SU ( N )table auxtha tappearhave twocolum nsorless.The SL (2, C )representatio ncanbere addirectlyfromthe SU ( N ) tableau(ifwe k eepcolumnsofheight N ):Theg eneral SU ( N )table auconsistsofa“rst

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863.REPRESENTATIONSOFSUPERSYMMETRYcolumnof height p andasecondofheight q ,where p + q isthenumberof D s;thecorresponding SL (2 C )representationisa( p Š q )i ndextotallysymmetricundottedspinor. Thereforethisrepresentationof SL (2, C ) SU ( N )hasdi mensionality ( p Š q +1) p Š q +1 p +1 N p N +1 q .(3. 4.12) c.Geometryof”atsuperspace Thecovariant deriva tives de“ne thegeometryof”at superspace.Wewrite themasasup ervector: DA=( D D€ a).(3.4 .13) Ingeneral,in”atorcurvedspace,acovariantderivativecanbewrittenintermsofcoordinatederivatives M zM andconn ectionsA: DA DA MM+A( M )+A( T )+A( Z ).(3.4 .14) TheconnectionsaretheLorentzconnection A( M )=A M +A€€ M€€,( 3.4.15a) isospinco nnection A( T )=Ab cTc b,(3. 4.15b) andcentralchargeconnection A( Z )=1 2 (A bcZbc+Abc Zbc).(3.4 .15c) TheLorentzgenerators M actonlyon tangentspace indices.(Althoughthedistinction isunimportantin ”atspace,wedistinguishcurved,orcoordinateindices M N ... fromcova riantortangentspaceindices A B ... .Incurveds uperspaceweu sua llywrite thecovariantderivativesas A= EA MDM+A, DM M ADA, i.e.,weusethe ”at superspacecovariantderivativesinsteadofcoordinatederivatives:seechapter5for deta ils.) In”atsuperspace,inthevectorrepresentation,from(3.4.6)we“ndthe”atvielbein

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3.4.Covariantderivatives87DA M= 0 0 0 € € 01 2 i a m m€1 2 i €€m am a m ,(3. 4.16) andthe”atcentralchargeconnection A bc= Š1 2 ( C[ b a c ],0,0), Abc= Š1 2 (0, C€€ [ b€c ] a,0),(3 .4.17) allother”atconnectionsvanishing.Wecandescribethegeometryofsuperspacein termsofcovariant torsionsTAB C, curvaturesRAB( M ),and “eldstrengthsFAB( T )and FAB( Z ): [ DA, DB} = TAB CDC+ RAB( M )+ FAB( T )+ FAB( Z )(3. 4.18) From(3.4 .16-17),we“ndthat ”at superspacehasnonvanishingtorsion T € c= i a b €€(3.4.19) andnonvanishingcentralcharge“eldstrength F cd= Ca [ cb d ], F€ € cd= C€€[ c ad ] b,(3. 4.20) allothertorsions,curvatures,and“eldstr engthsvanishing.Hence”atsuperspacehasa nontrivial geometry. d.Casimiro pera tors Thecompletesetofoperatorsthatcommutewith P a, Q and Q€ (andtransformcovariantlyunder Jand J€€)is { DA, M M€€, Y ta b, d d d d } .(Ex ceptfor DA, whichisonlycovariantwithrespecttothesuper-Poincar ealgebra,a lltheseoperators arecovariantwithrespecttotheentiresuperconformalalgebra.Notethatthematrix operators M Y t d d d d actonlyontangentspaceindices.)ThustheCasimiroperators (groupinvariants)canallbeexpressedinte rmsoftheseoperators.Followingthediscussionofsubsec.3.3.a.3,itissucienttoconstruct:

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883.REPRESENTATIONSOFSUPERSYMMETRYP[ aW b ]= P[ af b ], f a1 2 [ Da Da€] Š i ( €M Š € M€€), W aŠ AP a= f a+ Yi a,(3. 4.21) Wa b= Š m2ta bŠi 4 €([ Da Db€] Š1 N a b[ Dc Dc€]), W5= m2Y Ši 4 €[ Da Da€](3. 4.22) Wa b a= ta bi aŠ1 4 ([ Da Db€] Š1 N a b[ Dc Dc€]); W5 a= Š Yi aŠ1 4 [ Da Da€](3. 4.23) whereweh aveused P2= Š m2for Wa b,and P€Q = €D = € =0for Wa b c(the masslesscase:seesubsec.3.3.a.3).

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3.5.Constrainedsuper“elds893.5.Constraine dsuper “elds Theexistenceofcovariantderivativesallowsustoconsiderconstrainedsuper“elds;thes implest(andformanyapplicationsthemostuseful)isachiralsuper“eld de“nedby D€ =0.(3 .5.1) Weobservethatt heconstraint(3.5 .1)impliesthatonachiralsuper“eld D =0and ther efore { D D } =0 Z =0. Inachiralrepresentation,theconstraintissimplythestatementthat(+)isindependentof thatis(+)( x , )=(+)( x ).Ther efore,inavectorrepresentation, ( x , )= e1 2 U(+)( x ) eŠ1 2 U=(+)( x(+), ),(3.5 .2) where x(+)isthechiralcoordinateof(3.3.27b).Alternatively,onecanwriteachiral super“eldintermsofageneralsuper“eldbyusing D2 N +1=0: = D2 N( x , )(3. 5.3) Thisformofthesolutiontotheconstraint(3.5.1)isvalidinanyrepresentation.Itis themostg eneralpossible;seesec.3.11. Similarly,wecande“neantichiralsuper“elds;theseareannihilatedby D .Note that ,thehermitianconjugateofachiralsuper“eld,isantichiral.Thesesuper“elds maycarryexternalindices. *** Thesupersymmetrygeneratorsarerepresentedmuchmoresimplywhentheyact onchiralsuper“elds,particularlyinthechiralrepresentation(3.3.25),thanwhenthey actongeneralsuper“elds.Forthesuper-Poincar ecasew ehave: Q = i ( Š1 2 b Zba), Q€ = a €, P a= i a, J= Š i1 2 ( x( € )€+ a ( a )) Š iM, J€€= Š i1 2 x (€€ )Š i M€€,(3. 5.4)

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903.REPRESENTATIONSOFSUPERSYMMETRYwhere Zab=0(asex plainedabove)but Zabis unrestricted. Ifwethinkof Zabasa partialder ivativewithrespecttocomplexcoordinates ab, i.e., Zab= i ab ,thenachiral super“eldisafunctionof x , andisi ndependentof .Inthesupe rconformal case, Zabmustvani sh,and,forconsistencywiththealgebra,achiralsuper“eldmust have no dottedindices(i.e., M€€=0).Onchiralsuper“elds,t heinvers ion(3.3.33)takes theform II ...( x )= xŠ 2 df€... ...( x, )= xŠ 2 df€... € ...( x, ), f€= i ( x )Š 1x€, x a= xŠ 2x a, = ixŠ 2x€a ;(3. 5.5) (notethat dŠ=0andh ence d = d+).Thegeneratorsofthesuperconformalalgebraare nowjust( 3.5.4), S€ = Š x€a S = Š a G K a= x€G ;(3. 5.6a) with G = J + [+ i (1 2 x a a+2 Š N )], = Š i ( x a a+1 2 +2 Š N + d d d d ), A =1 2 Š (4 Š N )Š 1Nd d d d Ta b=1 4 ([ b a ] Š1 N a b[ ]).(3.5.6b) Thecommutatoralgebrais,ofcourse,unchanged.Notethattheexpressionfor A containsaterm(1 Š1 4 N )Š 1d d d d ;thisimpliesthatfor N =4,either d d d d vanishes,o rtheaxial char gemustbedroppedfromthealgebra(seesec.3.2.e).Theonlyknown N =4theoriesareconsistentwiththisfact: N =4 Ya ng -M illshasnoaxialchargeand N =4conformalsuper gravityhas d d d d =0.Wefurth ernotethatconsistencyofthealgebraforbids theadditionofthematrixoperator ta bto Ta binthecas eofconfo rmalchiralsuper“elds. Thismeansthatconformalchiralsuper“eldsmustbeisosinglets,i.e.,cannotcarryexternalisospinindices.

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3.5.Constrainedsuper“elds91*** For N =1,acomple x“eldsatis fyingtheconstraint D2=0isc alledalinear super“eld.Areallinearsuper“eldsatis“estheconstraint D2G = D2G =0.Wh ilesuch objectsappearinsometheories,theyarelessusefulfordescribinginteractingparticle mult ipletsthanchiralsuper“elds.Acomplexlinearsuper“eldcanalwaysbewrittenas = D€€,whereasareal linearsuper“eld canbewri ttenas G = D D2+ h c ..

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923.REPRESENTATIONSOFSUPERSYMMETRY3.6.Componentexpansions a. -expansions B ecausethesquareofanyanticommutingnumbervanishes,anyfunctionofa “nitenumberofanticommutingvariableshasaterminatingTaylorexpansionwith respecttothem.Thisallowsustoexpandasuper“eldintermsofa“nitenumberof ordi naryspacetimedependent“elds,or components. Forgen eral N ,thereare4 N i ndepe ndentanticommutingnumbersin ,and thus4 N i =0 4 N i =24 Ncomponentsinanunconstrainedscalarsuper“eld.Forexample,for N =1,arealsc alarsuper“eldhasthe expansion V = C + + € €Š 2M Š 2 M + €A aŠ 2Š 2 € €+ 2 2D(3.6.1) with16realcomponents.Similarly,achiralscalarsuper“eldinvectorrepresentation hastheexpansion: = e1 2 U( A + Š 2F ) eŠ1 2 U= A + Š 2F + i1 2 € aA + i1 2 2 € a+1 4 2 2 A (3.6.2) with4independentcomplexcomponents. Theseexpansionsbecomecomplicatedfor N > 1super “eldsbutfortunatelyare no tn eed ed .H ow ev er ,w eg ivesomeexamplestofamiliarizethereaderwiththecomponentcontentofsuchsuper“elds.Forinstance,for N =2,ina ddition tocarrying Lorentzspinorindices,super“eldsarerepresentationsof SU (2).Arealscala r-isoscalar super“eldhastheexpansion V ( x , )= C ( x )+ + € € Š 2 M( )Š 2 abM( ab )Š 2€€ M(€€ )Š 2 ab M( ab )+ a b€( Wa b €+ a bV€)+ ...

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3.6.Componentexpansions93+ 4N + 4 N + ... + 2 2€€h€€+ ... + 4 4D( x )(3. 6.3) where Wa a €=0,wh ileachiralscalarisospinorsuper“eldhastheexpansion(inthechiralrepresentation) (+) a( x )= Aa+ b ( Cab+ ( ab ) ) Š 2 Fa ( )Š 2 bc( F( abc )+ CabFc) Š 3 b ( a b+ a b )+ 4D a,(3. 6.4) where a a =0.Thespinan diso spinofthecomponent“eldscanbereadfromthese expressions. Generalsuper“eldsarenotirreduciblerepresentationsofextendedsupersymmetry. As we discussinsec.3.11,chiralsuper“eldsareirreducibleundersupersymmetry(except forapossiblefurtherdecompositionintorealandimaginaryparts);wepresenttherea systematicwayofdecomposinganysu per“eldintoitsi rreducibleparts. Thesupersymmetrytransformationsofthe component“eldsfollowstraightforwardlyfromthet ransformationsofthesuper“ elds.Thus,forexample,for N =1,from V =[ i ( Q + Q ), V ]= C + + ... (witha cons tant spinorparameter )we “nd: C ( x )= Š ( + € €), ( x )= M Š €( i1 2 aC + A a), €( x )= € M Š ( i1 2 aC Š A a), M ( x )= Š €( €+ i1 2 a), A a( x )= Š ( C €+ i1 2 €)+ €( C€€+ i1 2 € €), ( x )= ( CD+ i1 2 €A€) Š i1 2 €€ M ,

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943.REPRESENTATIONSOFSUPERSYMMETRY D( x )= i1 2 a( €+ €), etc .,(3.6 .5) Similarly,forachiralsuper“eldwe“nd: A = Š = Š €i €A + F F = Š €i €.(3. 6.6) b.Projection Formanya pplications,the -expansionsjustconsideredareinconvenient;an alternativeistode“necomponentsbyprojectionofanexpressionasthe -independent partsofitssuccessivespinorderivatives.Weintroducethenotation X | toindicatethe i ndependentpartofanexpression X .The n,forexample,wecande“nethecomponents ofachirals uper“eldby A ( x )=( x , ) | ( x )= D( x , ) | F ( x )= D2( x , ) | .(3. 6.7) Thesupersymmetrytransformationsoftheco mponent“eldsfollowfromthealgebraof thecovariantderivatives D ;weuse( Š iQ ) | =( D ) | and { D, D€} = i €to “nd A = i ( Q + Q ) | = Š ( D + D ) | = Š D | = Š = i ( Q + Q ) D | = Š ( D + D ) D | =( D2Š €i €) | = F Š €i €A F = i ( Q + Q ) D2 | = Š DD2 | = Š €i €.(3. 6.8)

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3.6.Componentexpansions95Explicitcomputationofthecomponents showsthat,inthisparticularcase,the componentsinthe -expansionareidenticaltothosede“nedbyprojection.Thisisnot necessarilythecase:Forsuper“eldsthatarenotchiral,somecomponentsarede“ned withboth D sand D s;forthesecomponents,therei sanambiguityst emmingfromhow the D sand D sareorder ed.Forexample,the 2 componentofarealscalarsuper“eld V couldbede“nedas D2 DV | D DDV | ,or DD2V | .These de“nitionsdieronlyby spacetimederivativesofcomponentslowerdowninthe -expansion(de“nedwithfewer D s).Ingeneral,theywillalsodi erfromcomponentsde“nedby -expansionsbythe samederivativeterms.Thesedierencesarejust“eldrede“nitionsandhavenophysical signi“cance. Usually,o neparticularde“nitionofcomponen tsispref erable.Forexample,one modelthatwewillconsider(seesec.4.2.a)dependsonarealscalarsuper“eld V which transformsas V= V + i ( Š )underagaugetransformationthatleavesallthe physicsinvariant(hereisachiral“eld).Inthiscase,ifpossible,weselectcomponents thataregaugeinvariant;intheexampleabove, D2 DV | isthepreferredchoice. Ifthesuper“eldcarriesanexternalLorentzindex,theseparationintocomponents requiresreductionwithrespecttotheLorentzgroup.Thus,forexample,achiralspinor super“eldhastheexpansioninthechiralrepresentation(whereitonlydependson ): (+)( x )= + ( CD+ f) Š 2.(3. 6.9) Usingprojections,wewou ldde “nethecomponentsby =| D=1 2 D| f=1 2 D( )| = D2| .(3. 6.10) For N > 1asim ilarde“nitionofcomponentsby projectionispossible.Inthis case,inadditiontoreductionwithrespecttotheexternalLorentzindices,onecanfurtherre ducewithrespectto SU ( N )i ndices.

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963.REPRESENTATIONSOFSUPERSYMMETRYTheprojectionmethodisalsoconvenientfor“ndingcomponentsofaproductof super“elds.Forexample,theproduct=12ischiral,andha scomponents | =12| = A1A2, D | =( D1)2| +1( D2) | = 1 A2+ A12 D2 | =( D21)2| +( D1)( D2) | +1( D22) | = F1A2+ 1 2 + A1F2.(3. 6.11) Similarly,thecomponentsoftheproduct= 12canbeworkedoutinastraightforward manner,usingtheLeibnitzruleforderivatives. c.Thetransformationsuper“eld ThetransformationsofPoincar esupersy mmetry(translationsand Q -supersymmetrytransformations)areparametrizedbya4-vector aandaspinor respectively.Itis po ssibletoviewthese,alongwiththeparameter r ofR-symmetrytransformations generatedby A in(3.2.12,3.3.34a),ascomponentsofan x -independentrealsuper“eld a1 2 [ D€, D] | i D2D | r 1 2 D D2D | ,(3. 6.12) andtowritethesupersymmetrytransformationsintermsof andthecovariant deriva tives DA: = i ( aP a+ Q+ € Q€+2 rA ) = Š [( i D2D ) D+( Š iD2 D€ ) D€+(1 2 [ D€, D] ) €+ iw (1 2 D D2D )],(3.6.13) where1 2 w istheeigenvalueoftheoperator Y (thematrixpartoftheaxialgenerator A ). Thesetransformationsareinvariantundergaugetransformations = i ( Š ), chiraland x -independent.Consequently,theydependonlyon a, ,andthec omponent r TheR-transformationswithparameter r areaxialrotations ( x , )= eŠ iwr( x eir eŠ ir ).(3.6 .14)

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3.6.Componentexpansions973.7.Superintegration a.Berezinintegral Toconstr uctmanifestlysupersymme tricallyinvariantactions,itisusefultohave anotionof( de“nite)integrati onwithrespectto .Thee ssentialpropertieswerequire ofthe Berezin integralaretranslationinvarianceandlinearity.Considera1-dimensional anticommutingspace;thenthemostgeneralformafunctioncantakeis a + b .The mostgeneralformthattheintegralcantakehasthesameform:d ( a + b )= A + B where A B arefunctionsof a b .Imposing linearityandinvarian ceundertranslations + leads uniquelytotheconclusionthatd ( a + b ) b .The normalizat ionoftheintegralisarbitrary.Wechoose d =1(3.7 .1) a nd,aswefoundabove, d 1=0.(3 .7.2) Wecand e“nea -function:Werequire d ( Š )( a + b )= a + b (3.7.3) and “nd ( Š )= Š (3.7.4) Theseconceptsgeneralizeinanobviouswaytohigherdimensionalanticommuting spaces;for N -extendedsupersymmetry,d2 N d2 N picksoutthehighest component oftheintegrand,anda -functionhastheform 4 N( Š )=( Š )2 N( Š )2 N.(3. 7.5) Wede “ne 4+4 N( z Š z) 4( x Š x) 4 N( Š ).Wethushave d4+4 Nz 4+4 N( z Š z)( z )

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983.REPRESENTATIONSOFSUPERSYMMETRY= d4xd4 N4( x Š x) 4 N( Š )( x )=( z)(3. 7.6) Wenote thatallthepropertiesoftheBerezinintegralcanbecharacterizedbysayingitisidenticaltodierentiation: d f ( )= f ( ).(3.7 .7) Thishasanimportantconsequenceinthecontextofsupersymmetry:Becausesuperspaceactionsareintegratedoverspacetimeaswellasover ,anyspa cetimetotalderivati ve a ddedtotheintegrandisirrelevant(moduloboundaryterms).Consequently,inside aspa cetimeintegral,intheabsenceofcentralchargeswecanreplaced = by D Thisallowsustoexpandsuperspaceactionsdirectlyintermsofcomponentsde“nedby projection(seechap.4,whereweconsiderspeci“cmodels).Insidesuperspaceintegrals, wecanint egrate D byparts, b ecaused4 N = 2 N 2 N =0(since 2 N +1=0). Sincesupersymmetryvariationsarealsototalderivatives(insuperspace),wehaved4xd2 N Q = d4xd2 N Q€ =0,andth usforanygeneralsuper“eldthefollowingisasupersymmetryinvariant: S= d4xd4 N .(3.7 .8) Inthecaseofchiralsuper“eldswecande“neinvariantsinthechiralrepresentationby S= d4xd2 N ,(3.7 .9) sinceisafunctionofonly x aand .Infact, thisde “nitionisrepresentationindependent,sinceth eoperator U usedtochangerepresentationsisaspacetimederivative,so onlythe1partof e1 2 Ucontributesto S.Furthe rmore,ifweexpressintermsofageneralsuper“eldby= D2 N,we“nd S= d4xd2 N D2 N= d4xd4 N = S,(3. 7.10) since D€ = d € wheninsidea d4x integral.

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3.7.Superintegration99Similarly,thechiraldeltafunction,whichwede“neas 4( x Š x) 2 N( Š ) 4( x Š x)( Š 1)N( Š )2 Ninthechiralreprese ntation, takesthe followingfo rminarbitraryrepresentations: D2 N4+4 N( z Š z),(3.7 .11) whichisequivalentinthechiralrepresentation( D€ = € ),andi ngen eralrepresentations gives d4xd2 N [ D2 N4+4 N( z Š z)]( z ) = d4xd4 N4+4 N( z Š z)( z ) =( z )(3. 7.12) b.Dimensions SincetheBerezinintegralacts likeaderivative(3.7.7),italso scales likeaderivative;thusith asdime nsion[d ]=[ D ].However,from(3.4.9 ),weseetha tthe dimensionsof D m1 2 ,andco nsequently,ageneralintegralhasdimensiond4xd4 N m2 N Š 4andachiralintegralhasdimensiond4xd2 N mN Š 4.Inparticular,for N =1,wehaved4xd4 d8z mŠ 2andd4xd2 d6z mŠ 3. c.Superdeterminants Finally,weusesuperspaceintegralstode“nesuperdeterminants(Berezinians). Considera( k n )by( k n )dimensi onalsupermatrix M witha k by k dimensionalevenevenpart A ,a k by n dimensionaleven-oddpart B ,an n by k dimensionalodd-even part C ,andan n by n dimensionalodd-oddpart D : M = A C B D (3.7.13) wheretheentriesof A D arebosonicandthoseof B C arefermionic.Wede“nethe superdeterminantbyanalogywiththeusualdeterminant:

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1003.REPRESENTATIONSOFSUPERSYMMETRY( sdetM )Š 1= K dkxdkxdn dneŠ z tMz,( 3.7.14a) where z t=( x), z = x ,(3. 7.14b) and K isanormalizationfacto rchosentoe nsurethat sdet (1)=1.Theexponent xAx + xB + Cx + D canbewri tten,aftershiftsofintegrationvariableseitherin x orin ,intwoequiva lentforms: xAx + ( D Š CAŠ 1B ) or x( A Š BDŠ 1C ) x + D .Integrationov erthebosonicvariablesgivesusaninverse determinantfactor,andintegrationoverthe fermionicvariablesgivesadeterminantfactor.Weobtain sdetM intermsofordinarydeterminants: sdet ( M )= detA det ( D Š CAŠ 1B ) = det ( A Š BDŠ 1C ) detD .(3. 7.15) Thisformulahasanumberofusefulprope rties.Justaswiththeordinarydeterminant,thesuperdeterminantoftheproductofseveralsupermatricesisequaltothe productofthesuperdeterminantsofthesupermatrices.Furthermore, ln ( sdetM )= str ( lnM ), (3.7.16a) wherethesupertraceofasupermatrix M isthetraceofthee ven-ev enmatrix Aminus thetraceoftheodd-oddmatrix D : strM trA Š trD (3.7.16b) An ar bitraryin“nitesimalvariationof M inducesavariationofth esuper determinant: ( sdetM )= exp [ str ( lnM )] =( sdetM ) str ( MŠ 1 M )(3. 7.17)

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3.8.Superfunctionaldierentiationandintegration1013.8.Superfunctionaldierentiationandintegration a.Dierentiation Inthissectionwediscussfunctionalcalculusforsuper“elds.Webeginbyreviewingfunctionaldierentiationforcomponent“elds:Byanalogywithordinarydierentiation,functionaldierentiationofafunctional F ofa“eld A canbede“nedas F [ A ] A ( x ) = 0lim F [ A + xA ] Š F [ A ] ,(3. 8.1) where xA ( x)= 4( x Š x).(3.8 .2) Thisis not thesameasdividing F by A .The derivativecanalsobede“nedforarbitraryvariationsbyaTaylorexpansion: F [ A + A ]= F [ A ]+ A F [ A ] A + O (( A )2),(3.8 .3) wheretheproduct(,)oftwoarbitraryfunctionsisgivenby ( C B )= d4xC ( x ) B ( x ).(3.8 .4) Inparticular,from(3.8.2)we“nd ( xA B )= B ( x ).(3.8 .5) Thisde“nitionallowsaconvenientprescriptionforgeneralizeddierentiation.For example,incurvedspace,wheretheinvariantproductis( C B )= d4xg1/2CB ,the normaliz ation( A B )= B ( x )co rrespondstothefunctionalvariation A ( x)= gŠ 1/2( x ) 4( x Š x).Generally,achoiceof xisequivalenttoachoiceofthe product(,).Inparticula r,for(3.8.2,4)wehavethefunctionalderivative A ( x ) A ( x) = 4( x Š x).(3.8 .6) Incurvedspace,usingtheinvariantproduct,wewouldobtain gŠ 1/2( x ) 4( x Š x).Note thattheinnerpro ductisnot alwayssymmetric:In( C B ), C transformscontragredientlyto B .Forex ample,if A isacovariantvector,thequantityontheleft-handsideof

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1023.REPRESENTATIONSOFSUPERSYMMETRYtheinner-productisacovariantvector,whilethatontherightisacontravariantvector; if A isanisospinor, F A isacomplex-conjugateisospinor;etc. Insuperspace,thede“nitionsforgeneralsuper“eldsareanalogous.Theproduct (,)isd4+4 Nz ( z )( z )=d4xd4 N ( x )( x ),andthus ( z ) ( z) = 4+4 N( z Š z)= 4( x Š x) 4 N( Š ).(3.8 .7) (Appropriatemodi“cationswillbemadein curvedsuperspace.)However,forchiral super“eldswehave (,)= d4+2 Nz = d4xd2 N ,(3. 8.8) sinceandessentia llydependononly x aand ,not € .Thevariatio nisthe refore de“nedintermsofthechiraldeltafunction: z( z)= D2 N4+4 N( z Š z)(3. 8.9) sothat ( z ) ( z) = D2 N4+4 N( z Š z),(3.8 .10) andthecomplexconjugaterelation ( z ) ( z) = D2 N4+4 N( z Š z).(3.8 .11) (Again,appropriatemodi“cationswillbema deincurvedsuperspace.)Furthermore, variations ofchiralintegralsgivetheexpectedresult ( z) d4xd2 N f (( z ))= d4xd2 N f(( z )) D2 N4+4 N( z Š z) = d4xd4 N f(( z )) 4+4 N( z Š z)= f(( z)).(3.8.12) Whenthefunctionaldierent iationisonanexpressionappearinginachiralintegral with d2 N ,the D2 Ncanalwaysbeusedt oconvertittoa d4 N integral,afterwhichthe full -functioncanbeusedasin(3.8.12).

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3.8.Superfunctionaldierentiationandintegration103Thisresultcanalsobeobtainedbyexpre ssingintermsofageneralsuper“eld, as= D2 N:wehave ( z ) ( z) = D2 N( z ) ( z) = D2 N ( z ) ( z) = D2 N4+4 N( z Š z).(3.8 .13) Wecanthusi dentify with for= D2 N. Thesede“nitionscanbeanalyzedintermsofcomponentsandcorrespondtoordinaryfunctionaldierentiationofthecomponent“elds.Wecannotde“nefunctionaldifferentiationforconstrainedsuper“eldsotherthanchiralorantichiralones.Forexample,foralinearsuper“eld( whichcanbewrittenas= D€ €)thereis nofunctional derivati vewhichisb othlinearandascalar. b.Integration Inchapters5and6wediscussquantizationofsuper“eldtheoriesbymeansof functionalintegration.Weneedtode“neonly integralsofGaussians,asallotherfunctionalintegralsinperturbationtheoryarede“nedintermsofthesebyintroducing sourcesanddierentiatingwithrespecttothem.Thebasicintegralsare IDVed4xd4 N1 2 V2=1 ,( 3.8.14a) ID ed4xd2 N1 2 2=1,(3. 8.14b) ID ed4xd2 N 1 2 2=1,(3. 8.14c) where,e.g., IDV =i IDVi,for Vithecomponentsof V .B ecauseasuper“eldhasthe samenumberofboseandfermicomponents,manyfactorsthatappearinordinaryfunctionalintegralscancelforsuper“elds.Thuswecanmakeanychangeofvariablesthat doesnotinvolve both exp licit sand swithoutg eneratinganyJacobianfactor, b ecauseunlessthebosonsandfermionsmixnontrivially,thesuperdeterminant(3.7.14) isequaltoo ne.Forexample,achangeofvariables V f ( V X )where X isanarbitraryexternals uper“eldgeneratesnoJac obianfactor;thesameistrueforthechangeof

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1043.REPRESENTATIONSOFSUPERSYMMETRYvariables V V aslongas isapurelybosonicoperator.NontrivialJacobian determinantsariseforchangesofvariablessuchas V D2V or V V where is ba ck groundcovariant,e.g.,insupergravityorsuper-Yang-Millstheory,andhencecontainsspinorderivatives. To provetheprecedingassertions,weconsiderthecasewithone ;thege neralcase canbeprovenbychoosingoneparticular an dp roceedinginductively.Weexpandthe super“eldwithrespectto as V = A + ;sim ilarly,weexpandthearbitraryexternal super“eldas X = C + .Thenwecan expandthenewvariable f ( V X )as f ( V X )= f ( A C )+ [ fV( A C ) | + fX( A C ) | ](3. 8.15) where fV| fA ( f | ) A ,etc.TheJa cobianofthistransformationis sdet f V = sdet fA( A C ) 0 fAA+ fACfA( A C ) = det ( fA) det ( fA) =1.(3. 8.16) Inparticular,theexternalsuper“eld X canbeanonlocaloperatorsuchas Š 1. An i mmediateconsequenceoftheprecedingresultisthatsuper“eld -functions ( V Š V) i ( ViŠ V i)( 3.8.17a) ar ei nv ar ia nt under -nonmixingchangesofvariables: ( f ( V ))=f ( ci)=0 ( V Š ci).(3.8 .17b) Ingeneral,ifnontrivialoperatorsappearintheactions,thefunctionalintegralsare nolongerconstant.We“rstintroducethefollowingconvenientnotation: V ,t d4x d4 N Vt d2 N t d2 N t ,(3. 8.18) where V ,,and themselvescanst andforseveralsuper“eldsarrangedascolumnvectors.Wenextconsideractionsoftheform

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3.8.Superfunctionaldierentiationandintegration105S =1 2 tO O ,(3.8 .19) wherethe nonsingular operator O O issuchthatthecomponentsofthecolumnvector O O havethesamechiralityasthecorrespondingcomponentsof.Theseactionsgivethe “eldequations S = O O ,(3.8 .20) dueto theintegrationmeasureschosenforthede“nitionoftheintegrals(3.8.18). Wede “ne,forcommuting, ( det O O )Š1 2 ID eS,(3. 8.21) with S givenby(3.8.19).Foranticommutingweobtain( det O O )1 2 .Then(3. 8.14)can bewri ttenas det I I =1.(3. 8.22) Fromthede “nition (3.8.21)wehaveID 1ID 2e1 TO O 2=( det O O )Š 1.(3. 8.23) Wealsohave ( det O O1)( det O O2)= det ( O O1O O2).(3.8 .24) Thiscanbeprovenasfollows:Weconsidertheaction(1 tO O12+3 tO O24).(3.8 .25) Thefunctionalintegraloftheexponentialofthisactionisequaltothatof(1 tO O1O O22+3 t4),(3.8 .26) ascanbeseenfromthe“eldrede“nitions 2 O O22,4 O O2 Š 14,(3. 8.27)

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1063.REPRESENTATIONSOFSUPERSYMMETRYwhoseJacobianscancel. Asanimportantexampleweconsiderthe N =1casewith oneandone and no V : = O O = 0 D2 D20 .(3. 8.28) Thisoperatorsatis“estheidentity O O2= .(3.8 .29) Therefore,from(3.8.24)wehave ( det O O )2= det (3.8.30) andhencetheintegraloftheexponentialoftheaction S =1 2 [ d4xd2 (1 D2 1+2 D2 2)+ h c .] = d4xd4 ( 11+ 22)(3. 8.31) isequalt othatof S =1 2 [ d4xd2 + h c .].(3. 8.32) Inthesamemannerwehavethefollowingequivalence: d4xd4 m 2 m +1 i =1 d4xd4 ii.(3. 8.33) Asanotherexampleweconsiderthecaseofachiralspinor: = € O O = 0 i €D2i € D20 ,(3. 8.34) with O O2= 2.(3.8 .35)

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3.8.Superfunctionaldierentiationandintegration107Therefore d4xd4 €i € 1 2 [ d4xd2 + h c .].(3. 8.36)

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1083.REPRESENTATIONSOFSUPERSYMMETRY3.9.Physical,auxiliary,andgaugecomponents Insection3.6wediscussedthecomponent“eldcontentofsupersymmetrictheories.However,the“eldcontentofatheorydo esnotdetermineitsphysicalstates.Conversely,agiv ensetofphysicalstatescanbedescribedbydierentsetsof“elds. Givenasetof“eldsandtheirfreeLagrangian,wecanclassifyany component ofa “eldas oneofthreetypes:(1) physical, withapropagatingdegreeoffreedom;(2) auxiliary, withanequationofmotionthatsetsitidenticallyequaltozero;and(3) gauge, not appearingintheLagrangian.(Super)Fieldsc ancontainallthreekindsofcomponents; o-shellrepresentations(ofthePoincar eorsupersy mmetrygroup)containonlythe“rst two; andon-shellrepresentationscontai nonlyt he“rst.Wealsoclassifyany “eld asone ofthreetypes:(1)physical,containingphysicalcomponents,butperhapsalsoauxiliary and/orgaugecomponents;(2)auxiliary,containingauxiliary,butperhapsalsogauge, components;and(3) compensating, containingonlygaugecomponents. Thesimplestexampleofthisistheconventionalvectorgauge“eldofelectromagnetism.Theexplicitseparationisnecessarilynon(Poincar e)covariant,andismostconvenientlyp erformedina light-cone formalism.Inthenotatio nof(3. 1.1)wetreat xŠ= xŠ€Šasthetimecoordinate,and x+, xT, xTasspacecoordinates.Wearethus freetoconstructexpressionsthatarenonlocalin x+(i.e.containinginversepowersof +),sincet hedynamicsisdescribedbyevolutionin xŠ€Š.(Infact,t heformalis mclo sely resembles nonrelativistic “eldtheo ry,with xŠactingasthetimeand +asthe ma ss.) Thevectorgauge“eld A€transformsas A€= € .(3. 9.1) Bymakingthe“eldr ede“nitions(by A f ( A )wemean A = f ( A)andthen drop alls) A+ A+, AT AT+( +)Š 1TA+, AŠ AŠ+( +)Š 1( ŠA+Š T ATŠ TAT);(3.9 .2) weobtain thenewtransformationlaws

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3.9.Physical,auxiliary,andgaugecomponents109 AT= AŠ=0, A+= + .(3. 9.3) Furthe rmore,theLagrangian IL = Š1 2 FF,(3. 9.4) where F=1 2 ( €A )€(3.9.5) intermsofthe oldA ,b ecomes IL = AT ATŠ1 4 AŠ( +)2AŠ.(3. 9.6) Thus,thecomplexcomponent ATde scribesthetwophysical(propagating)polarizations, therealcomponent AŠisauxiliary(ithasnodynamics;itsequationofmotionsetsit equaltozero),andtherealcomponent A+isgauge.Inthisformalismtheobvious gaugechoiceis A+=0 (thelight-conegauge),since A+doesnotappearin IL .Howev er, gaugecomponentsareimportantforLorentzcovariantgauge“xing:Forexample, ( €A€)2 ( +AŠ+2( +)Š 1 A+)2. Wecanp erformsimilarrede“nitionstoseparat earbit rary“eldsintophysical,auxiliary,andgaugecomponents.Anyorigin alcomponentthattransformsunderagauge transformationwitha +oranonderivativetermcorrespondstoagaugecomponentof therede“ned“eld.Anycomponentthattransformswitha Štermcorrespondstoan auxiliarycomponent.Oftheremainingcomponents,somewillbeauxiliaryandsome physical(dependingontheaction),organizedinawaythatpreservesthetransverse SO (2)Lorentzcovarian ce.Fortheknown“eldsappearingininteractingtheories,the componentswithhighestspinarephysicalandtherest(whenthereareany:i.e.,for physical spin2or3 2 )a reauxiliary.Theseargumentscanbeappliedinalldimensions. Anexamplethatillustratestheseparationbetweenphysicalandauxiliary(butnot gauge)componentswithouttheuseofnonlocal,noncovariantrede“nitionsisthatofa massivespinor“eld: IL = €i €Š1 2 m ( + € €).(3.9 .7) Since and maybeconsideredasindependent“eldsinthefunctionalintegral(and,

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1103.REPRESENTATIONSOFSUPERSYMMETRYinfact,mustbeconsideredindependentlocallyafterWickrotationtoEuclideanspace), wecanmakethefo llowing nonunitary(butlocalandcovariant)rede“nition: € €Š1 m i €.(3. 9.8) TheLagrangianbecomes IL =1 2 m ( Š m2) Š1 2 m € €.(3. 9.9) Wethus “ndthat representstwophysicalpolarizations,while co nt ainstwoauxiliary components. Thesameanalysiscanbemadeforthesimplestsupersymmetricmultiplet:the massivescalarmultiplet,describedbyachiralscalarsuper“eld(seesection4.1).The actionis S = d4xd4 Š1 2 m ( d4xd2 2+ d4xd2 2).(3.9 .10) Wenowr ede“ne +1 m D2;(3.9 .11) a nd,usingd2 = D2,weobtaint heac tion S =1 2 m d4xd2 ( Š m2) Š1 2 m d4xd2 2.(3. 9.12) (Notetha tthere de“nitionof prese rvesitsantichirality D =0.)Nowc ontains onlyphysicaland c ontainsonlyauxiliarycomponents;eachcontainstwoBosecomponentsandtwoFermi.Ascanbecheckedusingthecomponentexpansionof,theoriginalaction(3. 9.10)containsthespinorLagrangianof(3.9.7),whereas(3.9.12)contains theLagrangian(3.9.9).Italsocontainstwoscalarsandtwopseudoscalars,oneofeach bein gaphysical“ eld(withkine ticoperator Š m2)a nd theotheranauxiliary“eld (withkineticoperator1).Formoredetailofthecomponentanalysis,seesec.4.1.

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3.9.Physical,auxiliary,andgaugecomponents111Aswediscussinsec.4.1,auxiliary“eldsareneededininteractingsupersymmetric theoriesforseveralreasons:(1)Theyfacilitatetheconstructionofactions,sincewithout themthekineticandvariousinteractiontermsarenotseparatelysupersymmetric;(2) b eca us eo ft hi s, actionswithoutauxiliary“eldshavesupersymmetrytransformations thatarenonlinearandcouplin gdep endent,andmakediculttheapplicationofsupersy mmetryWardidentities(e.g.,toproverenormalizability);and(3)auxiliary“eldsare necessaryformanifestlysupersymmetricquantization.Compensating“elds(seefollowingsection)arealsonecessaryforthelattertworeasons.Althoughtheydisappearfrom theclassicalaction,theyappearinsupersymmetricgauge-“xingterms.

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1123.REPRESENTATIONSOFSUPERSYMMETRY3.10.Compensators Inoursubsequentdiscussions,wewilloftenusecompensating“eldsorcompensators.Theseare“elds thatenteratheoryinsuchawaythattheycanbe algebraically gaugedaway.Thus,inacertainsense,theyaretrivial:Thetheorycanalwaysbewrittenwitho utthem.However,theyfrequentlysimplifythestructureofthetheory;inparticular,they canbeusedtowritenonlinearlyrealizedsymmetriesinalinearway.This isoftenimportantforquantization.Anotherapplication,whichisparticularlyrelevant tosupergravity,arisesinsituationswhere oneknows howtowriteinvariantactionsfor systemstransformingunderacertainsymmetrygroup G (e.g.,thesuperconformal gr o up):Ifonewantstowriteactionsforsystemstransformingonlyunderasubgroup H (e.g.,thesuper-Poincar egro up),onecanenlargethesymme tryofsuchsystemstothe fullgr oupbyintroducingcompensators.Afterwritingtheactionforthesystemswith theenlargedsymmetry,onesimplychooses a gauge,thusbreakingthesymmetryofthe actiondownto thes ubgroup H Asimpleexamp leinordinary“eldtheoryisfa kescal arelectrodynamics.The usualkineticaction foracomplexscalar z ( x ) S =1 2 d4x z a az (3.10.1) hasagl obal U (1)symmetry: z= ei z .Thissy mmetrycanbegaugedtriviallybyintroducingarealcomp ensatingscalar ,a ssumedtotransformunderalocal U (1)transformationas = Š .Wecanthen constructacovariantderivative a= eŠ i aei = a+ i a thatcanbeusedt ode “nealocally U (1)invariantaction S =1 2 d4x z a az (3.10.2) Fakespinorel ectrodynamicscanbeobtainedbyanobviousgeneralization. a.Stueckelbergformalism Inthepreviousexample,thecompensatorservednousefulpurpose.TheStueckel be rgformalismprovidesafamiliarexampleofacompensatorthatsimpli“esthetheory. Webeginwit htheL agrangianforamassivevector A a: IL = Š1 8 F a bF a bŠ m2( A a)2,(3. 10.3)

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3.10.Compensators113F a b= [ aA b ].(3. 10.4) Thepropagatorforthistheoryis: D a b= Š1 Š m2 ( a bŠ1 2 m2 a b)(3. 10.5) Wecanr ecastthetheoryinanimprovedformbyintroducinga U (1)compensator thatmakestheaction(3.10.3)gaugeinvariant.Wede“ne A a= A a+1 m a (3.10.6) where A aand tr an sformunder U (1)gaugetransformations: A a= a = m .(3. 10.7) Intermsofthese“elds,the gaugeinvariant Lagrangianis(droppingtheprime): IL = Š1 8 F a bF a bŠ m2( A a)2Š m aA aŠ ( a )2.(3. 10.8) Wenowc hoosea gaugebyaddingthegauge“xingterm ILGF= Š1 4 ( aA aŠ 2 m )2(3.10.9) an d “nd: IL + ILGF=1 2 A a( Š m2) A a+ ( Š m2) .(3. 10.10) Thepropagatorscanbetriviallyreadofrom(3.10.10):for A a, D a b= Š a b( Š m2)Š 1,andfor ,D= Š1 2 ( Š m2)Š 1.Theyhaveb e tterhighenergy behaviorth an(3.10.5).Thus,byintroducingthecompensator ,wehavesim p li“edthe structureofthetheory.Wenotetha tthecomp ensatordecoupleswhenever A aiscoupledtoaconservedsource(i.e .,inagaugeinvariantway). b.CP(1)model Anotherfamiliarexampleisthe CP (1)nonlinear -model,whichdescribesthe Goldstonebosonsofan SU (2)gaugetheoryspontaneouslybrokendownto U (1).Itconsistsofarealscalar“eld andacomplex“eld y subjectto theconstraint

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1143.REPRESENTATIONSOFSUPERSYMMETRY| y |2+ 2=1(3. 10.11) Thegroup SU (2)canberealized nonlinearly onthese“eldsby =1 2 ( y + y ) y = Š 2 i y Š Š1 2 Š 1( y Š y ) y .(3. 10.12) where ,and arethe(constant)parametersoftheglobal SU (2)transformations. ThesetransformationsleavetheLagrangian IL = Š [( a )2+ | ay |2+1 4 ( y ay )2](3. 10.13) invariant,butbecausethetransformationsarenonlinearthisisfarfromobvious. Wecangiveadesc riptionofthetheorywherethe SU (2)isrepresent edlinearlyby introducinga localU (1)invariancewhichisrealizedbyacompensating“eld .U nder thislocal U (1), transformsas ( x )= ( x ) Š ( x ).(3. 10.14) Wede “ne“elds ziby z1= eŠ i z2= eŠ i y .(3. 10.15) B ecauseofthisde“nitiontheytransformunderthelocal U (1)as z i= ei zi.(3. 10.16) Theconstraint(3.10.11)becomes | z1|2+ | z2|2=1.(3. 10.17) Ignoringtheconstraintthe SU (2)actslin earlyonthese“elds(seebelow): z1= i z1+ z2 z2= Š i z2Š z1.(3. 10.18) Thecomplicatednonlineartransformations(3.10.12)ariseinthefollowingmanner: whenwe“xthe U (1)gauge z1= z1 (3.10.19)

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3.10.Compensators115thelinear SU (2)transformations(3.10.18)donotpreservethecondition(3.10.19).Thus wemustadda  gauge-restoring U (1)transformationwithparameter i ( x )= Š1 2 Š 1( z1Š z1)= Š i Š1 2 Š 1( z2Š z2).(3. 10.20) Thecombinedlinear SU (2)transformationandgaugetransformation(3.10.16)with nonlinearparameter(3 .10.20)preservesthegaugecondition(3.10.19)andareequivalent to(3.10.12). To writeanactioninvariantunderboththeglobal SU (2)andthelocal U (1)transformationsweneedacovariantderivative forthela tter.Byanalogywithour“rst examplewecouldwrite a= eŠ i aei = a+ i a .(3. 10.21) Amanifestly SU (2)invariantchoiceintermsofthenewvariablesis a= aŠ1 2 zi azi= a+ i a Š1 2 y ay .(3. 10.22) Thisdiersfrom(3.10.21)bythe U (1) gaugeinvariant term y ay ;oneisalwaysfreeto chan geacovariantderivativeby a ddingcovarianttermstotheconnection.(ThisissimilartoaddingcontortiontotheLorentzconnectionin(super)gravity;seesec.5.3.a.3.) ThenamanifestlycovariantLagrangianis IL = Š| azi|2= Š| azi|2Š1 4 ( zi azi)2.(3. 10.23) Inthegauge(3.10.19)thisLagrangianbecomesthatof(3.10.13). Weconsid ernowanotherapplicationofcompensators:Theconstraint(3.10.17)is awkward:Itmake sthetransf ormations(3.10.18)implicitlynonlinear.Wecanavoid thisbyintroducinga s econd compensating“eld.Weobservethatneithertheconstraint northeLagrangianareinvariantunderscaletransformations.However,wecanintroduceascaleinvarianceintothetheorybywriting zi= eŠ Zi(3.10.24)

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1163.REPRESENTATIONSOFSUPERSYMMETRYintermsof new“elds Ziandthecompensator ( x ).Theconstraintandtheaction, writteni nte rmsof Zi, ,w ill be in va riantunderthescaletransformations Z i= eZi, = + .(3. 10.25) The SU (2)transformationsof Ziarenowthe(truly)lineartransformations (3.10.18).The U (1)andthescaletransformationscanbecombinedintoasinglecomplexscaletransforma tionwithparameter = + i (3.10.26) Z i= eZi, = + .(3. 10.27) Theconstraint(3.10.17)becomes Z Z = e2 (3.10.28) wherew ewrite Z Z | Z1|2+ | Z2|2.Inte rmsofthenewvariablestheLagrangianis IL = Š| a( eŠ Zi) |2= Š| a( eŠ Zi) |2Š1 4 eŠ 4 ( Zi aZi)2.(3. 10.29) Substitutingfor thesolutionoftheconstraint(3.10.28),amanifestly SU (2)inv ariant procedure,leadsto IL = Š| aZi Z Z |2Š1 4 ( Zi aZi)2( Z Z )2 = Š 1 Z Z ( i kŠ Zi ZkZ Z )1 2 ( a Zi)( aZk)(3. 10.30) ThislastformoftheLagrangianisexpressedintermsofunconstrained“elds Zionly.It ismanifestlyglobally SU (2 )i nv ar ia nt an da ls oi nv ar ia nt underthelocalcomplexscale transformations(3.10.27).Wecanusethisinvariancetochooseaconvenientgauge.For example,wecanchoosethegauge Z1=1;orwecanchoo seagaugeinwhichweobtain (3.10.13).Oncewechooseagauge,the SU (2)transformationsbecomenonlinearagain. Thesetwocompensatorsallowedustorealizeaglobalsymmetry( SU (2))ofthe systemlinearly.However, theyplay dierentroles: ( x ),the U (1)compensator,gauges

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3.10.Compensators117aglobalsy mmetryofthesystem,whereas ( x ),thescaleco mpensatorintroducesan altogethernewsymmetry.Forthe U (1)invariancewei ntrodu cedaconnection,whereas forthescaleinvarianceweintroduced ( x )dir ectly,withoutaconnection.Intheformer case,theconnectionconsistedofapuregaugepart,andacovariantpartchosentomake it manifestlycovariantunderasymmetry( SU (2))ofthesystem;hadwetriedtointroduce ( x )dir ectly,wewouldhavefounditdiculttomaintainthe SU (2)invar ian ce.In thecaseofthescaletransformationsnosuchd icultiesarise,andaconnectionisunnecessary.Asweshallsee,bothkindsofcompensa torsappearinsupers ymmetrictheories. c.Cosetspaces Compensatorsalsosimplifythedescriptionofmoregeneralnonlinear -models. Weconsid eramodelwith“elds y ( x )thatarepoints ofacosetspace G / H ;theyt ransformnonlinearlyundertheglobalactionofagroup G butlinearlywithrespecttoa subgroup H .Byintro ducinglocaltransformationsofthesubgroup H viacompensators ( x ),werealize G linearly,andthuseasily“ndaninvariantaction. Thegeneratorsof G are T S ,where S arethegeneratorsof H and T arethe remaininggenerators,with T S antihermitian.Since H isas ubgroup,thegenerators S closeundercommutation: [ S S ] S .(3. 10.31) Werequireina ddition thatthegenerators T carryarepresentationofthe H ,thatis [ T S ] T .(3. 10.32) (Thisisalwaystruewhenthestructureconstantsaretotallyantisymmetric,sincethen theabsen ceof[ S S ] T termsimpliestheabsenceof[ T S ] S terms.) Wecouldwrite y ( x )= e ( x ) TmodH butinsteadweintroducecompensating“elds ( x ),andde “ne“elds z ( x )thatare elementsofthewholegroup G : z = e ( x ) Te ( x ) S e(3.10.33) (where= ( x ) T + ( x ) S providesanequivalentparametrizationofthegroup).The new“elds z tr an sformunder globalG -transformationsand localH -transformations: z= gzhŠ 1( x ), gG hH (3.10.34)

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1183.REPRESENTATIONSOFSUPERSYMMETRY(whereagainwecanuseanexponentialparametrizationfor g and h ( x )ifwewish). Thelocal H transformationscanbeusedtogaugeawaythecompensators and reduce z tothecosetvariables y .Ifwechoos ethe gauge =0,thent hegl obal G -transformationswillinducelocalgauge-restoring H -transformationsneededtomaintain =0:For gH dueto (3.10.32),weuse h ( x )= g : eT= ge TgŠ 1(3.10.35) andthust he“elds y tr an sformlinearlyunder H .For gG / H ,the gaugerestoringtransformationiscomplicatedanddependsnonlinearlyon ,andth usthe“elds y transform nonlinearlyunder G / H To “ndaglobally G -andlo ca lly H -invariantLagrangian,weconsiderthefollowing quantity: zŠ 1 az a+ A aS + B aT a+ B aT .(3. 10.36) U nderglobal G -transformations,both aand B aar ei nv ar ia nt ; underlocal H -transformationswehave ( zŠ 1 az )= hzŠ 1 a( zhŠ 1) = h ahŠ 1+ hzŠ 1( az ) hŠ 1= h ahŠ 1+ h ( A aS + B aT ) hŠ 1= h ( a+ B aT ) hŠ 1(3.10.37) Becauseof(3.10.31), hShŠ 1 S and h ahŠ 1 S ;b ecauseof(3.10.32), hThŠ 1 T ; hence A atransformsasaconnectionforlocal H transformations( atransformsasa covariantderivative),and B atransformscovariantly.Therefore,aninvariantLagrangian is IL = Š1 4 tr ( B aB a)(3. 10.38) Ifwechoosethegauge ( x )=0,thisb ecomesacomplicatednonlinearLagrangianfor the“elds y ( x ).Wecanalsoc o uplethissystemtoother“eldstransforminglinearly under H byreplacin gallderiva tiveswith a. Finally,from(3.10.33)wehave zŠ 1 az = a+ eŠ S( ae S)+ eŠ S(eŠ T ae T)e S

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3.10.Compensators119= a+( a ) S +( a ) T + ... (3.10.39) andhence a= a+ a S + ... and B a= a T + ... = ayT + ... .Thisisw hatwe expect:Thecovariantderivativehastheusualdependenceonthecompensator,andthe Lagrangian(3.10.38)hasaterm Š1 2 tr ( ay )2,whichisa ppropri ateforaphysical“eld.

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1203.REPRESENTATIONSOFSUPERSYMMETRY3.11.Projectionoperators a.General Theanalysisofmanyaspectsofthesuperspaceformulationofsupersymmetric th eoriesrequiresanunderstandingoftheirreduciblerepresentationsof(o-shell)supersy mmetry(physicalandauxiliarycomponents).Weneedtoknowhowtodecomposean arbitrarysuper “eldorproductofsuper“eldsintosuchrepresentations.Inthissection wedescribeap rocedureforconstructingprojectionoperatorsontoirreduciblerepresentationsofsupersymmetryforgeneral N Thebasicideaisthatageneralsuper“eldcanbeexpandedintoasumofchiral super“elds.Achiralsuper“eldthatisirreducibleunderthePoincar eandinte rnalsymmetrygroupsisalsoirreducibleundero-shellsupersymmetry(exceptforpossibleseparationintorealandimaginaryparts,whichwecall bisection). Thus,thisexpansionperformsthedecomposition. To showthatchiralsuper“eldsareirreducibleundersupersymmetryuptobisection,wetrytoreduceachiralsuper“eld byimposing somecovariantconstraint =0.Ifwedonotconsid errealityconditions(bisection),wecannotallowconstraints relatingto .Theonlycovariantoperatorsavailableforwritingconstraintsarethe spinorderi vatives Da Da€andthespacetimederivative €.Inmomen tums pace, sinceweareo-shell,allrelationsmustbetrueforarbitrarymomentum,andhencewe canfreelydivideoutanyspacetimederivativefactors.Therefore,anyconstraintwe writedowncanbereducedtoaconstraintthatisfreeofspacetimederivatives.Ifthe constraintco ntainedany D spinorderivatives,sinceischiral, D =0,bymovingthe D stotherightwecouldconvertthemtosp acetimederivatives,whichwehavejust arguedcanberemoved.(Forexample D DD = iD .) Wethusco ncludethatanypossibleconstraintoninvolvesonlyproductsofthe spinorderi vatives Da .Howev er,byapplyingasucientnumberof D stotheconstraint,wecanconvertallofthe D stospacetimederivatives ;hen ce,anyconstrainton i ndependentof w ouldsetitselftozero(o-shell!).Thereforemustbeirreduc ible.Thisargumentisanalogoustotheproofinsection3.3thatirreduciblerepresentationsofsupersymmetrycanbeobtainedbyrepeatedlyapplyingthegenerators Qa€to theCliordvacuum | C > de“nedby Qa | C > =0:ins teadof | C > Q ,and Q with

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3.11.Projectionoperators121Q | C > =0,wehave, D ,and D with D =0,resp ectively. Theonlyfurtherreductionwecanperformistoimposearealityconditiononthe super“eld.Achiralsuper“eldofsuperspin s (thespincontentofitsexternalLorentz i ndices)hasa single maximumspincomponentofspin smax= s +1 2 N residingatthe N [ a1... aN]( 1... N)levelofthesuper“ eld.(Thisismosteasilyseeninthechiralrepresentation,where achirals uper“elddependsonlyon .There ductionofproductsof sinto irreduciblerepresentationsi sdonebythemeth oddescribedforther eductionofproducts ofspinorderivativesinsec.3.4.Sincethe maximumspincomponenthasthemaximum numberof sy mmetrizedSL (2 C )i ndices,itmusthavethemaximumnumberof antisymmetrizedSU ( N )i ndices,i.e .,itmusthave N i ndicesofeachtype.Termswithfewer s havefewer SL (2 C )i ndices,whereastermswithmore scannotbeantisymmetricin N SU ( N )i ndices,andhencecannotbesymmetricin NSL (2 C )i ndices.Forexamplessee (3.6.1-4)).Onlyifwecanimposearealityconditiononthehighestspincomponentcan weimposearea lityconditionontheentiresup er“eld.Thisispossiblewhen smaxisan integer.(Acomponen t“eldwithanoddnumb erofWeylindicescannotsatisfyalocal realitycondition.) a.1.Poincar eprojectors Webeginwitht hedecompositionofanarbitrarysp inorintoirreduciblerepresentationsofthePoincar egroupinor dinaryspacetime,bothbecauseitisoneofthestepsin thesuperspacedecomposition,andbecauseitillustratessomeofthesuperspacefeatures. Thisreductionismosteasilyperformedbyconvertingdottedindicesintoundottedones withtheformaloperator€= Š i € Š1 2 ,r educingunder SU (2)(bysymmetrizingand antisymmetrizing,i.e.,takingtraces),andconvertingformerlydottedindicesbackwith .(Thisinsuresthatnofractionalpowersof remain.Wegenerallyconsider Š1 2 to behe rmitian,sincewemainlyareconcernedwith = m2> 0.)Explicitly,wewritefor eachindex =€€, €=€, (  )=( ), =.(3. 11.1) Thus,forexample,avector adecomposesinthefollowingmanner:

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1223.REPRESENTATIONSOFSUPERSYMMETRY a=€ =€1 2 ( C + ( )) =€1 2 (€C€+( € )€) =1 2 Š 1[ €( €€) Š €( ( € )€)] =[(L+T)] a,(3. 11.2) whereLandTarethelo ngitudinalandtransverseprojectionoperatorsforafour-vector. Theprojectionscanbewrittenintermsof “eldstrengthsS and F: (L) a= Š 1€S S =1 2 €€, (T) a= Š 1€F, F=1 2 ( € )€.(3. 11.3) The“eldstrengthsarethemselvesirreduciblerepresentationsofthePoincar egro up. TheprojectionsL=LandT=Ta reinvariantundergaugetransformations =T and =L respectively.The“eldstrengthshavethesamegaugeinvarianceastheprojections: L a= Š 1 a S =0imp lies S =0,andsi milarly T a= Š 1€ F=0imp lies F=0. a.2.Super-Poincar eprojectors Projectionsofsuper“eldscanbewritteni nte rmsof“eldstrengthsinsuperspace aswell.Wewill“ndthatprojectionsofageneralsuper“eldcanbeexpressedinterms of chiral “eldstrengthswithgaugeinvariancesdeterminedbytheprojectionoperators. Thus,forasuper“eldwithdecomposition=(nn),anysingletermn=nhas a gaugeinvariance =i = nii.Eachproj ectioncanbewri tteni ntheform n= D2 N Š n( n )wherethechiral“eldstrengths( n )= D2 NDnarePoincar eand SU ( N )i rreducibleandhavethesamegaugeinvarianceasn:0= n= D2 N Š n ( n )implies ( n )=0b ecauseandhence arei rreducible.

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3.11.Projectionoperators123Thesameindexconversionusedin(3.11.1)canbeusedtode“netheoperationof rest-frameconjugation onacomponent“eldor genera lsuper “eld1... i€i +1...€2 sby 1... i€i +1...€2 s=1€1... i€ii +1€i +1... 2 s€2 s 2 s... i +1€i...€1, =.(3. 11.4) Forexample,wehave: = ,=€ €, H€=€€ H€.(3. 11.5) Weextend thistochiralsuper“eldsandde“nearest-frameconjugationoperator K K whichpreserveschirality,byusinganextrafactor Š1 2 N D2 Ntoconv erttheantichiral (complexconjugatedchiral)s uper“eldbacktoachiralone(andsimilarlyforantichiral super“elds).Wede“ne K K 1... i€i +1...€2 sa1... aib1... bi= D2 N Š1 2 N1... i€i +1...€2 sb1... bia1... ai, K K ...= D2 N Š1 2 N ..., K K (...)= ( K K ...), K K2=1,[1 2 (1 K K )]2=1 2 (1 K K ),(3. 11.6) where D€ ...= D ...=0.Forex ample,foran N =1chiral spinor, K K = Š D2i € €,(3. 11.7) Wecand e“ne self-conjugacy or reality under K K ifwerestrictourselvestosuper“elds thatarerealrepresentationsof SU ( N )with smax= s +N 2 integral(thelatterisrequired toinsurethatonlyintegralpowersof appear).Therealityconditionis K K ...= ...andthesplitti ngofachiralsuper“eldintorealandimaginarypartsissimply ...=1 2 (1 K K )....(3. 11.8) Inthepreviousexample,ifwei mposetherealitycondition K K =,contr actboth sideswith Dandusetheantichiralityof €,we “ndtheequivalentcondition: D= D€ €(3.11.9)

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1243.REPRESENTATIONSOFSUPERSYMMETRYTheserealchiralsuper“eldsappearinmanymodelsofinterest.Forexample,isoscalar realchiralsuper“eldswith2 Š N undottedspinorindicesdescribe N 2Y ang-Mills gaugemultiplets.Similarsuper“eldswith4 Š N undottedspinorindicesdescribethe conformal“el dstren gth of N 4supergravity. Tod ecomposeageneralsuper“eldintoirreduciblerepresentations,we“rstexpand itintermsofchiralsuper“elds.Inthe chiralrepresentation ( D€ = € )aTaylorseriesin gives ( x , )=2 N n =0 1 n n €1... €n( n ) €n... €1( x ), (3.11.10a) wherencanberewrittenas ( n ) €1... €n( x )= Dn €1... €n( x , ) | =0,(3. 11.10b) or,using { D€ € } = € € and { € € } =0(which implies 2 N +1=0), ( n ) €1... €n( x )=( Š 1)N D2 N 2 N Dn €1... €n( x , ).(3. 11.10c) However, isnotcovariant,andhenceneitheristheexpansion(3.11.10).Wecangeneralize(3.11.10):For anyoperator € ( )whichobeys { D€ € } = € € { € € } =0,(3. 11.11) wecanwrite ( x , )=2 N n =0 1 n n €1... €n ( n ) €n... €1( x ), (3.11.12a) where ( n ) €n... €1( x )=( Š 1)N D2 N 2 N Dn €n... €1 ( x , ).(3. 11.12b) Ifwechoose( x , ) ( x , ( )),weobtain,substituting(3.11.12b)into(3.11.12a): ( x , )=( Š 1)N 2 N n =0 1 n n €1... €n D2 N 2 N Dn €n... €1( x , ),(3. 11.13) forany satisfying(3.11.11).Amanifestlysupers ymmetricoperatorsatisfying(3.11.11)

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3.11.Projectionoperators125is € = Š i € D = € Š i € .(3. 11.14) Substituting(3.11.14)into(3.11.13),we“nd ( x , )= Š N 2 N n =0 1 n Dn 1... n Š i 1 €1 ... Š i n €n D2 ND2 N Dn €n... €1( x , ),(3. 11.15) whereweh aveused( Š i € D )2 N=( Š )Š ND2 N.Pus hing Dnthrough D2 Ntothe D2 N,we “nd(reorderingthesumbyreplacing n 2 N Š n ) = Š N1 (2 N )! C 1... 2 N2 N n =0 ( Š 1)n 2 N n D2 N Š n 2 N... n +1 D2 NDn n... 1( x , ) = Š N 2 N n =0 1 n ( Š 1)nD2 N Š n 1... n D2 NDn 1... n( x , ).(3. 11.16) This“nalexpressioncanbecomparedtothenoncovariant expansionin(3.11.10). Thechiral“elds D2 NDnarethecovaria ntan alogsofthe(2 N Š n )s.Wethusobtain 1=2 N n =0 1 n ( Š 1)n Š ND2 N Š n 1... n D2 NDn 1... n.(3. 11.17) Forexample,in N =1this istherelation 1= D2 D2 Š D D2D + D2D2 .(3. 11.18) Eachterminthesumisa(reducible)proj ectionoperatorwhichpicksoutthepart ofasuper “eldappearinginthechiral“eldstrength D2 NDn(whichisi rreducible under SL (2 N C ) butreducibleunder SU ( N ) Poincar e,andpossiblyalsounder K K ). Wethus havetheprojectionoperatorsn, n =0,1,. ..,2 N : n= 1 n ( Š 1)n Š ND2 N Š n 1... n D2 NDn 1... n,

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1263.REPRESENTATIONSOFSUPERSYMMETRY2 N n =0 n=1.(3. 11.19) Inparticular,0= Š ND2 N D2 Nand2 N= Š N D2 ND2 Nprojectoutthean tichiraland chiralpartso fresp ectively.Theprojectors(3.11.19) satisfyanumberofrelations: Orthon ormality mn= mnm(notsummed)(3.11.20) followsfrom D2 NDn D2 N=0 unless n =2 N andhencemn=0for m = n ;then m=1imp liesn=n m=2 n.Therearerel ationsbetweenthes:nis equaltothetransposeandtothecomplexconjugateof2 N Š nn=t 2 N Š n=1 (2 N Š n )! Š ND2 N Š n 1... 2 N Š n D2 NDn 1... 2 N Š n,(3. 11.21) n=* 2 N Š n=1 (2 N Š n )! ( Š 1)n Š N Dn €1... €2 N Š nD2 N D2 N Š n €1... €2 N Š n.(3. 11.22) Combining(3.11.21)and(3.11.22),we“ndanotherformofn: n= n= 1 n Š N Dn €1... €nD2 N D2 N Š n €1... €n.(3. 11.23) Thecomplexconjugationrelation(3.11.22)impliesthathalfofthesareredundantfor realsuper“elds: V = V 2 N Š nV =* n V = (nV ). Reduction ofthesintoirreducibleprojectionoperatorsisnoweasy: (1)Algebraicallyreduce D2 NDn under SU ( N ) Poincar e(wheremayhavefurther isospi norandWeylspinorindices); (2)Whent hereducedchiral“eldstrength D2 NDnisinareal representationof SU ( N ) andhas s +1 2 N integral,furtherreducebybisection,i.e.multiplicationby1 2 (1 K K ). Toperform( 1)itisconvenientto“rstreduce D2 NDnbyusingt heto talantisymmetryof the D s(sees ec.3.4),andthenreducethetensorproductoftheirreducible

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3.11.Projectionoperators127representationsof D2 NDnwiththerepresentationofthesuper“eldasusual.Ifwe onlywanttopreserve SO ( N ),furtherre ductionisperformedi nstep1;f orstep2, D2 NDnisalway sinareal representationof SO ( N ). Althoughncontainstheproductof2 ND sand2 N D sandisthu sinitssimplestform,n ,obtai nedbydir ectly introducing 1 2 (1 K K )infrontofthe D2 N,contains 2 ND sand4 N D sinthe K K term,andcanbefurthersimpli“ed.Aftersomealgebra we “nd: For n N : K K D2 NDn 1... nb1... bn=( Š 1)2 sn 1 2 ( n Š N ) D2 ND2 N Š n 1... nC11... Cnna1... an,(3. 11.24) or K K D2 NDn 1... 2 N Š nb1... b2 N Š n=( Š 1)2 sn 1 2 ( n Š N ) D2 NC11... C2 N Š n2 N Š nD2 N Š n 1... 2 N Š na1... a2 N Š n,(3. 11.25) where2 s extraWeylspinorindices,andextraisospinorindices,reducedasinstep1,are impliciton. For n N : K K D2 N D2 N Š n €1...b1...=( Š 1)2 sn 1 2 ( N Š n )D2 N Dn€1...C€1€1...a1...,(3. 11.26) or K K D2 N D2 N Š n €1...b1...=( Š 1)2 sn 1 2 ( N Š n )D2 NC€1€1... Dn €1...a1....(3. 11.27) Asanexampleofthi ssimp li“cation,weconsiderthe N =1chiral“e ldabove(3.11.7)for thesp ecialcasewhenitisa“eldstrengthofarealsuper“eld V := D2DV K K = Š D2i € D2 D€V = D2DV =.(3. 11.28) Wenowcoll ectourresults:Thesuperprojectorstakethe“nalform Ifbisectionispossible: n N :n i =1 n Š N Dn €1... €n1 2 (1 K K ) IPiD2 N D2 N Š n €1... €n,

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1283.REPRESENTATIONSOFSUPERSYMMETRYn N :n i =1 (2 N Š n )! Š ND2 N Š n 1... 2 N Š n1 2 (1 K K ) IPi D2 NDn 1... 2 N Š n,(3. 11.29) Ifbisectionisnotpossible: n i=eithero ftheabovewith1 2 (1 K K )dro pped,(3 .11.30) wherethe IPiare SU ( N ) Poincar eproj ectorsactingontheexplicitindices(including thoseofthesuper“el d).Wehavechosentheparticularformsofnfrom(3. 11.19,21-23) thatminimizethenumberofindicesthatthe IPiacton.Thechiralexpansion,besides itssimplicity,hastheadvantagethatthechiral“eldstrengthsappearexplicitly,andthe superspinandsuperisospinoftherepresentationontowhichprojectsarethoseofthe chiral“e ldstrength. b.Examples b.1.N=0 Webeginbyg ivingafewPoincar ep rojectionoperators.Theprocedurefor“ndingthemwas discussedinconsiderabledetailins ubsec.3. 11.a.1,soherewesimplylist results.Scalarsandspinorsareirreducible(nobisectionispossibleforaspinor: s +N 2 =1 2 isnotaninteg er).A(real)vectordecomposesintoaspin1andaspin0projection(see(3. 11.2,3)).Foraspinor-vector € =€ wehave: =1 3! ( )+1 3! ( ( ) C+ ( ) C)+1 2 C andhence € =(3 2 +T1 2 +L1 2 ) € , where 3 2 € = Š Š 1€w, w=1 6 ( €€ ); T1 2 € = Š Š 1€[ Cr+ Cr], r=1 6 ( € )€ ; L1 2 € = Š 1€s, s=1 2 a a .(3. 11.31)

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3.11.Projectionoperators129Forareal two-i ndextensor h a c=€€h wehave h =1 4! h( )+( C ( q ) + C ( q ) )+ CCq + C ( C ) r +1 4 ( Ch ( | | )+ Ch( | | ) ),(3. 11.32) where q= Š1 32 ( h( | | ) + h( | | ) + h( | | ) + h( | | ) ), r = Š1 24 h( ( ) ), q =1 4 h .(3. 11.33) Thereforethecompletedecompositionofthethetwo-indextensorisgivenby h€ €=(2, S+1, S+L 0, S+T 0, S+1, A ++1, A Š) h€ €, wheretheprojectorsarelabeledbythespin(2,1,0),thesymmetricandantisymmetric partof h a b( S and A ),longit udinalandtransverseparts( L and T ),andsel f-dualand anti-self-dualparts(+and Š ).Theexplicitformoftheprojectionoperatorsis 2, Sh€ €= Š 2€€w, w=1 4! ( €€h€ )€; 1, Sh€ €= Š 2€€[ C ( w ) + C ( w ) ], w= Š1 32 (€[ € )h( € )€+ € )h( € )€]; L 0, Sh€ €= Š 2€€S S =1 4 €€h€ €; T 0, Sh€ €= Š 2( CC€€ + €€) T T = Š1 12 ( € )€h€ €; 1, A +h€ €= Š 1€€l+ ( ), l+ ( )= Š1 4 h( € )€; 1, A Šh€ €= Š 2€€lŠ ( ), lŠ ( )= Š1 4 ( € )€h€ €.(3. 11.34) (The “eldstrengths wand T areproportionaltothelinearizedWeyltensorand scalarcurvaturesrespectively.)Fromthisdecomposition,weseethatthetwo-index

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1303.REPRESENTATIONSOFSUPERSYMMETRYtensor“eldconsistsofirreduciblespins2+1+1+1+0+0. b.2.N=1 Weconstr ucttheirreducibleprojectionoperatorsforacomplexscalarsuper“eld .From(3.9 .26-32)wehave,forthecaseswithoutbisection( s +1 2 N = s +1 2 ishalfintegral,sothat s isintegral) 0= Š 1D2 D2,1= Š Š 1D D2D,2= Š 1 D2D2.(3. 11.35) Sincehasnoexternalindiceswecangodirectlytostep2.Thechiral“eldstrengths D2and D2D2donotsati sfytheconditionthat s +1 2 isintegral,whereas D2D does.For N =1,theco ndition ofbeinginarealisospinrepresentationistriviallysatis“ed,andthatmeansthat1needstobebisected: 1=1++1 Š, 1 = Š Š 1D1 2 (1 K K ) D2D.(3. 11.36) Therefore,from(3.11.26-7), 1 = Š Š 1D D2D1 2 ( ),(3. 11.37) andthus0,1 and2completelyreduce.Theseirreduciblerepresentationsturn outtodescribetwoscalarandtwovectormultiplets,respectively. Wegivenextth ed ecomposit ionofthespinorsuper“eld.To “ndtheirreduciblepartsof1wePoincar ere ducethechiral“eldstrength D2D=1 2 [ C D2D+ D2D( )].Thisgivestheprojectionsn sforsuperspin s ofthischiral“eldstrength: 1,0=1 2 Š 1D D2D,1,1= Š1 2 Š 1D D2D( ).(3. 11.38) Thelatterirreduciblerepresentationisaconformalsubmultipletofthe(3 2 ,1)multiplet(seesection4.5).For0and2wemustbi sect: 01 2 = Š 11 2 (1 K K ) D2 D2= Š 1D21 2 ( D2 i € €),

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3.11.Projectionoperators13121 2 = Š 11 2 (1 K K ) D2D2= Š 1 D21 2 ( D2 i € €).(3. 11.39) Equivalentformsare: 01 2 = Š Š 1D1 2 ( D D2 D€D2 €), 21 2 = Š 1 D2D1 2 ( D D€ €).(3. 11.40) Finallywedecomposetherealvectorsuper“eld H€.B ecauseofitsrealitybisectionisunnecessary.Poincar eproj ectionisperformedbywriting H€=€Hand (anti)symmetrizingintheindicesofthechira l“eldstr engths.Toensurethattheprojectionoperatorsmaintaintherealityof H€,wecombi nethe2swiththe0s,since from(3.9 .24)2H€=(0H€).Weobtain T 0,1H€=1 2 Š 1€{ D2, D2} H( ), L 0,0H€=1 2 Š 1€{ D2, D2} H T 1,3 2 H€= Š1 6 Š 1€D D2D( H ), T 1,1 2 H€=1 6 Š 1€( D D2DH( )+ D D2DH( )), L 1,1 2 H€= Š1 2 Š 1€D D2DH ,(3. 11.41) where T and L denotetransverseandlongitudinal.Reexpressing Hintermsof H€, we “nd T 0,1H€=1 2 Š 2€{ D2, D2} ( €H )€, L 0,0H€=1 2 Š 2€{ D2, D2} cH c, T 1,3 2 H€=1 6 Š 2€D D2D( €H )€,

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1323.REPRESENTATIONSOFSUPERSYMMETRYT 1,1 2 H€=1 6 Š 2€( D D2D( €H )€+ D D2D( €H )€), L 1,1 2 H€= Š1 2 Š 2€D D2D dH d.(3. 11.42) b.3.N=2 Webeginbygiv ingtheex pressi onsfor SL (4, C ) C sintermsofthoseof SU (2)and SL (2, C ): C = CabCcdCCŠ CadCcbCC, C € € € €= CabCcdC€€C€€Š CadCcbC€€C€€, C = CabCcdCCŠ CadCcbCC, C € € € €= CabCcdC€€C€€Š CadCcbC€€C€€.(3. 11.43) Wede “nethe SU (2) Poincar ere ducti onof D2 asfollows: D2 = CD2 ab+ CbaD2 D2 € €= C € € D2 ab+ Cba D2€€, D2 = CCacCdbD2 cd+ CbaD2 D2 ab=1 2 Da Db = D2 ba=( D2 ab), D2 =1 2 CbaDa Db = D2 = Š ( D2€€).(3. 11.44) Thesetof(possibly)reducibleprojectionoperatorsis: 0,0= Š 2D4 D4,4,0= Š 2 D4D4, 3,1 2 = Š 2D D4D3 ,1,1 2 = Š Š 2D3 D4D 2,0= Š 2CcaCbdD2 ab D4D2 cd,

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3.11.Projectionoperators1332,1= Š 2D2 D4D2 .(3. 11.45) Inwriting2,0and2,1wehavetaken2de“nedby(3.11. 19)andused(3.11.44)tofurtherreduceit.Weca nnowd ecompose N =2super“elds. Westartwit hacomplex N =2scal arsuper“eld.Weneednotbisecttheterms obtainedfrom1,1 2 and3,1 2 .Bis ectingtherest,we“ndeightmoreirreducibleprojections. 0,0 = Š 21 2 (1 K K ) D4 D4= Š 2D41 2 ( D4 ), 4,0 = Š 21 2 (1 K K ) D4D4= Š 2 D41 2 ( D4 ), 2,0 = Š 2CcaD2 ab1 2 (1 K K ) D4CbdD2 cd= Š 2CcaCbdD2 ab D4D2 cd1 2 ( ), 2,1 = Š 2D2 1 2 (1 K K ) D4D2 = Š 2D2 D4D2 1 2 ( ).(3. 11.46) Wegivetwom oreresultswithoutdetails:Forthe N =2v ectormult iplet,describedby areals calar-isovectorsuper“eld Va bwe “nd 0,0,1 Va b= Š 2D4( D4 ) Va b, 1,1 2 ,3 2 Va b=1 3! Š 2CdbD3 c D4Ce ( aDc Vd ) e, 1,1 2 ,1 2 Va b=1 3 Š 2CdbD3 c D4De Cc ( aVd ) e, 2,1,1Va b= Š 2D2 D4D2 Va b, 2,0,2Va b= Š1 4! Š 2CbgCceCfdD2 ef D4D2 ( cdVa hCg ) h, 2,0,1Va b= Š1 4 Š 2CcdCbeD2 d ( a | D4( D2 | e ) fVc f+ D2 cfV| e ) f), 2,0,0Va b=1 3 Š 2CbcD2 ac D4CfeD2 deVf d,(3. 11.47) wheretheprojectionoperatorsarelabeledbyprojectornumber,superspin,superisospin, and K K conjugation .A gain,toconstructrealprojectionoperators,thecomplex

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1343.REPRESENTATIONSOFSUPERSYMMETRYconjugatemustbeaddedforthe0sand1s(0 0+4,1 1+3).Finally, forthespinor-isospinorsuper“elda (theunconstrainedprepotentialof N =2supergravit y)we “nd 0,1 2 ,1 2 a = Š 2D4 D4a 4,1 2 ,1 2 a = Š 2 D4D4a 2,1 2 ,3 2 a =1 3! Š 2D2 de D4Cb ( dCe | cD2 bc| a ) 2,1 2 ,1 2 a =2 3 Š 2CadCecD2 de D4D2 cbb 2,3 2 ,1 2 a =1 3! Š 2D2 D4D2 ( a ), 2,1 2 ,1 2 a =2 3 Š 2D2 D4D2 a 1,1,1 a =1 8 Š 2 Db€D4€( Š 1 D3 ( b (€€ )a ) Š+ i D( a (€ b )€ )), 1,1,0 a =1 8 Š 2 Da€D4€( Š 1 D3 b (€€ )b Š+ i Db (€ b€ )), 1,0,1 a = Š1 8 Š 2 Db€D4€( Š 1€ D3 ( b€a ) Š+ i D( a€ b )€), 1,0,0 a = Š1 8 Š 2 Da€D4€( Š 1€ D3 b€b Š+ i Db€ b€), 3,1,1 a =1 4 Š 2Db D4D2 Cc ( a( Dc b ) Db )€ c€), 3,1,0 a =1 4 Š 2CabDb D4D2 ( De e De€ e€), 3,0,1 a =1 4 Š 2CbdDb D4CacD2 dc( De e De€ e€), 3,0,0 a =1 12 Š 2CabDb D4CcdD2 ce( Dd e De€ d€). (3.11.48a)

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3.11.Projectionoperators135Thereare22irreduciblerepresentations.Onesimpli“cationispossible:Using(3.9.21) insteadof(3.9.25)forjustthe“rsttermin1 K K ,we “nd 1,1,1 a =1 8 Š 2( Š D3 b D4D( b ( a ) )Š+ i € Db€D4 D( a (€ b )€ )), 1,1,0 a =1 8 Š 2( Š D3 a D4De ( e )Š+ i € Da€D4 De (€ e€ )), 1,0,1 a = Š1 8 Š 2( D3 b D4D( b a ) Š+ i € Db€D4 D( a€b )€), 1,0,0 a = Š1 8 Š 2( D3 a D4De e Š+ i € Da€D4 De€e€).(3. 11.48b) b.4.N=4 Webeginbyde “ningasetofirreducible D -operators: D2 = CD2 ab+ D2 [ ab ] D3 = Cd cbaD3 d +( CD3 [ ac ] b Š CD3 [ ab ] c ) D4 = Cd cbaD4 +1 2 ( CCCcdefD4 [ ab ] [ ef ]Š CCCbcefD4 [ ad ] [ ef ]) +( CCeacdD4 b e Š CCeabdD4 c e + CCeabcD4 d e ) D5 = Cd cbaD5 d +( CD5[ ac ] b Š CD5[ ab ] c ) D6 = CD6 ab+ D6[ ab ] .(3. 11.49) Theysatisfythefollowingalgebraicrelations D2 ab= D2 ba, D6 ab= D6 ba, D4 a a = D4 [ ab ] [ cb ]= Ca bcdD3 [ ab ] c = Ca bcdD5[ ab ] c =0.(3. 11.50) All SL (2, C )i ndicesonthe Dnsaretotallysymmetric.Wealsohave D4 1... 4=1 4! C 1... 8D4 5... 8,

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1363.REPRESENTATIONSOFSUPERSYMMETRYD4 1... 4=1 4! C 1... 8D4 5... 8,(3. 11.51) andthese imply D4 = Cd cbaD4 +1 2 ( CCCcdefD4 [ ef ] [ ab ]Š CCCbcefD4 [ ef ] [ ad ]) +( CCeacdD4 e b Š CCeabdD4 e c + CCeabcD4 e d ),(3. 11.52) ascanbeveri“edbysubstitutingexplicitvaluesfortheindices. Weconsid ernowacomplexscalar N =4super“eld and “nd“rst 0,0,1= Š 4D8 D8,8,0,1= Š 4 D8D8, 1,1 2 ,4= Š 4 D€ D8 D7 €,7,1 2 4= Š 4D D8D7 2,0,10= Š 4 D2 abD8 D6 ab, 2,1,6=1 2 Š 4 D2[ ab ]€€D8 D6 [ ab ]€€, 3,3 2 4= Š 4 D3 a€€€D8 D5 a€€€, 3,1 2 ,20=1 3! Š 4 D3[ ab ] c€D8 D5 [ ab ] c€, 4,2,1= Š 4 D4€€€€D8 D4€€€€, 4,1,15= Š 4 D4 a b€€D8 D4 b a€€, 4,0,20= Š 4 D4 [ cd ] [ ab ]D8 D4[ cd ] [ ab ], 5,3 2 ,4= Š 4D3 a D8D5 a 5,1 2 20=1 3! Š 4D3 [ ab ] c D8D5[ ab ] c 6,0, 10= Š 4D2 ab D8D6 ab,

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3.11.Projectionoperators1376,1,6=1 2 Š 4D2 [ ab ] D8D6[ ab ] ,(3. 11.53) wherethesuperisospinquantumnumberherereferstothedimensionalityofthe SU (4) representation.Theonlyprojectorsthatneedbisectionaretherealrepresentationsof SU (4 ): th e1 ,6 15, an d 20 .W e “nd: 0,0,1 = Š 4D81 2 ( D8 2 ), 8,0,1 = Š 4 D81 2 ( D8 2 ), 4,2,1 = Š 4 D4€€€€D8 D4€€€€1 2 ( ), 2,1,6 =1 2 Š 4 D2[ ab ]€€D81 2 ( D6 [ ab ]€€ 1 2 Ca bcd D2[ cd ]€€ ), 6,1,6 =1 2 Š 4D2 [ ab ] D81 2 ( D6[ ab ] 1 2 Ca bcd D2 [ cd ] ), 4,1,15 = Š 4 D4 a b€€D8 b a€€1 2 ( ), 4,0,20= Š 4 D4 [ cd ] [ ab ]D8 D4 [ ab ] [ cd ]1 2 ( ),(3. 11.54) andatotalof22irreduciblerepresentations.The6isarealrepresentationonlyifwe useadualitytransformationinth erest-fram econju gation(3.11.4): X[ ab ]=1 2 Ca bcd X[ cd ].Thiso ccursforrank1 2 N antisymmetrictensorsof SU ( N )when N isamulti pleof4.

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1383.REPRESENTATIONSOFSUPERSYMMETRY3.12.On-shellrepresentationsandsuper“elds Insection3.9wediscussedirreduciblerepresentationsofo-shellsupersymmetry intermsofsuper“elds;herewegivethecorresp ondinganalysisofon-shellrepresentations.We“r stdiscussthedescriptionofon-shellphysicalcomponentsbymeansof“eld strengths.Wethendescribea(non-Lorent z-covariant)subgroupofsupersymmetry, whichwecall on-shell supersymmetry,underwhich(reducibleorirreducible)o-shell representationsofordinary(oro-shell)su persy mmetrydecomposeintomultipletsthat containonlyoneofthethreetypesofcomponentsdiscussedinsec.3.9.Byconsidering representationsofthissmallergroupintermsof on-shell super“elds(de“nedinasuperspacewhichisanon-Lorentz-covariantsubspaceoftheoriginalsuperspace),wecanconcentrateonjustthephysicalcomponents,andthusonthephysicalcontentofthetheory. a.Fieldstrengths Forsimp licitywerestrictourselvestomassless“elds.(Massive“eldsmaybe treatedsimilarly.)Itismoreconvenienttod escribethephysicalcomponentsintermsof “eldstrengthsratherthangauge“elds:EveryirreduciblerepresentationoftheLorentz group,whenconsideredasa “eldstrength, satis“escertainuniquec onstraints(Bianchi identities)plus“ eldequations,andcorrespondstoauniquenontrivialirreduciblerepresentationoft hePoincar egroup(a zeromasssinglehe licitystate).Ontheotherhand,a givenirreduciblerepresentationoftheLorentzgroup,whenconsideredasagauge“eld, maycorrespondtoseveralrepresentationsofthePoincar egro up,dependingontheform ofitsgaugetransformation. Speci“cally,any“eldstrength 1... 2 A€1...€2 B, totallysymmetric inits2 A undotted i ndicesandinits2 B dottedindices,hasmassdimension A + B +1ands atis“est heconstraintsplus“eldequations 1€1... 2 A€1...€2 B= €11... 2 A€1...€2 B=0,(3. 12.1a) 1... 2 A€1...€2 B=0.(3. 12.1b) TheKlein-Gordonequation(3.12.1b)projectsontothemasszerorepresentation,while (3.12.1a)projectontothehelicity A Š B state.TheKlein-Gordonequationisaconsequen ceoftheothersexceptwhen A = B =0.Tosolvethesee quationswegoto

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3.12.On-shellrepresentationsandsuper“elds139momentumspace:Then(3.12.1b)sets p2to0(i.e., ( p2)),andwemaychoosethe Lorentzframe p+€+= p+€Š=0, pŠ€Š =0.Inthisfra me(3.12.1a)statesthatonlyonecomponentof isnonvan ishi ng: + ... +€+ ...€+.Since each+i ndexhasah e licity1 2 andeach €+hashe licity Š1 2 ,theto talhelicityof is A Š B ,andofitsc omplexco njugate B Š A .Intheca seswhere A = B wemaychoose real(sinceithasanequalnumber of dottedandundottedindices),sothatitdescribesasinglestateofhelicity0. Themostfamiliarexamplesof“eldstrengthshave B =0: A =0istheusual descriptionofascalar, A =1 2 aWeyls pinor, A =1describesa v ector(e.g .,the photon), A =3 2 thegravitino,andthecase A =2istheWeylt ensorofthegraviton.Sinceweare describingonlytheon-shellcomponents,wedonotsee“eldstrengthsthatvanishon shell:e.g.,ingravitytheRiccitensorvanishesbytheequationsofmotion,leavingthe Weyltensor astheonlynonvanishingpartoftheRi emanncurvaturetensor.(Thishappensbeca use,alth oughthesetheoriesareirreducibleonshell,theymaybereducibleo she ll;i.e.,the“eldequationsmayeliminatePoincar erepresentations note liminatedby (o-shell)constraints.)Themostfamiliarexampleof A B =0isthe“elds trengthof thesecond-rankantisymmetrictensorgauge“eld:( A B )=(1 2 ,1 2 )(sees ec.4.4.c).Some lessfamiliarexamplesarethespin-3 2 representationofspin1 2 ,( A B )=(1,1 2 ),thespin-2 representationofspin0,( A B )=(1,1 ),andthehigher-derivativerepresentationofspin 1,( A B )=(3 2 ,1 2 ).Generally,t heo-shelltheorycontainsmaximumspinindicatedby thei ndicesof : A + B Althoughtheanalogousanalysisforsupersymmetricmultipletsisnotyetcompletelyunderstood,theon-shellcontentofsuper“eldscanbeanalyzedbycomponent projection.Inparticular,acompletesuper“eldanalysishasbeenmadeofon-shellmultipletsthatcontainonlycomponent“eldstrengthsoftype( A ,0).This issucientto describeallon-shellmultiplets:Theorieswith“eldstrengths( A B )des cribethesame on-shellhelicitystatesastheorieswith( A Š B ,0),andareph ysicallyequivalent.They onlydierbytheirauxiliary“eldcontent.Furthermore,type( A ,0)theorie sa llowthe mostgeneralinteractions,whereastheorieswith B =0“eldsarege nerallymore restrictedintheformoftheirself-interactionsandinteractionswithexternal“elds.(In somecases,theycannotevencoupletogravity.)

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1403.REPRESENTATIONSOFSUPERSYMMETRYBeforedisc ussingthegeneralcase,weconsideraspeci“cexampleindetail.The mult ipletof N =2superg ravity(seesec.3.3.a.1)withhelicities2,3 2 ,1,isdes cribedby component “eldstrengths ( x ), a ( x ), ab ( x ).Theyha vedime nsion3,5 2 ,2 respectively,andsatisfythe component Bianchiidentitiesand“eldequations(3.12.1). Weintr o ducea super“eld strength F(0) ab ( x )thatcontain sthelowe stdimensioncomponent “eldstrengthatthe =0le vel: F(0) ab ( x ) | = CabF(0) ( x ) | = ab ( x ).(3. 12.2) Werequirethat all thehighercom ponentsof F(0)arecomponent“eld strengthsofthe theory(ortheirsp acetimederivatives;super“eldstrengthscontainnogaugecomponents and,onshell,noauxiliary“elds).Thus,forexample,wemusthave Da ( F(0) ab )| =0, whereas Cc ( dDc F(0) ab ) | = D €F(0) ab | =0.Sinceasup er“eldthatvanishesat =0 vanishesid entically(asfo llowsfromthesupersymmetrytransformations,e.g.,(3.6.5-6)) Cc ( dDc F(0) ab ) = D €F(0) ab =0.Fromthese argumentsitfollowsthatthesuper“eld equationsandBianchiidentitiesare: D €F(0) ab =0, D F(0) ab = c [ aF(1) b ] D D F(0) ab = c [ ad b ]F(2) D D D F(0) ab =0;(3. 12.3) where F(1)( x )and F(2)( x )aresupe r“eldscontainingthe“eldstrengths b ( x )and ( x )atthe =0leve l.Bya pplyingpowersof D and D€ totheseequationswe recoverthecomponent“eldequationsandBianchiidentities,andverifythat F(0) ab containsnoextracomponents. Genera lizationtotherestofthesupermultipletsinTable3.12.1isstraightforward: Weintro duceasetofsuper“eldswhichat =0arethecomp onent“eldstrengths(asin (3.12.1))thatdescribethestatesappearinginTable3.3.1:Thesesuper“eldssatisfyaset ofBianchiidentitiesplus“eldequations(asintheexample(3.12.3))thatareuniquely determinedbydimension alanalysisandLorentz SU ( N )covar ian ce.

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3.12.On-shellrepresentationsandsuper“elds141 he licityscalarmultipletsuper-Yang-Millssupergravity +2 F+ 3/2 Fa +1 FFab + 1/2 FFa Fabc 0 FaFabFa bcd-1/2 Fab€Fabc€Fa bcde€-1 Fa bcd€€Fa bcdef€€-3/2 Fa bcdefg€€€-2 Fa bcdefgh€€€€ Table3. 12.1.Fieldstrengthsintheoriesofphysicalinterest Wenowconsi derarbitrarysupermultipletsoftype( A ,0).Thereare twocases: Foranon-shel lmulti pletwithlowestspin s =0,thesuper“eldstr engthhastheform F(0) a1... am,N 2 m N ,andis totallyantisymmetric inits mSU ( N ) isospin i ndices.If thelowestspin s > 0,thesup er“eldstrengthhastheform F(0) 1... 2sandis totallysymmetric inits2 sWeylspinor i ndices.Totreatbothcasestogether,for s > 0wewrite F(0) a1... aN1... 2s= Ca1... aNF(0) 1... 2s.Thent hesuper“eldstren gthhastheform F(0) a1... am1... 2sandistotallyantisymmetricinitsisospinorindicesandtotallysymmetric initsspino ri ndices.Ithas(mass)dimension s +1. Thissuper“eldcontainsalltheon-shellcomponent“eldstrengths;inparticular,at =0,itcontainsthe “eldstrengthoflowestdimension(andthereforeoflowestspin). For s =0,thesuper“el dstrengthd escribeshe licitiesm Š N 2 ,m Š N +1 2 ... ,m 2 ,andits hermitianconjugated escribeshe licitiesŠ m 2 ,Š m +1 2 ... ,N Š m 2 .Since m N ,some he licitiesappearinboth F(0)and F(0).For s 0,thesuper “eldstrengthdescribeshelicities s s +1 2 ... s +N 2 ,anditshe rmitianconjugatedescribeshelicities Š ( s +N 2 ), ... Š s .Inthiscase,po sitivehe licitiesappearonlyin F(0)andnegative he licitiesonlyin F(0).Forbothcasest hesuper“eldstr engthtogetherwithitsconjugate describe(perhapsmu ltiple)helicities s ( s +1 2 ), ... ( s +m 2 ).

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1423.REPRESENTATIONSOFSUPERSYMMETRYThehigher-spincomponent“eldstrengthsoccurat =0inthesup er“elds F( n )obtainedby a pplying nD s(for n > 0)or Š n D s(for n < 0)to F(0).Theyare totally antisymmetricintheir m Š n isospinorindicesandtotallysymmetricintheir2 s + n spin i ndices,andsatisfythefollowingBianchiidentitiesand“eldequations: n > 0: Dn n... 1F(0) a1... am1... 2s=1 ( m Š n )! b1[ a1... bnanF( n ) an +1... am] 1... 2s 1... n,(3. 12.4a) n < 0: Dn€ Š n...€ 1F(0) a1... am= F( n ) a1... amb1... bŠ n€1...€Š n,(3. 12.4b) with m Š N n m ;inparticular,for s > 0, DF(0)=0.Theseeq uation sfo llowfrom therequirementthat all componentsoftheon-shell super“eld strength(de“nedbyprojection)areon-shell component “eldstrengths.The =0compon entofthesuper“eld F(0)isthe lowestdimension component“eldstrength;thisdeterminesthedimensionand indexstructureofth esuper “eld.Thehighercomponentsofthesuper“eldareeither higherdimensioncomponent“eldstrengths,orvanish;thisdeterminesthesuper“eld equationsandBianchiidentities.Notethatthedierencebetweenmaximumandminimumhe licitiesinthe F( n )isalways1 2 N Inthespecialcase s =0, m even,and m =1 2 N wehaveina ddition to(3.12.4a,b) theselfconjugacyrelation F(0) a1... a1 2 N= 1 (1 2 N )! Ca1... aN F(0) a1 2 N... aN.(3. 12.4c) Forthisc aseonly, F(+ n )isrelatedto F( Š n );thisrelatio nfo llowsfrom(3.12.4c)for n =0, andfromspinorderivativesof(3.12.4c),using(3.12.4a,b),for n > 0.Eqs.(3.12.4a,b) are U ( N )covariant,where as,becausetheantisymmetrictensor Ca1... aNisnot phase invariant,(3.12.4c)isonly SU ( N )covaria nt;thus, self-conjugatemultipletshavea smallers ymmetry. b.Light-coneformalism Whenstudyingonlytheon-shellpropertiesofafree,masslesstheoryitissimpler torepres entthe“eldsinaformwherejustthephysicalcomponentsappear.As describedinsec.3. 9,weusealight-coneformalism,inwhichanirreduciblerepresentationofthePoincar egroupisgivenbyasi nglecomponent(complexexceptforzero

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3.12.On-shellrepresentationsandsuper“elds143he licity).Forsuper“eldswemakealight-conedecompositionof aswellas x .Weuse thenotation(see(3.1.1)): ( x+€+, x+€Š, xŠ€+, xŠ€Š) ( x+, xT, xT, Š xŠ),( a+, aŠ) ( a, a),(3. 12.5a) ( +€+, +€Š, Š€+, Š€Š) ( +, T, T, Š Š),( a+, aŠ) ( a, a).(3. 12.5b) (The spinor deriva tive ashouldnotbeconfusedwiththe spacetime deriva tive a). U nderthetransverse SO (2)partoftheLorentzgroupthecoordinatestransformas x = x, xT = e2 i xT, a = ei a, a = eŠ i a,andtheco rrespondingderivativestransformintheopp ositeway. Insec.3.11wedescribedthedecompositionofgeneralsuper“eldsintermsofchiral “eldstrengths,whichareirre ducib legaugeinvariantrepresentationsofsupersymmetry oshell.Althoughtheycontainnogaugecomponents,theymaycontainauxiliary“elds thatonlydropoutonshell.Toanalyzethedecompositionofanirreducible o-shell representationofsupersymmetryintoirreducible on-shell representations,weperforma nonlocal,nonlinear,nonunitarysimilaritytr ansformationonthe“eldstrengthsandall operators X : = eiH, X= eiHXeŠ iH; H =( ai a) T+ .(3. 12.6a) Weuset histransformationbecauseitmakessomeofthesupersymmetrygenerators independent of ainthechiralrepresentatio n.Droppingprimes,wehave Qa+= i a, Qa€+= i ( aŠ ai +), QaŠ= i ( a+ a T+ ), Qa€Š= i ( aŠ ai T+ i a + ).(3. 12.6b) Thus Q+and Q€+arelocalanddependonlyon a, a,and +, but not on a, a, Š,and T,whereas QŠand Q€Šarenonlocalanddependonall and a.Weexpandthe transformedsup er“eldstrengthinpowersof a(theexterna li ndicesofaresuppressed): ( x€, a, a)=N m =0 1 m ma1... am( m ) am... a1( x€, a),(3. 12.7)

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1443.REPRESENTATIONSOFSUPERSYMMETRYwherethe mthpower m,and thus ( m ),istot allyantisymmetricinisospinorindices. Each ( m )isarepresentationofa subgroupofsupersymmetrythatwecallonshellsupersymmetry,andthatincludesthe Q+transformations,thetransverse SO (2) partofLorentztransformations(an daco rrespondingconformalboost), SU ( N )(or U ( N )),andallfourtranslations(aswellasscaletransformationsinthemasslesscase). Althougheach ( m )isarealizationofthefullsupersymmetrygroupo-shellaswellas on-shell,on-shellsupersymmetryisthemaximalsubgroupthatcanberealizedlocally (andintheinteractingcase,linearly). Theremaininggeneratorsofthefullsupersymmetrygroup(includingtheother Lorentzgenerators,thatmix awith a)mixthev arious ( m )s.Inparticular, QŠand Q€Šallowustodistinguishphysicalandauxiliaryon-shellsuper“elds:Auxiliary“elds vanishon-s hell,andhencemusthavetr ansformationsproportionalto“eldequations. WegotoaL orentzframewhere T=0.Inthi sfra me, QaŠ= i aand Q€Š= i ( a+ i a + ).The QŠand Q€Šsupersymmetryvariationofthehighest acomponentof, ( N )Š ( N ) i + ( N Š 1)(3.12.8a) isproportionalto ,w hi ch identi“esitasanauxiliary“eld.Setting ( N )to zeroonshell,weiteratetheargument:thevariationofthenextcomponentof, ( N Š 1)Š ( N Š 1) i + ( N Š 2)(3.12.8b) isagainproportionalto ,etc.W e “ndthat only (0)hasavariation not prop ortional to .Thisi denti“esitasthephysicalon-shellsuper“eld. Thus,on-shell,reducesto (0).(Allother ( m )vanish.)IntheL orentzframe chosen above( T=0), QŠand Q€Švanishwhen actingon (0),andth usthissuper“eldis alocalrepr esentati onofthe full supersymmetryalgebraonsh e ll,namely,itdescribes themultipletof physicalpolarizations.Byexpandingactionsin ,itcanbeshownthat (0)representsthemultipletofphysicalcomponentswhiletheother ( m )srepres entmultipletsofauxiliarycomponents.

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3.12.On-shellrepresentationsandsuper“elds145Wecanalsod e“ne(chiralrepresentation)spinorderivatives Da, Dathatare covariantundertheon-shellsupersymmetry: Da= a+ ai +, Da= a; { Da, Db} = a bi +.(3. 12.9) Whenabisectioncondi tionisimposedonthechiral“eldstrength(i.e.,isreal, asdiscussedinsec.3.11),wecanexpressthe conditionintermsoftheon-shellsuper“elds.Forsuperspin s =0,theco ndition D2 N = 1 2 N(3. 12.10) b ecomes DN ( m ) a1... am= iN m Š1 2 N( i +)N Š m1 ( N Š m )! CaN... a1( N Š m ) aN... am +1(3.12.11) (where DN1 N CaN... a1DN a1... aN)andsim ilarlyforsuperspin s > 0.Ingeneral,anonshellrepresentationcanbereducedbyarealityconditionoftheform DN ( i +)1 2 N whenthemiddle( 1 2 N)compo nentof hashelicity0(i.e.,isinvariantundertransverse SO (2)Lorentztransformations).(Comparethediscussionofrealityofo-shell representationsinsec.3.11.) Puttingtogethertheresultsofsec.3.11andthissection,wehavethefollowing reductions:generalsuper“elds(4 N s;physical+auxiliary+gauge) chiral “eld strengths(2 N s;physical+auxiliary=irreducibleo-shellrepresentations) chiral on-shellsu per“elds( N s;physical=irreducibleon-shellrepresentations).Allthree typesofsuper “eldscansatisfyrealityconditions;th erefore,thesmallesttypeofeachhas 24 N,22 N,and2Ncomponents,respectively(whentherealityconditionisallowed),andis area lscalarsuper“eld.Allotherrepresentationsare(realorcomplex)super“elds with(Lorentzorinternal)indices,andthushaveanintegralmultipleofthisnumberof components.Thesecountingargumentsforo-shellandon-shellcomponentscanalso beobtainedby theusualoperatorarguments(o-shell,thecountingisthesameasfor on-shellmassivetheories,since p2 =0), butsuper“eldsallowanexplicitconstruction, andarethusmoreusef ulforapplications. Similarargumentsapplytohigherdimensions:Wecanusethesamenumbers th ere(buttakingintoaccountthedierenceinexternalindices),ifweunderstand4 N  tomeanthenumberofanticommu tingcoor dinatesinthehigherdimensionalsuperspace.

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1463.REPRESENTATIONSOFSUPERSYMMETRYForsimpl esupersy mmetryin D < 4,becausec hiralitycannotbede“ned,thecountingof statesisdierent.

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3.13.O-shell“eldstrengthsandprepotentials1473.13.O-shell“eldstrengthsandprepotentials Wehaveshownh owsuper“elds canbereducedtoirreducibleo-shellrepresentations(sec.3.11),whichcanb eredu cedfurthertoon-shellsuper“eldstrengths(sec. 3.12).To“ndasuper “elddescriptionofagivenmultipletofphysicalstates,weneedto reversetheprocedure:Startingwithan onshellsuper“eldstrength F(0)thatdescribes themultiplet,weneedto“ndthe oshellsuper“eldstrength W thatreducesto F(0)on shell,andthen“ndasuper“eld prepotential inte rmsofwhich W canbeexpressed. Thereisnounambiguouswaytodothis:Thesame F(0)isdescribedbydierent W s, andthesame W isdescribedbydierents.How ever,foraclassoftheoriesthat in cludesmanyofthemodelsthatareunderstood,weimposeadditionalrequirementsto reducetheambiguityand“ndauniquechiral“eldstrengthandafamilyofprepotentials foragivenmultiplet. Themultipletsweconsiderhaveon-shellsuper“eldstrengthsofLorentzrepresentationtype( A ,0)(superspin s = A )andare isoscalars:F(0) 1... 2 s.From( 3.12.4b),this impliesthatthe F(0)sare chiral andthereforecanbegeneralizedtoo-shellirreducible (uptobisection)“eldstrengths W1... 2 s, D€ W1... 2 s=0.Physi ca lly,the W scorrespond to“eldstrengthsofconformallyinvariantm odels.(Theyt ransforminthesamewayas Ca1... aN:as SU ( N )scalars butnot U ( N )scalars). Inthephysicalmodelswherethese super“eldsarise,thechiralityandbisectionconditionson W arelinearizedBianchi identities.Wecanuset heprojectionoperatoranalysisofsec.3.11tosolvetheidentities byexpr essingthe W sintermsofappropriateprepotentials. Whenthereisnobisection( s +1 2 N notaninteger),the W saregeneralchiral super“elds: W1... 2 s= D2 N1... 2 s.The1... 2 ssmaybeexpressedintermsofmorefundamentalsuper“elds.Aninterestingclassofprepotentialsarethosethatcontainthe lowest superspins:Inthatcase,the W shavetheform N 2 s Š 1: W1... 2 s=1 (2 s )! D2 NDN ( 1... NN +1€1... N + M€MN + M +1... 2 s)€1...€M, N 2 s Š 1: W1... 2 s=1 ( Š M )!(2 s )! D2 NDN [ a1... aŠ M] [ b1... bŠ M] ( 1... 2 s Š 12 s) b1... bŠ Ma1... aŠ M(3.13.1) whereisanarbitrary(complex)super“eldand M = s Š1 2 ( N +1).

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1483.REPRESENTATIONSOFSUPERSYMMETRYIf W isbis ected( s +1 2 N integer,(1 Š K K ) W =0), thenmust beexpr essedin termsofa real prepotential V thathasmaxim umsuperspin s W hasaformsimilarto (3.13.1): N 2 s : W1... 2 s=1 (2 s )! D2 NDN ( 1... NN +1€1... N + M€MVN + M +1... 2 s)€1...€M, N 2 s : W1... 2 s=1 ( Š M )! D2 NDN [ a1... aŠ M] [ b1... bŠ M] 1... 2 sVb1... bŠ Ma1... aŠ M(3.13.2) where M = s Š1 2 N Whetherornot W isbis ected,ambiguityremainsintheprepotentials, V ,since th eymaystillbeexpressedasderivativesofmorefundamentalsuper“elds:Thisleadsto varianto-shellm ultiplets(seesec.4.5.c).O urexpression (3.13.1)forintermsof isanexampleofsuchana mbiguity:T hereisnoapriorireasonwhymustta kethe specialform, unless itisobtainedasasubmulti pletofabisectedhigherN mult iplet(as, forexample,inthecaseofthe N =1spin3 2 ,1mult iplet(sec.4. 5.e),which isas ubmultipletofthe N =2superg ravitymultiplet).Modulosuchambiguities,theexpressions for W intermsofand V arethemostgenerallocalsolutionstotheBianchiidentities constraining W (i.e.,chirality,andifpossible,bisection).

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Contentsof 4.CLA SSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS 4.1.Thescalarmultiplet149 a.Renormalizablemodels149 a.1.Actions149 a.2.Auxiliary“elds151 a.3.R-invariance153 a.4.Super“eldequations153 b.Nonlinear -models154 4.2.Yang-Millsgaugetheories159 a.Prepotentials159 a.1.Linearcase159 a.2.Nonlinearcase162 a.3.Covariantderivatives165 a.4.Fieldstrengths167 a.5.Covariantvariations168 b.Covarian ta pproac h 170 b.1.Conven tionalconstraints171 b.2.Representation-pre servingconstraints172 b.3.Gaugechiralr epresentation174 c.Bianchiidentities174 4.3.Gauge-invariantmodels178 a.Renormalizablemodels178 b.CP(n)m odel s 179 4.4.Superforms181 a.General181 b.Vectormu ltiplet185 c.Tensormu ltiplet186 c.1.Geometricformulation186 c.2.Dualitytransformati ontochiralmultiplet190 d.Gauge3-formmultiplets193 d.1.Re al3-form193 d.2.Comple x3-for m 195 e.4-formmu ltiplet197 4.5.Othergaugemultiplets198 a.GaugeWess-Zu minomodel198

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b.Thenonminimalscalarmultiplet199 c.Morevariantmultiplets201 c.1.Vectormultiplet201 c.2.Tensormultiplet203 d.Super“eldLagran gemultipliers203 e.Thegravitinoma ttermultiplet206 e.1.O-shell“eldstrengthandprepotential206 e.2.Compensators208 e.3.Duality211 e.4.Geometricformulations212 4.6.N-extendedm ultiplets216 a.N=2multiplets216 a.1.Vectormultiplet216 a.2.Hypermultiplet218 a.2.i.Freetheory218 a.2.ii.Interactions219 a.3.Tensormultiplet223 a.4.Duality224 a.5.N=2super“eldLagrangemultiplier227 b. N=4Yang-Mills228 b.1.Minimalf ormulation228 b.2.L agrangemultiplierformulation229

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4.CLA SSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS Inthischapterwediscussinteracting“eldtheoriesthatcanbebuiltoutofthe super“eldsofglobal N =1Poincar esupersy mmetry.Thisrestrictsustotheories describingparticleswithspinsnohigherthan1.Thesimplestdescriptionofsuchtheoriesisintermsofchiralscalarsuper“eldsforparticlesofthescalarmultiplet(spins0 and1 2 ),andrealscalargaugesuper“eldsforp articlesofthevectormultiplet(spins1 2 and1).However,otherdescriptionsarepossible;wetreatsomeoftheseinageneral frameworkprovidedbysuperforms.Wedescribe N =1theori esandalsoextended N 4theoriesi nte rmsof N =1 super“elds.Ourprimarygoalistoexplainthestructureofthesetheoriesinsup erspace.Wedonotdiscussphenomenologicalmodels. 4.1.Thescalarmultiplet a.Renormalizablemodels Thelowestsuperspinrepresentationofthe N =1supersy mmetryalgebraiscarriedbyachiralscalarsuper“eld.Insec.3.6wedescribeditscomponentsandtransformations.Inthechiralrepresentationwehave(+)= A + Š 2F ,with complex scalarcomponent“elds A =2Š1 2 (A+iB), F =2Š1 2 (F+iG),andthetransformationsof (3.6.6). a.1.Ac tions To “ndsuperspaceactionsforthechiralsup er“eldweusedimensionalanalysis: Thesuper“eldcontainstwocomplexscalarsdieringbyoneunitofdimension(recall that hasdimension Š1 2 );however,itcontainsonlyon espi nor,andwerequirethis spinortohavetheusualphysicaldimension3 2 .Therefo re,weshouldassignthesuper“elddimension1.Thisleadsustoauniquechoiceforafree(quadratic)masslessaction withnodimensionalparameters: Skin= d4xd4 (4. 1.1) (seesec.3.7.afor ades criptionoftheBerezinintegral).Uptoanirrelevantphasethere

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1504.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSisa uniquemasstermandauniqueinteractiontermwithdimensionlesscouplingconstant: S( m )= d4xd2 (1 2 m 2+ 3! 3)+ d4xd2 (1 2 m 2+ 3! 3).(4.1 .2) TheresultingactiondescribestheWess-Zuminomodel. Alloftheintegralsare i ndependentoftherepresentation(vector,chiralorantichiral)inwhichthe“eldsaregiven;theintegrandsindierentrepresentationsdierby total x -derivat ives(fromthe eUfactors,see( 3.3.26)),thatvanishupon x -integration. Wecanexpressthea ctioninitscomponentformeitherbystraightforward -expansionandintegration,orby D -projection.Intheformerapproach,wewritefor example,intheantichiralrepresentationfor = ( Š ),and( Š )= eU(+): Skin= d4xd4 ( Š )eU(+)= d4xd4 [ A + € €Š 2 F ] e €i €[ A + Š 2F ],(4.1 .3a) andaftersomealgebraobtain Skin= d4x [ A A + €i €+ FF ].(4.1 .3b) Itissimplertousetheprojectiontechnique;wewrited4xd4 = d4x D2D2and Skin= d4xd4 = d4x D2[ D2] | = d4x [ D2D2+( D2 )( D2)+( D€ )( D€D2)] | .(4. 1.4) Usingtheidentities DD2= Di and D2D2= ,whichfollowfromthechiralityof ,andthede“nitionofthecomponents(3.6.7),weobtain(4.1.3b). Toevaluatech iralintegralsbyprojectionwewrite,foranyfunction f () d4xd2 f ()= d4xD2f ()

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4.1.Thescalarmultiplet151= d4x [ f()( D )2+ f() D2] | = d4x [ f( A ) 2+ f( A ) F ].(4.1 .5) Inparticularweobtainforthemassandinteractionterms S( m )= d4x { m [ 2+ AF ]+ [ A 2+1 2 FA2]+ h c } .(4. 1.6) (Witho utlossofgenerality,wecanchoose m and real.) Wecoulda ddalinearterm anditshe rmitianconjugatetotheaction(4.1.2). Suchatermwouldaddtothecomponentactionalinear F + F term.However,in theWess-Zuminomodelsuchatermcanalwaysbeeliminatedfromtheactionbyaconstantshift + c .Lineart ermsdohoweverplayanimportantroleinconstructing modelswithspontaneoussupersymmetrybreaking(seesec.8.3). a.2.Auxiliary“elds Thecomponent“eld F doesnotdescribeanindependentdegreeoffreedom;its equationofmotionisalgebraic: F = Š m A Š1 2 A2.(4. 1.7) Ifweeliminatethe auxiliary “eld F fromtheactionandthetransformationlaws,we“nd S = d4x [ A ( Š m2) A + €i €+ m ( 2+ 2) Š1 2 m ( A A2+ AA2) Š1 4 2A2 A2+ ( A 2+ A 2)],(4.1.8) and A = Š = Š €i €A Š ( m A +1 2 A2).(4.1 .9) ThereforetheWess-Zuminoactiongivesequalmassestothescalarsandthespinor, cubicandquarticself-interactionsforthescalars,andYukawacouplingsbetweenthe

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1524.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSscalarsandthespinor,allgovernedbyacommoncouplingconstant. Aftereliminating F ,thesupersy mmetrytransformationso fthespinor arenonlinear;thismakesananalysisofthesupersymmetryWardidentities without theauxiliary “e ldsdicult.Thisisnottheonlyproblemcausedbyeliminatingauxiliarycomponent “elds:Thetransformationsarenotonlynonlinear,butalsodependentonparametersin theLagrangian,anditisdiculttodiscoverfurthersupersymmetrictermsthatcouldbe addedtothecomponentLagrangian(e.g.,gaugecouplings).Furthermore,equation (4.1.7)isnotitselfsupersymmetric unless theequationofmotionofthespinorissatis“ed;onlythenis F = Š €i €(4.1.10a) thesameas F ( A )= ( Š m A Š1 2 A2)=( m + A ) € €.(4. 1.10b) Forthisreaso n,formulationsofsupersymmetricth eoriesthatlackthecomponentauxiliary“eldsareoftencalled on-s hell supersymmetric.Indeed ,ifwecalcula tethecommuta toroftwosupersymmetrytransformation sactingonthe spinor,we“ndthatthe “elds A ,formarepres entationofthealgebra(i.e.,thealgebracloses)onlyifthe spinorequationofmotionissatis“ed. TheWess-Zuminomodelcanbegeneralizedtoincludeseveralchiralsuper“elds. Themostgeneralactionthatleadstoaconventionalrenormalizabletheoryis S = d4xd4 ii+ d4xd2 P(i)+ h c .,(4.1 .11) wherePisapolynomialofmaximumdegree3inthe“elds.Thecomponentactionhas theform S = d4x [ Ai Ai+ i€i € i+ FiFi] + d4x [Pi( A ) Fi+Pij( A )1 2 i j+ h c .],(4. 1.12) where

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4.1.Thescalarmultiplet153Pi= P Ai ,Pij= 2P Ai Aj .(4. 1.13) Inparticular,eliminationoftheauxiliary“eldsgivesthescalarinteractionterms(the scalarpotential U ): Š U ( Ai)= Ši | Pi|2.(4. 1.14) Asaconsequenceofsupersymmetry(see(3.2. 10))thepotentialispositivesemide“nite. Theaction(4.1.11)canalsobeinvariantunderaglobalinternalsymmetrygroupcarried bythes. a.3.R-invariance AnadditionaltoolusedtostudythesemodelsisR-symmetry(3.6.14).Thisisthe chiralsy mmetrygeneratedbyrotating and byopposi te phases(sothatd4 is invari ant butd2 isnot)andbyrotatingdierentchi ralsuper“eldsbyrelatedphases: ( x , ) eŠ iwr( x eir eŠ ir ).(4.1 .15) Itmaybe,butisnotalways,possibletoassignappropriateweights w tothevarious super“eldstomakethetotalactionR-invariant.Forexample,withonlyonechiralmultiplet,R-invaria nceholdsifeitheramassoradimensionlessself-couplingispresent,but notboth:Theappropriatetra nsformationsw eightsare w =1and w =2 3 respectively. Withmorethanonechiralmultiplet,itisp ossibletowriteR-symmetricLagrangians havingbothmassandinteracti onterms:Achiralself-interac tiontermisR-invariantif itstotalR-weight w =2( i.e.,thesumoftheR-weightsofeachsuper“eldfactoris2). a.4.Super“eldequations Fromtheactionf orachirals uper“eld,weobtaintheequationsofmotionbyfunctionaldierentiation(see(3.8.10,11)).Forexample,includingsources,wehave S = d4xd4 + { d4xd2 [P()+ J ]+ h c } ,(4. 1.16) fromwhich,using( 3.8.9-12),wederivetheequations D2 +P()+ J =0 ,( 4.1.17a)

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1544.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSD2+ P( )+ J =0.(4. 1.17b) Weconsid er“rstthemassivenoninteractingcaseP()=1 2 m 2.M ul tiplying(4.1.17a) by D2,we “nd D2 D2 + mD2+ D2J =0.(4. 1.18) Substituting(4.1.17b)into(4.1.18)andusingthechiralityof( D2 D2 = ),we obtain ( Š m2) = m J Š D2J .(4. 1.19) Similarly,we“nd ( Š m2)= mJ Š D2 J .(4. 1.20) andtheseequationscanbereadilysolved. Forarbit raryP(),wederivetheequationsofmotionforthecomponent“eldsby projectionfromthesuper“eldequations.Successivelyapplying D sto(4.1.17a)we“nd F +P( A )+ JA=0 i € €+P( A ) + J =0 A +P( A ) 2+PF + JF=0(4.1 .21) aswouldbeobtainedfromthecomponentLagrangian. b.Nonlinear -models Ifrenormalizabilityisnotanissue,wecanconstructgeneralsupersymmetric actionsbytakingarbitraryfunctionsof, ,andtheirderivatives,andintegratingover superspace.Aninterestingclassofsupersy mmetricmodelsthatcanbeconstructedout ofchiralsuper“eldsisthegeneralizednonlinear -model.Inordinaryspacetime,ageneralizednonlinear -modelisdescribedby“elds ithatarethecoordinatesofanarbitrarymanifold.Theactionofsuchamodelis S= Š1 4 d4xgij( ai)( aj),(4.1 .22)

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4.1.Thescalarmultiplet155where gij( i)isthemetric tensor de“nedonthemanifold.Thesupersymmetricgeneralizationofthesemodelsisdesc ribedbychiralsuper“eldsiandtheirco njugates iwhicharethecomplexcoordinatesofanarbitraryK¨ ahlermanifold(seebelow).(Weuse agrouptheo reticconvention:Upperandlowerindicesarerelatedbycomplexconjugation,andallfactorsofthemetricarekeptexplicit.)Theactiondependsonasinglereal function IK (, )de“neduptoarbitraryadditivechiralandantichiraltermsthatdo notcontribute: S= d4xd4 IK (i, j).(4.1 .23) Thecomponentcontentofthisactioncanbeworkedoutstraightforwardlyusingthe projectiontechnique;we“nd S= Š1 2 d4x 2IK Ai Aj ( aAi)( a Aj)+ ... .(4. 1.24) Thishastheform(4.1.22)ifweidentify2IK Ai Aj asthemetric gij.Acomplexm anifold whosemetriccanbewritten(locally)intermsofapotential IK iscalledK¨ ahler;thusall four-dimension alsupersymmetricnonlinear -modelsarede“nedonK¨ ahlermanifolds. Conversely,anybosonicnonlinear -modelwhose“eldsresideonaK¨ ahlermanifoldcan beextendedtoasupers ymmetricmodel.Theremainingtermsin(4.1.24)providecouplingsbetweenthescalar“ eldsandthespinor“elds. K¨ ahlergeometryisaninterestingbranchofcomplexmanifoldtheorythatmathematicianshaveinvestigatedextensively.Herewediscussonlythoseaspectsrelevantto subsequenttopics(e.g.,sec.8.3.b).Wede“ne IKj1... jni1... im= i1 ... im j1 ... jn IK .( 4.1.25a) Inparticular,themetricis IKi j= 2IK i j .(4. 1.25b) Equivalently,wecanwritethelineelementas ds2= IKi jd id j.(4. 1.25c)

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1564.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSThemetric,liketheaction(4.1.23),isinvariantunderK¨ ahlergaugetransformations IK IK +()+ ( )(4.1.26) oftheK¨ ahlerpotential IK .Fie ldrede “nitions= f ()d e“ne holomorphic coordinate tr an sformationsonthemanifold;underthese,theformofthemetric(4.1.25b,c)ispreserved,whereasunderarbitrary nonholomorphic coordinatetransformations,ingeneral termsoftheform gijd id jand gijd id jaregeneratedinthelineelement.The nonhermitianmetriccoecients gij, gijare not relatedto IKijand IKij.Whenw orking withsuper“elds,sinceiischiral,onlyholomorphiccoordinatetransformationsmake obvioussense;however,wecanperformarbitrarycoordinatetransformationsonthe scalar“elds Ai. Usingthegaugetransformations(4.1.26)andholomorphiccoordinatetransformations,itispo ssiblet ogot oa normalgauge where,atanygivenpoint0, 0,evaluated at = =0, IKi1... im= IKj1... jn=0 foralln m ,( 4.1.27a) IKj i1... im= IKj1... jni=0 foralln m > 1,(4.1 .27b) IKi j= i j,(4. 1.27c) with i j=(1, 1,... Š 1, Š 1,. ..)depe ndingonthesignatureofthemanifold.Iftheidescribephysicalmattermultiplets, i j= i j.Inano rmalgauge,alltheconnections vanishatth epoint0,theRiemannc urvaturetensorhastheform: Ri j k l= IKik jl,( 4.1.28a) withallothercomponentsrelatedbytheusualsymmetriesoftheRiemanntensoror zero,andhencetheRiccitensorissimply: Rk j= IKik ji.(4. 1.28b) Inageneralgauge,theconnectionis ij k= IKij l( IKŠ 1)l k(4.1.29a) where( IKŠ 1)l kistheinverseofthemetric IKk l;allothe rcomponentsa rerelatedby complexconjugationorarezero.Thecontractedconnectionis,asalways,

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4.1.Thescalarmultiplet157i ij j=[ lndetIKk l]i.(4. 1.29b) TheRiemanntensorinageneralgaugeis Ri j k l= IKik jlŠ ( IKŠ 1)m nIKik mIKn jl(4.1.30a) andtheRiccite nsorhasthesimpleform Rk j Ri j k lIKl i=[ lndetIKi l]k j.(4. 1.30b) Manifoldscanhavesymmetries,or isometries. OnaK¨ ahlermanifold,anisometry ofthemetricis,ingeneral,aninvarianceoftheK¨ ahlerpotential IK uptoaK¨ ahler gaugetransformation(4.1.26).Onecanrequiretheisometrytobean invariance ofthe potentia l.(Actually,thisiso nlytrueifthereisapointon themanifoldwheretheisometrygroupisunbroken,i.e.,thetransformati onsdonotshiftthepoint.)This(partially) “xestheK¨ ahlergaugeinvariance:Itisnolongerpossibletogotoanormalgauge (4.1.27).However,holomorphiccoordinatetransformationsstillmakeitpossibleto choose normal coor dinates, wherethemetric IKi jsatis“es(4.1 .27c),anditsholomorphic deriva tives( IKi1j)i2... im IKj i1... imsatisfy(4.1.27b)(likewisefortheantiholomorphic de rivatives)buttheconditions(4.1.27a)are not satis“ed. Inarbitrarycoordinatesystems,the isometri esactonthecoordinatesas i=AkAi, i= AkAi(4.1.31) wherethesarein“nitesimalparameters(= areconsta nt unlessweintroduce gauge“eldsandgaugetheisometrygroup;supersymmetricgaugetheoriesarediscussed intheremai nderofthischapter),andthe k(, )sare Killingvectors. Thesesatisfy K illingsequations: kAi ; j+ kAj ; i= kAi ; j+ kAj ; i=0 (4.1.32a) kAi ; j+( IKŠ 1)i kkAl ; kIKl j=0.(4. 1.32b) where ki ; j= ki j= ki j (4.1.32c) and

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1584.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSki ; j= ki j+ kkjk i= ki j + kkIKjk l( IKŠ 1)l i(4.1.32d) For holomorphic K illingvectors ki= ki(), ki= ki( ),( 4.1.32a)isatrivialityand (4.1.32b)followsdirectlyfrom IKikAi+ IKikAi=0,(4. 1.33) whichisjustthestatementthattheK¨ ah le rp ot entialisinvariantundertheisometries. (Actually,invarianceuptogaugetransformations(4.1.26)sucestoimply(4.1.32b).) Wecanalsowrite thetransformations(4.1.31)as i=AkAj j i, i= AkAj j i;( 4.1.34a) Thisformexponentiatestogivethe“nitetransformation: i= exp (AkAj j )i, i= exp ( AkAj j ) i.(4. 1.34b) Forthecasesw henthereexistsa“xedpointonth emanif old,wecanchooseaspecial coordinatesystem(thatingeneralis not compatiblewithnormalc oordinates )wherethe transformations(4.1.31,34)takethefamiliarform i= i A( TA)i jj, i= Š i j A( TA)j i(4.1.35a) or,for“nitetransformations, i=( ei ATA)i jj, i= j( eŠ i ATA)j i.(4. 1.35b) Inarbitrarycoordinates,thenotionofmultiplyingvectorsby i isrepres entedby mult iplicationbyatwoindextensorcalledthe complexstr ucture. Ithastheproperty thatitssquareis Š 1 aKroneck erdelta.ForaK¨ ahlermanifold,thecomplexstructure iscovariantlyconstantandpreservesthemetric. Itmayhappenthatthereexistnontrivial nonholomorphic coordinatetransformationsthat do preservetheformofthe metric(4.1.25b,c);thenonecanshowthatthe manifoldis hyperK¨ ahler. Suchmanifoldshavethreelinearlyindependentcomplexstructuresandarelocallyquaternionic.Theyareeven(complex)dimensional;all hyperK¨ ahlermanifoldsareRicci”at,thoughtheconverseistrueonlyinfour(real) dimensions(twocomplexdimensions).

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4.2.Yang-Millsgaugetheories1594.2.Yang-Millsgaugetheories a.Prepotentials Ingeneral,wecan“ndaformulationofanysupersymmetricgaugetheoryeither byst udyingo-shellrepresentationstode rivethefree(linear)theoryintermsof unconstrained gaugesuper“eldsor prepotentials, orbypostulating covariant derivati vesand imposing covariant constraintsonthemuntilallquantitiescanbeexpressedintermsof asinglei rreduciblerepresentationofsupersy mmetry.Intheformercase,wemustconstructcovariantlytransformingderivativesoutoftheunconstrained“eldsandgeneralize tothenonlinearcase,whereasinthelattercasewemustsolvethecovariantconstraints intermsofprepotentials.Westudybotha pproachesandexhibittherelationbetween them. a.1.Linearcase Fromtheanalysiso fs ec.3.3.a.1,the N =1v ectormultipletconsistsofmassless spin1 2 andspin1physicalstates.Wedenotethecorrespondingcomponent“eld strengthsby f.A ccordingtothediscussionofsec.3.12.a,theselieinanirreducib leon-shellchiralsuper“eldstrength(0) ,whichsati s“esthe“eldequationsand Bianchiidentities D(0) =0.Theco rrespondingirreducibleo-shell“eldstrengthisa chiralsuper“eld W, D€W=0,satisfy ingthe bisectioncondition( s +1 2 N =1 2 +1 2 is aninteger) K K W= Š W,whichcanb ewri tten(see(3.11.9)) DW= Š D€ W€.(4. 2.1) (Wehavea Š signinthebisectionconditiontoobtainusualparityassignmentsforthe components.)Therefore,by(3.13.2),itcanbeexpressedintermsofanunconstrained realscalarsuper“eldby W= i D2DV W€= Š iD2 D€V V = V ,(4. 2.2) andthist urnsouttobethesimplestdescriptionofthecorrespondingmultiplet. Thede“nitionof Wis in va riantunder gaugetransformations withachiralparameter

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1604.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSV= V + i ( Š ), D€= D =0.(4 .2.3) Laterwegeneralizethistoanonabeliangaugeinvariance,butforthemomentweanalyzethesimplestcas e.Theprepotential V canbeexpandedincomponentsbyprojection: C = V | = iDV | €= Š i D€V | M = D2V | M = D2V | A€=1 2 [ D€, D] V | = i D2DV | €= Š iD2 D€V | ,D=1 2 D D2DV | .(4. 2.4a) (Toavoidconfusionwith D,w ed en otetheDauxiliary“eldbyD.)Asdisc ussedin sec.3.6.b,thereissomechoiceintheorderofthe D swhichsimplyamountsto“eld rede“nitions.Theparticularformwechosein(4.2.4a)issuchthatthephysicalcomponentsareinvariantunderthegaugetransformations(exceptforanordinarygauge transformationofthevectorcomponent“eld).Bymakingasimilarcomponentexpansion 1= | ,= D | ,2= D2 | ,(4. 2.4b) we “nd C = i ( 1Š 1), =, M = Š i 2, A€=1 2 €(1+ 1), =0, D=0.(4. 2.5) Thus,allthecomponentsof V canbegaugedawayby nonderivative gaugetransformationsexceptfor A a, andD.Thev ectorandspinorarethephysicalcomponent“elds ofthemultiplet;Disanauxiliary“eld.They(andtheirderivatives)aretheonly

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4.2.Yang-Millsgaugetheories161componentsappearingin W: = W| f=1 2 D( W )| ,D= Š1 2 iDW| i € €= D2W| .(4. 2.6) Thesymmetricbispinor fanditsconjugate f€€aretheselfdualandantiself-dual partsofthecomponent“el dstrengthoft he gauge“eld A a.The gaugeinwhich A ,Daretheonly nonzerocomponentsof V iscalledthe Wess-Zuminogauge. Theremaining gaugefreedomistheusualabeliangaugetransformationofthevectorcomponent“eld. TheWess-Zumino(WZ)gauge breaks supersymmetry:Thesupersymmetryvariationsof and M violatethegaugecondition C = = M =0,e .g., = i M + i €( i1 2 €C Š A€),(4.2 .7) does not vanishintheW Z gauge.Wecande“netransformationsthatpreservetheWZ gaugebyaugmentingtheusualsupersymmetrytransformationswithgauge-restoring gaugetransformations.Thus,insteadof V = i ( € Q€+ Q) V ,(4. 2.8) wetake WZV = i ( € Q€+ Q) V + i ( Š )WZ= i ( € Q€ WZ+ Q WZ) V ,(4. 2.9) whereWZischosentorestoretheWZgaugeconditionbycancelingthetermsin V thatviolateit.Speci“cally, WZ=0requires WZ( DV ) | =0.(4. 2.10) Using D2V | =0and { D€, D} V | = aV | =0(inthe WZ gauge),wehave WZ= DWZ| = i € D€DV | = i €A€.(4. 2.11) Sim ilarly,from WZM =0we “nd

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1624.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS2 WZ= D2WZ| = Š € €.(4. 2.12) Finally,from WZC =0,we “ndthat1 WZ= 1 WZ.There mainingrealscalarinWZis theusualcomponentgaugeparameterforthevectorgauge“eld(see(4.2.5)). TheWZgaugepreservingsupers ymmetrytransformationsare A a= Š i ( €+ €), = Š f+ i D, D=1 2 €( €Š €).(4.2 .13) Thecommutatoralgebraofthesetransformationsclosesonlyuptogaugetransformationsofthevector“eld. Theneedforgauge-restoring transformationsmakessupersymmetricquantizationintheWZgaugeimpossible. The(vector)gauge-“xingprocedure,bybreakinggaugeinvarian ce,alsobreakssupersymmetry. Fromtherequi rementthatthephy sicalcomponents A aand havecanonical dimension,weconcludethat V hasdimensionzero.Bydimensionalanalysisandgauge in va rianceunderthetransformationswe“ndtheaction S = d4xd2 W2=1 2 d4xd4 VD D2DV .(4. 2.14) Replacing d2 by D2andusing(4.2.6),weobtainthecomponentaction S = d4x [ Š1 2 ff+ €i €+D 2].(4.2 .15) Wehave notaddedthehermitianconjugateto S ; ImS isatotalderivativeandcontri butesonlyasurfaceterm( d4x a b c df a bf c d+spi norialterms).The“eldDis cl ea rlyauxiliary. a.2.Nonlinearcase Thenonabeliangeneralizationcanbemotivatedbystartingwithaglobalinternal symmetryandmakingitlocal.Forthispurposeweconsideramultipletofchiralscalar “eldstransformingaccordingtosomerepre sentationofaglobalgroupwithgenerators TAandconstantparameters A:

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4.2.Yang-Millsgaugetheories163= ei = ATA, TA= TA.(4. 2.16) Weextend thistoalocaltransformationinsuperspace.Clearly,tomaintainthechiralityofthelocalparametersshouldbechiral .Wetherefo reconsidertransformationsof theform = ei ,=ATA, D€=0,(4 .2.17) andcorrespo ndingly,fortheantichiral ,transformingwiththecomplexconjugaterepresentation, = eŠ i = ATA, D =0.(4 .2.18) TheLagrangian is invariantiftheparameters Aarereal.Forlocaltransformations = andwemustintro duceagau ge“eldtocovariantizetheaction.Thesimplestprocedureistointroduceamultipletofrealscalarsuper“elds VAtransformingin thefollowingfashion: eV= ei eVeŠ i V = VATA.(4. 2.19) Intheabeliancase,thistransformationisjust(4.2.3).Wecovariantizetheactionby d4xd4 eV.(4.2 .20) Thegauge“eld V actsasa converter ,cha ngingarepre sentationtoa representationofthegroup.Thus, ( eV)= ei ( eV),(4.2.21a) andsimilarly ( eV)=( eV) eŠ i .(4. 2.21b) Inthenonabeliancase,eventhein“nitesimalgaugetransformationsof V are highlynonlinear.Nonetheless,asinthea be liancasetheycanbeusedtoalgebraically gaugeawayallbutthephysicalcomponentsof VAandtakeustotheWess-Zumino gauge:Startingwithanarbitrary V ,weperform successivegaugetransformationsto gaugeaway C ,and M .Requiri ngthat the“rsttransformationgaugeaway C we “nd,byevaluating(4.2.19)at =0:

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1644.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS1= eV| = ei (1)eVeŠ i (1)| =( ei (1)| ) eC( eŠ i (1)| ),(4.2 .22) andhencewemustchoose(1) 1=(1)| = Š i1 2 C .The gauge C=0ispr eservedbyall furthertransformationswith Im 1=0.To gaugeaway wechooseas econdgauge transformation(2)(with(2) 1=0)byreq uiring 0= DeV| = D( ei (2)eVeŠ i (2)) | = DV|Š iD(2)| = Š i Š i (2) ,(4. 2.23) andhence(2) = D(2)| = Š .Fina lly,wecan“ndathirdtransformation(3)to gaugeaway M .IntheWZ gauge,theonlygaugefreedomleftcorrespondstoordinary gaugetransformationsofthevector“eld A a,withpar ameter= = ( x ). Asintheabeliancase,theWZgaugeisno tsupersy mmetric,andgauge-restoring transformationsarerequiredtode“netheWZ gaugesupersymmetrytransformations. Theparameterofthetransformationsisstill(4.2.11-12),butthetransformationsnow b ecomenonabelianandhencenonlinear.To“ndthem,wecomputethein“nitesimal gaugetransformationsof V :Webeginby de“ningthesymbol LVX =[ V X ],(4.2 .24) sothat eVXeŠ V= eLVX .(4. 2.25) From[ V eV]=0weobtain ( V ) eV+ V ( eV) Š eV( V ) Š ( eV) V =0 ,( 4.2.26a) or eŠ1 2 V( V ) e1 2 VŠ e1 2 V( V ) eŠ1 2 V+ eŠ1 2 V[ V eV] eŠ1 2 V=0,(4. 2.26b) andhence 2 sinh(1 2 LV)( V )= eŠ1 2 VLV( eV) eŠ1 2 V= LV[ eŠ1 2 Vi e1 2 VŠ e1 2 Vi eŠ1 2 V]

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4.2.Yang-Millsgaugetheories165= iLV[ cosh(1 2 LV)( Š ) Š sinh(1 2 LV)( +)],(4 .2.27) fromwhichi tfo llows V = Š1 2 iLV[ ++ coth(1 2 LV)( Š )] = i ( Š ) Š1 2 i [ V +]+ O ( V2).(4.2 .28) Fromthetransfo rmations(4.2.28)andtheparameter(4.2.11-12)we“ndthenonabelianWZgauge-preservings upersymmetrytransformations: A a= Š i ( €+ €), = Š f+ i D, D=1 2 €( €Š €),(4.2 .29) wherenow fisthesel f-dualpartofthe nonabelian “eldstrengthand €= €Š iA€.The nonlinearitycomesfromthegauge -covariant izationofthelinear transformations(4.2.13).Thecomponentsofthenonabelianvectormultipletarecovariantgeneralizationsoftheabeliancomponents;intheWZgauge,theyarethesameas (4.2.4a)(seealso(4.3.5)). a.3.Covariantderivatives Thegauge“eld V canbeusedtoconstructderivatives,gaugecovariant with respectto transformations A= DAŠ i A=( €, €),(4.2 .30) de“nedbytherequirement ( A)= ei ( A),(4.2.31) i.e., A= ei AeŠ i ,( 4.2.32a) or

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1664.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS A= i [, A].(4.2 .32b) Sinceischiral, € D€iscovariantwithoutfu rthermodi“cation: €= ei D€eŠ i = €.(4. 2.33) Theundottedspinorderivative Discovariantw ithres p ectto trans formations.We canuse eVtoconv ertitintoaderivativecovariantwithrespectto(see(4.2.21)); eŠ VDeVtransformscorrectly: =( ei eŠ VeŠ i ) D( ei eVeŠ i ) = ei eŠ VDeVeŠ i = ei eŠ i .(4. 2.34) Finally,weconstruct abyan alogywith(3.4.9): a= €Š i {, €} .Itscovariancefollowsfromthatof and €. Wesu mmarize: A=( eŠ VDeV, D€, Š i {, €} ).(4.2 .35) Thesederivativesarenothermitian.Theirconjugates Aarecovariantwith respectto transfo rmations: A=( D, eV D€eŠ V, Š i { €} ), A= ei AeŠ i .(4. 2.36) Thederivatives A( A)areca lled gaugechiral(antichiral)representation covariant derivatives.Theyarerelatedbyanon unitarysimilaritytransformation A= eVAeŠ V.(4. 2.37) Thisisanalogoustotherelationbetweenglobalsupersymmetrychiralandantichiral representations DA ( Š )= eUDA (+)eŠ U(4.2.38) of(3.4.8).

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4.2.Yang-Millsgaugetheories167Thegaugecovariantderivativesareusuallyde“nedintermsofvectorrepresentation DAs;ifweexpresstheseintermsofordi naryderivatives,(4.2.35)becomes A=( eŠ VeŠ1 2 Ue1 2 UeV, e1 2 U €eŠ1 2 U, Š i {, €} ).(4.2 .39) Byafurthersimilaritytransformation A eŠ1 2 UAe1 2 U,wegotoanewrepresentation thatischiralwithrespecttobothglobalsupersymmetryandgaugetransformations: A=( eŠ1 2 UeŠ VeŠ1 2 Ue1 2 UeVe1 2 U, €, Š i {, €} ).(4.2 .40) Wede “ne Vby e1 2 UeVe1 2 U= eU + V .(4. 2.41) Inthisform,itisclearthat Vgaugecovariantizes U : i €€ ... + i €( €Š iA€)+ ... .Thiscom binationtr ansformsas ( eU + V )= ei ( eU + V ) eŠ i = €=0.(4 .2.42) Therealsoexistsasymmetric gaugevectorrepresentation thattreatschiralandantichiral“eldsonthesamefooting.Sucharepresentationusesacomplexscalargauge“eld ,andrequiresalargergaugegroup.Wediscussthevectorrepresentationinsubsec. 4.2.b,wherethecovariantderivativesarede“nedabstractly,andwhereitentersnaturally. a.4.Fieldstrengths Thecovariantderivativesde“ne“eldstrengthsbycommutation: [ A, B} = TAB CCŠ iFAB,(4. 2.43) with V = VATA,and TAintheadjointrepresentation.Fromtheexplicitformofthe covariantderivatives(4.2.35)we“ndthatthetorsion TAB Cisthesam eoneas in”at globalsuperspace(3.4.19),andsome“eldstrengthsvanish: F= F€€= F€=0.(4. 2.44) Theremaining“eldstrengthsare F€ €= C€€ D2( eŠ VDeV)= iC€€W,

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1684.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSF€ €=1 2 ( C€€( W )+ C(€W€ )), W i D2( eŠ VDeV), W€ eŠ V W€eV eŠ V( Š W)eV.(4. 2.45) (Recallthat W€ ( W)implies W€=( Š W)(3.1.20).)Thusallt he“eldstrengthsof thetheoryare expressedintermsofasinglespinor Wthatisthenonlinearversionof (4.2.2).Itsatis“esBianchiidentitiesanalogousto(4.2.1): W= Š€W€.(4. 2.46) Itischiral,hasdimension3 2 ,andcanbeusedtoc onstructagaugeinvariantaction S =1 g2 tr d4xd2 W2= Š1 2 g2 tr d4xd4 ( eŠ VDeV) D2( eŠ VDeV), V = VATA, trTATB= AB.(4. 2.47) Asintheabeliancase,thisactionishermitianuptoasurfaceterm(seediscussionfollowing(4.2.15)). a.5.Covariantvariations Toderivethe“eldequatio nsfromtheaction( 4.2.47),weneedtovarytheaction withrespectto V .Howev er,since V isnotacovarianto bj ect,thisresultsinnoncovariant“eldequations(althoughmultiplicationbyasuitable(butcomplicated)invertible operatorcovariantizesthem).Inaddition,variationwithrespectto V iscomplicated b ecause V appearsin eVfactors.Wethereforede“ne acovariantva riationof V by V eŠ V eV= 1 Š eŠ LVLV V = V +....( 4.2.48) V satis“esthechiralrepresentationhermiticityconditionasin(4.2.37).Inpractice, wealwaysvary anactionwithrespectto V byexpr essingitsv ariationintermsof eV, andthenrewritingthatintermsof V .Wethusde “neacovariantfunctionalderivative F [ V ] V by(cf. (3.8.3))

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4.2.Yang-Millsgaugetheories169F [ V + V ] Š F [ V ] ( V F [ V ] V )+ O (( V )2).(4.2 .49) Wenowo btaintheequationsofmotionfrom: g2 S = itr d4xd4 ( eŠ VDeV) W= itr d4xd4 [ eŠ VDeV, V ] W= Š itr d4xd4 V W,(4. 2.50) whichgives g2 S V = Š i W=0.(4. 2.51) *** Attheendofs ec.3.6weexpressedsupersymmetryt ransformationsintermsofthe spinorderi vatives D.Usingthecovar iantderivati vesthatweh aveconstru cted,wecan writemanifestlygaugecovariantsupersymmetrytransformationsbyusingtheform (3.6.13)(for w =0)anda ddingthegaugetr ansformation = i D2(D ), (4.2.52a) whereAisde “nedin(4.2.30).Wethen“nd eŠ VeV=( W+ W€ €) =( WeŠ VDeV+ eŠ V W€eV D€) (4.2.52b) (where isareal x -independentsuper“eldthatcommuteswiththegroupgenerators, e.g., = D ).Since(4.2.52b)ismanifestlygaugecovariant,itpreservestheWessZuminogauge(butitisnotasymmetryofthe actionaftergauge-“xing).Thecorrespondingsupersymmetr ytrans formati onsfor covariantly chiralsuper“elds, €=0 witharbitraryR-weight w are = Š i 2[( ) + w ( 2 )].(4.2.52c)

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1704.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSb.Covariantapproach InthissubsectionwediscussanotherapproachtosupersymmetricYang-Mills theorythatreversesthedir ectionoftheprevioussection.Wepostulatederivatives tr an sformingcovariantlyunderagaugegroup,imposeconstraintsonthem,anddiscover thattheycanbeexpressedint ermsofprepotentials.Thisprocedurewillproveespeci allyusefulinstudyingsupergravityandextendedsuper-Yang-Mills,sowegivea detailedana lysisforthesimplercaseof N =1 su pe r-Yang-Mills. Westartwitht heordinarysuperspacederivatives DAsatisfying [ DA, DB} = TAB CDC,where TAB Cisthetorsionandhasonlyonenonzerocomponent T€ c(see(3.4.19)).ForaLiealgebrawithgenerators TAwecovarian tizetheder ivatives byintro ducingconnection“elds A= DAŠ i A,(4. 2.53) whereA=ABTBishermitianand A= Š ( Š )AA.Atthecomp onentlevelwehave = v+i 2 €v€+ ... a= w a+ ... ,( 4.2.54a) andhence = Š iv+i 2 €( €Š iv€)+ ... €= €Š i v€+i 2 ( €Š i v€)+ ... a= aŠ iw a+ ... ,(4. 2.54b) sothat thecompone ntderivati vesarecova riantized. Undergaugetransformationsthecovariantderivativesarepostulatedtotransform as A= eiKAeŠ iK,(4. 2.55) wheretheparameter K = KATAisarealsup er“eld. K = ( x )+ K(1) ( x )+ € K(1)€( x )+ ... .(4. 2.56) Thisisverydierentfromwhatemergedintheprevioussection:Insteadofchiral

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4.2.Yang-Millsgaugetheories171representationderivativestransformingwiththechiralparameter,wehave vectorrepresentation hermitianderivatives,transformi ngwith thehermitianparameter K .The asymmetricformoftheprevioussectionwillemergewhenwemakeasimilaritytransformationtogotothechiralrepresentation. Fo ri n“nitesimal K ,we “ndthecomponenttr ansformations: v€=[ €Š iv€, ] Š i €, w€=[ €Š iw€, ],(4.2 .57) where K | € [ D€, D] K | = €.Thecompon entgaugeparameter €canbe usedto gaugeaway Imv€algebraically;however,thecomponent“elds Rev€and w€bothre mainastwoaprioriindependentgauge“eldsforthe same componentgauge transformation.Toavoidthisweimpose constr aints onthecovariantderivatives. b.1.Conven tionalconstraints Fieldstrengths FABar ed e“nedby(4.2.43).Substituting(4.2.53)we“nd FAB= D[ AB )Š i [A,B}Š TAB CC.(4. 2.58) Inparticular, F€= D €+ D€Š i { €}Š i €.(4. 2.59) Ifweimposetheconstraint F€=0,(4. 2.60) (4.2.59) de“nes thev ectorconnection€intermsofthespinorc onnections.(Incomponents,this expresses w€intermsof v€and v.) Inanytheoryonecanaddcovarianttermstotheconnections(e.g.,(3.10.22)) withoutchangingthetransformationofthecovariantderivatives.Ifwedidnotimpose theconstraint(4.2.60)ontheconnectionsA,wecouldde “neequallysatisfactorynew connections A=(, €,€Š iF€)thati denticallysatisfytheconstraints.Forthis reason(4.2.60)iscalleda con ventional constraint .Itimp lies A=( €, Š i {, €} ).(4.2 .61)

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1724.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSThetheorynowisexpressedentirelyintermsoftheconnection.Howev er,it containsspin s > 1 gaugecovariantcomponent“elds,forexample ( )€ F( )€| = i [ D€, D( ] )| + ... .(4. 2.62) Italsocontainsasuper“eldstrength Fwhose -independentcomponent f= F| = D( )| + ... ,(4. 2.63) isadimensiononesymmetricspinor(equivalenttoanantisymmetricsecondranktenso r).Becauseofitsdimension,itcannotbetheYang-Mills“eldstrength.Althoughin pr inciplethetheorymightcontainsuch“elds(asauxiliary,notphysical,components),in thecovariantapproachtherearegenerallyfurthertypesofconstraintsthateliminate (many)suchcomponents. b.2.Representation-p reservingconstraints Toco uplescalarmultipletsdescribedbychiralscalarsuper“eldstosuper-YangM illstheory,wemustde“ne covariantl ychiral super“elds:Thecovariantderivatives transformwiththehermitianparameter K ,andall“el dsmusteitherbeneutralor transformwiththesameparameter.However, K isnotchiral,andgaugetransformationswillnotpreservechiralityde“nedwith D€.Instead wede “neacovaria ntlych iral super“eldby €=0,= eiK, =0, = eŠ iK.(4. 2.64) Thisimplies 0= { €, €} = Š iF€€.(4.2 .65) Consistencyrequiresthatweimposethe repres entation-preserving constraint F= F€€=0.(4. 2.66) Thiscanbewrittenas {, } =0.(4. 2.67) Themostgeneralsolutionis

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4.2.Yang-Millsgaugetheories173= eŠ De,=ATA,(4. 2.68) whereAisanarbitrary complex super“eld.Eq.(4.2.67)statesthat satis“esthe samealgebraas D,andthesol utionexpressesthefactthat theyareequivalentuptoa complex gaugetransformation.Hermitianconjugationyields €= e D€eŠ .(4. 2.69) Thus Aiscompletelyexpressedintermsoftheunconstrained prepotential bythe solutions(4.2.61,68,69)totheconstraints(4.2.60,66). The K gaugetransformationsarerealizedby ( e)= eeŠ iK.(4. 2.70) However,thesolutiontotheconstraint(4.2.67)hasintroducedanadditionalgauge invariance:Thecovariantderivatives(4.2.68)are invariant underthetransformation ( e)= ei e, D€=0.(4 .2.71) Therefore,thegaugegroupofislargerthanthatofA. Wede “nethe K -invarianthermitianpartofby eV= ee .(4. 2.72) The K gaugetransformationscanbeusedtogaugeawaytheantihermitianpartof. Inthisgauge,= =1 2 V ,andtra nsformationsmustbea ccompaniedbygaugerestoring K transformations: ( e)= ei eeŠ iK (), eŠ iK ()= eŠ eŠ i ( ei e2eŠ i )1 2 .(4. 2.73) In any gauge,thetransformationof V is ( eV)= ei eVeŠ i .(4. 2.74) Wehave de“nedcovariant lychiralsuper“eldsby(4.2.64).Wecanuse(see (4.2.69))toexpressthemintermsof ordinary chiralsuper“elds0(whichwecalledin sect.4 .2.a):

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1744.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS= e 0, D€0=0.(4. 2.75) Thefactor e converts K -transforming“eldsinto-transforming“elds: (0)=( eŠ )= ei 0.(4. 2.76) *** Ausefulidentityt hatfo llowsfromtheexplicitform(4.2.68)expresses in termsofanarbitraryvariation : =( eŠ ) e+ eŠ e=[ eŠ e].(4.2 .77) b.3.Gaugechiralrepresentation Wecanalsouset ogotogau gechiralrepresentationinwhich all quantitiesare K -inertandtransformonlyunder.Thisisa nalogoustoandnottobeconfusedwith thesupersymmetrychiralrepresentation( 3.3.24-27),(3.4.8).Wemakeasimilarity transformation 0 A= eŠ Ae =( eŠ VDeV, D€, Š i {0 0€} ), 0= eŠ 0= e = (0) eV.(4. 2.78) Thequantities 0 Aand0arethechiralrepresentation Aandoftheprevioussubsection.Wesometimeswritethechiralr epresentationhermitianconjugateof0as0toavoidconfusionwiththeordinaryhermitianconjugate 0 (0). Inthechiralrepresentationweseenotraceofor K :Only V andappear. However,wenecessarilyhaveanasymmetr ybetw eenchiralandantichiralobjects. c.Bianchiidentities Insubsection4.2.aweanalyzedthephysicalcontentofthetheoryusingcomponentexpansionsandtheWess-Zuminogauge.Alternatively,wecan“ndthe“eldcontentofthetheorybysolvingtheBianc hiidentities.ThesefollowfromtheJacobi

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4.2.Yang-Millsgaugetheories175identities: [ [ A[ B, C )}} =0 ,( 4.2.79a) whichimply [ AFBC )Š T[ AB | DFD | C )=0.(4. 2.79b) Normallytheseequationsaretrivialidenti ties.How ever,onceconstraintshavebeen imposedonsome“eldstrengths,theygiveinformationabouttheremainingones,andin particularallowonetoexpressallthe“eldsstrengthsintermsofabasicset.Wenow describetheprocedure. Wesolvetheeq uations( 4.2.79)subjecttotheconstraints(4.2.60,66)startingwith th eo nesoflowestdimension.Foreachequation,weconsidervariouspiecesirreducible undertheLorentzgroup,andseewhatrelationsareimpliedamongthe“eldstrengths. Thus,forexample,therelation[ {( } )]=0isidentica llysati s“edwhen F =0.From[ {, } €]+[ { €, ( } )]=0, we “nd F( )€=0,(4. 2.80) whichimplies,forsomespinorsuper“eld W, F €= Š iC W€.(4. 2.81) From[ {, } c]+ { [ c, ( ], )} =0we “nd C ( ) W€=0,(4. 2.82) whichimplies W€=0.(4. 2.83) From[ {, €} c]+ { [ c, ], €} + { [ c, €], } =0weobtain F€ €+ C € W€+ C€€W=0,(4. 2.84) whichseparatesintotwoequations: F€ €=1 2 ( C (€ W€ )+ C€€( W )) C f€€+ C€€f,(4. 2.85) and

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1764.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSW+ € W€=0.(4. 2.86) Thesecanbereexpressedas W= iCD+ f,D= D= Ši 2 W.(4. 2.87) Finally,[[ [ b], c ]]+[[ b, c], ]=0and[[ [ a, b], c ]]=0areau tomaticallysatis“edasaco nsequenceofthepreviousidentities.From(4.2.87)wealsoobtain €D=1 2 €W, €f= i1 2 ( €W ),(4. 2.88) and f=1 2 C ( i )€ W€.(4. 2.89) Therefore,allthe“eldstrengthsareexpressedintermsofthechiral“eldstrength W.Inparticular, thecommutatorsofthecovariantderivativescanbewrittenas: {, } =0, {, €} = i €, [ €, i €]= Š iC€€W, [ i a, i b]= i ( C€€f+ C f€€).(4.2 .90) Furthe rmore,theset F = { W,D, f} ,(4. 2.91) is closedundertheoperationofapplying and €:Onlys pacetimederivatives aof F aregenerated.Thesesuper“eldsarethenonlinearo-shellextensionofthesuper“eld strengths( n )ofsec.3.12.Thecovariantcomponentsarethe =0proj ectionsofthese super“elds.ThustheconstraintsandtheBianchiidentitiesdirectlydeterminethe“eld contentofthetheory. ***

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4.2.Yang-Millsgaugetheories177Theexistenceofageometricsuperspaceformulationintermsofa(constrained) connectionAisimportant.ForquantizedsuperYang-Millstheories,thegeometric(or covariant)formulationcanbecombined withthebackground“eldmethodtoderive improvedsuper“ eldpower-countinglaws.WecanalsouseAtogeneralizet heconcept ofthepath-orde redphasefactortosuperspace: IP [ e( i dzAA)],(4.2 .92) wherethedierentialsuperspaceelement dzAistobeinterpretedas d dzA for some parametrizationofthepath.(Inparticular,d is not aBer ezinintegral.)Ifwe chooseaclo sedpath,thisquantityde“nesasupersymmetricWilsonloop.ThusnonperturbativestudiesofordinaryYang-MillstheoriesbasedonthepropertiesoftheWilson loopshouldbeextendibleintosuperspace.(Thereisalsoamanifestlycovariantformof pathordering,expresseddirectlyintermsofcovariantderivatives:seesec.6.6.)

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1784.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS4.3.Gauge-invariantmodels a.Renormalizablemodels Inthissubsectionweconsiderpropertieso fsystemsofin teractingchiralandreal gaugesuper“eldswithactionsoftheform S = d4xd4 j(eV)j ii+ tr d4xd2 W2+[ d4xd2 P(i)+ h c .](4.3.1) (i nt hegauge-chiralrepresentation),invariantunderagroup G .Here Vi j= VA( TA)i jand( TA)i jisa(ingeneralreducible)matrixrepresentationofthegenerators TAof G Inthevectorrepresentation,(4.3.1)takestheform S = d4xd4 ii+ tr d4xd2 W2+[ d4xd2 P(i)+ h c .](4.3.2) whereweh aveused treŠ fe = trf inthechiralintegral,andrewrittentheactionin termsofcovariantlychiralsuper“elds.Thegaugecouplinghasbeensetto1,butcanbe restoredbytherescalings W gŠ 1W. S maybeR-symmetric,withthegaugesuper“eldtransf ormingas V( x , )= V ( x eŠ ir eir ). Anothertermcanbeaddedtotheaction:If G isabelian,orhasanabeliansubgroup,the Fayet-Iliopoulos term SFI= tr d4xd4 V = tr d4x D,(4. 3.3) isgaugeinvariant. Componentactionscanbeobtainedbytheprojectiontechniqueswehavediscussedbefore.Amoreecientand,upto“eldrede“nitions,totallyequivalentprocedureistode“ne covariantc omponents byproj ectingw ithcovariantderivatives.Thus, foracovariantlychiralsuper“eldwede“ne A = | = | F = 2 | .(4. 3.4) Sim ilarly,thecovariantcomponentsofthegaugemultipletcanbeobtainedbyprojection from W(here fdenotesthecomponent“eldstrength): = W| f=1 2 {( W )}| ,

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4.3.Gauge-invariantmodels179i € €=1 2 [ {, W} ] | ,D= Ši 2 {, W}| .(4. 3.5) Thecovariant deriva tive €| isthecovariantspace-timederivative.Toobtaincomponentacti onsbycovariantprojection,weusethefactthatonagauge invariant quantity D2D2= 22. Thecomponentactionthatresultsfrom(4.3.1)plus(4.3.3)takestheform S = d4x[Ai Ai+ ii € € i+ i Ai( )i j jŠ i € i( €)i jAj+ Ai(D)i jAj+ Fi Fi+ tr ( [ i €, €] Š1 2 ff+D 2) + tr D+(PiFi+1 2 Pij i j+ h c .)](4.3.6) where 1 2 a a,Pi,Pijarede“nedin(4.1.13),( )i j= ATA,e tc.Theauxiliary “eldDcanbeeliminatedalgebraica llyusingits“eldequations.Thisleadstointeractiontermsforthespin -zero“eldsofthechiralmultiplets: Š UD= Š1 4 [ Ai( TA)i jAj+ trTA]2(4.3.7) inadditiontothoseobtainedbyeliminating F (see(4.1.14)). b.CP(n)models Insec.4.1.bwediscussedsupersymmetricnonlinear -modelswrittenintermsof chiral andantichirals uper“eldsthatarethecomplexcoordinatesofaK¨ ahlermanifold. Somenonlinear -modelscanbewritten linearlyifweintroducea(classically)non-propagatinggauge“eld.Weconsiderheresupersymmetricextensionsofthebosonic CP ( n ) models.Thebosonicmodelsarestraightforwardgeneralizationsofthe CP (1)modelof sec.3.10.Theyarewrittenintermsof( n +1)complexsc alar“elds ziconstrainedby zi zi= c ;theacti oniswrittenbyintroducinganabeliangauge“eldwithnokinetic term: S = d4x [ | ( €Š iA€) zi|2+D( | zi|2Š c )],(4.3.8) whereDisaLagrangemultiplier“eld.Eliminating A€byitsc la ssical“eldequation,

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1804.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSwe “ndtheactiongivenin(3.10.23).ThisactionisstillinvariantunderthelocalU(1) gaugetransformation z ei z z eŠ i where ( x )isarealpa rameter.Itcanbe rewrittenintermsof n +1 unconstrained“elds Ziintheform(3.10.30). Inthesupersymmetriccase,themodelismostconvenientlydescribedintermsof ( n +1)chiral“ elds(andtheircomplexconjugates ),andasingleabeliangauge“eld V .Theacti on,whichisgloballysupersymmetric, SU ( n +1)invar iant,andlocally gaugeinvariant,is: S = d4xd4 (i ieVŠ cV ).(4.3 .9) NotethepresenceoftheFayet-Iliopoulosterm.Uponeliminatingthegauge“eld V by its“eldequationwe“nd S = d4xd4 cln (i i).(4.3 .10) Thisactionisstillinvariantunderthe(local)abeliangaugetransformation ei Wecanusethis invariancetochooseagauge,e.g.,i=( c ua).Incomponents,(4.3.10) givestheactiongeneralizing(3.10.30)forthe CP ( n )non linear -modelcoupledtoa spinor “eld. Theaction(4.3.9)hasastraightforwardgeneralization: S = d4xd4 ( i( eV)i jjŠ ctrV ),(4.3 .11) where,asin(4.3.1), V = VATAand( TA)i jisa(ingeneralreducible)matrixrepresentationofthegenerators TAofsomegroup.However,incontrastto(4.3.9),whenwevary (4.3.11)withrespectto V ,wegetan equationthatingeneraldoesnothaveanexplicit solution: eVTA Š ctrTA=0.(4. 3.12) (Toderive(4.3.12),weusethecovariantvariation(4.2.48) V = VATA eŠ V eV,and tr V = tr V .)

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4.4.Superforms1814.4.Superforms a.General Inordinaryspacetime,thereisafamilyofgaugetheoriesthatcanbeconstructed systematically;thesetheoriesareexpressedintermsof p -forms p=1 p dx m1/ / \ \dx m2/ / \ \.../ / \ \dx mp m1 m2... mpwherethedierentialssatisfy dx m/ / \ \dx n= Š dx n/ / \ \dx m.Thetowero ftheorie sbased onformsis:0=scalar,1= v ectorgauge“eld,2=tenso r gauge“eld,3=a ux iliary“eld,and4=not hing “eld.Theirgau getransformations,“eldstrengths,andBianchiidentitiesaregivenby gaugetransformation: p= dKp Š 1, fieldstrength: Fp +1= d p, Bianchii dentity: dFp +1=0.(4. 4.1) Here Kp,p, Fpare p -formgaugeparameters,gauge“elds,and“eldstrengthsrespectively,and d = dx m m.Byde “nition, Š 1-formsvanish,and5-forms(or(D+1)-formsin Ddimension s)vanishbyantisymmetry.TheBianchiidentitiesandthegaugeinvariance ofthe“eldstrengthsareautomaticconsequencesofthePoincar ele mma dd =0. Insuperspacethesameconstructionispossible,usingsuper p -forms: p=( Š 1)1 2 p ( p Š 1)1 p dzM1/ / \ \.../ / \ \dzMpMp... M1(4.4.2a) (notetheorde ringoftheindices),wherenow dzM/ / \ \dzN= Š ( Š )MNdzN/ / \ \dzM,(4. 4.2b) thecoecientsofthefo rmaresuper“elds,and d = dzMM.Thesameto werof gauge parameters,gauge“elds,“eldstrengths,andBianchiidentitiescanbebuiltup(now usingthe super Poincar ele mma dd =0).Anadvant ageofthisdescriptionof”atsuperspacetheoriesisthatitgeneralizesimmediatelytocurvedsuperspaceanddetermines thecouplingoftheseglobalmultipletstosupergravity. However,superformsdonotdescribeirreduc iblerepresentation sofsuper symmetry unlessweimposec onstraints.Tomaintaingaugeinvariance,theseconstraintsshouldbe

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1824.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSimposedo ntheco ecientsofthe“eldstrengthform;whentheconstraintsaresolved, thecoecientsofthe(gauge)potentialformareexpressedintermsofprepotentials.In table4.4.1theprepotentials Apcorrespondtothe constr ained super p -form Apandthe expressions dA pcorrespondto dAp. p ApdA p 0 i ( Š ) 1 Vi D2DV 21 2 ( D+ D€ €) 3 V D2V 40 Table4.4 .1.Simplesuper“elds(prepotentials)correspondingtosuperforms InthisTableandarechi raland V isreal.Therelation Ap= A4 Š pcorrespondsto Hodge dualityofthecomponentforms. Theconstrainedsuper p -formscorrespondtoparticularprepotentials Apwhether Apisagaugeparameter Kp,apotentialp,a“eldst rength Fp,oraBianchiidentity ( dF )p.Theex p licitexpressionsfor Apintermsof Aptakethesameformwhether A is K ,, F ,or dF .Thusthe prepotentialsgiverisetoatowero ftheoriest hatmim ics(4.4.1): Thegauge“eldstrengthandBianchiidentitiesatonelevelarethegaugeparameterand “eldstrengthatthenextlevel.If Ap Š 1, Ap,and Ap +1arethegaugeparameter Kp Š 1,the gauge“eldp,andthe“el dstrength Fp +1superforms,respectively,thenthegauge transformation,“eldstrength,andBianchiidentitiesofthe prepotentials are gaugetransformation: p= dK p Š 1, fieldstrength: Fp +1= d p, Bianchii dentity: dF p +1=0.(4. 4.3) TheLagrangiansforall p -formtheoriesarequadraticinthe“eldstrengths,without extraderivatives.Wediscussde tailsinthesubsectionsthatfollow.

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4.4.Superforms183Underasupersymmetrytransformationth esuperfo rmsarede“nedtotransformas ( z, dz)=( z dz ),(4.4 .4) where(cf.(3.3.15)) dz=( d , d €, dx €)=( d , d €, dx€Ši 2 ( €d + d €)).(4.4.5) Consequently,thecoecientsMN ...mixundersupersymmetrytransformationsandthis makesitdiculttoimposesupersymmetric constraintsonthem.Tomaintainmanifest supersymmetry,wethereforegotoatangentspacebasis,parametrizedbythedualsof thecovariantderivatives DAratherthanthedualsof M.Weusethe”at superspace vielbeins DA M(3.4.16): DA= DA MM=( D, D€, €),(4.4 .6) andthe dualforms A dzM( DŠ 1)M A.(4. 4.7) From(3.4 .18),the D ssati sfy D[ ADB ) M= TAB CDC M,(4. 4.8) andhence d A=1 2 C/ / \ \BTBC A.(4. 4.9) Inthis -basiswewr iteasuperformas p=( Š 1)1 2 p ( p Š 1)1 p A1/ / \ \.../ / \ \ApAp... A1.(4. 4.10) Wealsohave d dzMM= ADA.Theta ngentspacecoecientsAp... A1ofthe p -form donotmixundersupersymmetrytransformationsbecause Aisinvariant.Wecannow imposesupersymmetricconstraint soni ndividualcoecientsofaform. Inthisbasis,thecoecientsofthe“eldstrengthform(onwhichweimposethe constraints) Fp +1= d phavethefollowingexpressi onintermsofthegauge“elds: FA1... Ap +1=1 p D[ A1A2... Ap +1)Š1 2( p Š 1)! T[ A1A2| BB | A3... Ap +1),(4. 4.11) wherethetorsiontermscomefrom(4.4.9).TheBianchiidentityon F takesasimilar

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1844.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSappearance.Equation(4.4.11)istheessentialresultweneedforthediscussionofsubsecs.4.4 .b-e. Wenows ummarizesomeoftheresultsofsubsecs.4.4.b-e.Inparticular,wegive theexplicitexpressionsforthecoecientsofthesuperforms Apintermsoftheprepotentials Ap(oftable4.4.1)forall p .Inthecaseofp,theseex pressionsarefoundby solvingtheconstraintsoncertaincoecientsof Fp +1andchoosingasuitable K -gauge ( = dK ).Theexpressionsfor K followfromthenewinvariancefoundwhensolving theseconstraints.Theexpressionsfor F fo llowfromsolvingthoseBianchiidentities dF that explicitly expressonepartof F intermsofanotherinthepresenceoftheconstraints.F inally,for dF ,theex p licitexpressionscorrespondtotheremainingpartofthe Bianchiidentitiesthatarenotalgebraicallysoluble.(Forclari“cation,seesubsecs.4.4.be,wheretheexpressionsareworkedoutindetail.)We“nd: p =0: A =1 2 ( A + A ); p =1: A= i1 2 D A A a=1 2 [ D€, D] A ; p =2: A= A€=0, A b= iC A€, A a b=1 2 ( C€€D( A )+ C D(€ A€ )); p =3: A= A€= A c=0, A€ c= T€ c A A b c= Š C€€C ( D ) A A a b c= d a b c[ D€, D] A ; p =4: A= A€= A€€= A d= A€ d= A€ c d=0, A c d=2 C€€C ( C ) A A b c d=2 a b c d D€ A A a b c d=2 i a b c d( D2 A Š D2 A ),(4.4 .12) whereforeven p D€ A =0,and forodd p A = A .

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4.4.Superforms185Forexample,inthecaseof thevectormultipletofs ec.4.2,wefoundthevector representationpotentialsAgivenbythecase p =1above,with A = V (inthe v ector representation,andinthegaugewhere= =1 2 V ;then K =1 2 (+ ),asgiven aboveby p =0); the“eldstrengths FABby p =2above,with A= W= i D2DV (by table4.4.1);andtheremainingBianchiidentity dF on Wby p =3above,with A =1 2 ( DW+ D€ W€)( againbytable4.4.1; dF =0thusre ducesto A ( W)=0).Further exampleswillbederivedintheremainderofthissection.(Notethatanaction writtenintermsofasuper0-formdoes not describethe most general chiralmu ltiplet theory:T he“eldstrength FA= DAalwayshasth einvariance = k ,where k isa realconstant.Here F = [ i ( Š )]=0for = k .Thisinvar ianceexcl udesmass terms,and hasconsequencesevenforthefreemasslessmultipletwhenitiscoupledto supergravity.) b.Vectormultiplet Asanintroduction,wedescribetheabelianvectormultipletinthelanguageof superforms.Webeginwitharealsuper1 Š form 1= + € €+ a a,(4. 4.13) withgaugetransformation 1= dK0,where K0isa0 Š form(scalar).The“eld strengthisasuper2 Š form F2= d 1,withsuper “eldcoecientsthatfollowfrom (4.4.11): F = D( ), F ,€= D €+ D€Š i €, F b= D bŠ b, F a b= Š1 2 C€€( € )€+ h c ..(4.4 .14) Weimposeaconvent ionalconstraint F ,€=0whicha lgebraicallydetermines€.We furtherrestricttheformbyim posingthec onstraint(4.2.44) F =0.Theso lutionto theconstraintsis

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1864.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS= iD, €= Š i D€ a= Š i ( D €+ D€),(4.4 .15) andtheprepotentialtransformsas = Š iK0+ .(4.4 .16) Thetransformationsareaninvarianceof1introducedbysolvingtheconstraints. Itisalwaysobvious,byexaminingequationssuchas(4.4.14),whatconventional constraintscanbeimposed.Findingadditionalconstraintsismoredicult.Ingeneral, ifwewishtode scribeamultipletthatcontainsa componentp -form,werequirethatit bethe =0compon entofasuper p -formcoecientwithonly v ectorindices (e.g.,in (4.4.14) F a b| istheYang-Mills“eldstrength),andthereforewewillnotconstrainthis coecient.Forthesamereasonweassigndimension2tothiscoecient,andthisdeterminesthedimensionofthesuperform.Asaconsequence,coecientswithmorethan twospi norindiceshavetoolowdimensiontocontaincomponent“eldstrengths(orauxiliary“elds),andmustbeconstrainedtozero.Wealsoconstraintozerocoecients thatcontainatthe =0levelcomponentfo rmsthatarenotpresentinthemultiplet. c.Tensormultiplet c.1.Geometricformulation Theantisymmetric-tensorgaugemultipletcontainsamongitscomponent“eldsa second-rankantisymmetrictensor(2-form).Todescribeitinsuperspaceweconsidera super2-form2: Š 2=1 2 / / \ \ + €/ / \ \ ,€+ b/ / \ \ b+1 2 b/ / \ \ aC€€( )+ h c ., (4.4.17) wherew ehaveusedthesymmetriesoftowrite a b= C€€( )+ h c ..Thegauge variations 2= dK1are = D( K ), ,€= D K€+ D€KŠ iK€,

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4.4.Superforms187 b= DK bŠ bK, ( )= Š1 2 ( €K )€.(4. 4.18) The“eldstrengthsfollowfromthede“nition(4.4.11): F , =1 2 D( ), F ,€= D( )€+ D€ + i ( )€, F , c= D( ) c+ c , F ,€ c= D € c+ D€ c+ c ,€Š iC€€( )Š iC(€€ ), F b c= C€€( D( )Š1 2 ( € )€ )+ C( D(€€ )+1 2 (€€ ), ), F a b cŠ a b c dF d= Š i ( C€€CF€Š CC€€F€), F€= Š i ( € (€€ )Š €( )).(4.4 .19) wherewehaveused(3.1.22). Wecanimposet woconv entionalconstraints.The“rst, F ,€=0,(4. 4.20) gives ( )€= i [ D( )€+ D€ ],(4.4 .21) whichimplies €= iC €+ i [ D ,€+1 2 D€ ],(4.4 .22) foranarbitraryspinor €.Thes econdconventionalconstraint, F( ,€ )€=0,(4. 4.23) gives

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1884.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS( )= Š i1 4 [ D( € )€+ D€( )€Š ( € )€],(4.4 .24) whichimplies ( )=1 2 D( )Š1 2 D2 Š1 2 i ( € ),€.(4. 4.25) Thepotential ,€ispuregauge:Itcanbegaugedtozerousing(4.4.18).Toeliminate theremaini ngunwantedphysicalstateswechoosetwoadditionalconstraints F , = F , c=0.(4. 4.26) The“rstimplies ispuregauge,andth es econdimposes D €=0, D€=0.(4. 4.27) Inthegauge = ,€=0,allofABisexpr essedintermsof;thusthesuper“eld isthechiralspinorprepotentialthatdescribesthetensorgaugemultiplet. Theconstraintsalsoimplythatallthenonvanishing“eldstrengthscanbe expressedintermsofasingleindependent“eldstrength G = Š1 2 ( D+ D€ €).(4.4 .28) Forexample, F ,€ c= i €€G = T€ cG .(4. 4.29) G isalinearsuper“eld: D2G =0 .I ti si nv ar ia nt undergaugetransformationsoftheprepotential = i D2DL L = L .(4. 4.30) Projectingthecomponentsofwehave: =| t=1 2 D( )| =( )| A+ i B= Š D| = D2| (4.4.31) Thecomponentsofthegaugeparameterthatenter are: L= i D2DL | ,

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4.4.Superforms189L(1)= D D2DL | = L(1), L( )= i1 2 D( D2D )L | =1 2 ( €[ D ), D€] L | Š1 2 ( €L )€, L€= L€.(4. 4.32) Thecomponents andBcanbealgebraicallygaugedawayby Land L(1)respectively, whereas L€istheparameteroftheusualgaugetransformationforthetensorgauge “eld t.Thespinor isthephysicalspinorofthetheory(uptotermsthatvanishin theWZgauge).Thegaugeinvariantcomponentsarefoundbyprojectingfromthe“eld strength G : A= G | = DG | =1 2 ( Š i € €), f a= F a| =[ D€, D] G | = i ( €tŠ € t€€), D2G = D2G =0.(4. 4.33) Sincethereisonlyonephysicalspinorinthemultiplet, G hasdimensi onone.This determinesthekineticactionuniquely: Sk= Š1 2 d4xd4 G2.(4. 4.34) Thecorrespondingcomponentactionis Sk= d4x [1 4 A A+1 4 ( f a)2+ €i €].(4.4 .35) Notethatnoneofthe“eldsisauxiliary.Thephysicaldegreesoffreedomarethoseof thescalarmultiplet .Onshe ll,theonlydierenceisther eplacementofthephysical pseudoscalarbythe“ eldstrengthofthean tisymmetrictensor.

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1904.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSc.2.Dualitytransformationtochiralmultiplet Wecanwritetw o“rsto rderactionsthatareequivalentto Sk.Intro ducingan auxiliarysuper“eld X ,wed e“ne S k= d4xd4 [1 2 X2Š GX ]. (4.4.36a) Varying X andsubstitutingtheresultbackinto S k,wereobtain Sk.Wealsoseethat thetensormultipletis classicallyequivalenttoachiralscalarmultiplet:Varying,we obtain D2DX =0,whichis solvedby X = + D€ =0 Substitutionbackinto S kyieldstheusualkinetica ctionforachiralscalar (b ecause ischiraland G islinear, d4xd4 G =0).B ecausethesame“rstorderactioncanbeusedtodescribethetensormultipletandthechiralscalarmultiplet,wesaythattheyare dual toeachother. Alternativel y,wecanwrite S k= d4xd4 [ Š1 2 X2+( + ) X ].(4.4 .36b) Varying X andsubstitutingtheresultbackinto S k,weobtaint heusualkineticaction forthechir alscalar ;varying ,we “nd D2X = D2X =0,whichis solvedby X = G Substitutionbackinto Syields Sk(4.4.34). Thetensormultipletadmitsarbitrary(nonrenormalizable)self-interactionswitha dimensionalcouplingconstant : S = 2 d4xd4 f ( Š 1G ).(4.4 .37) Thecomponentactioncontainsquarticfermi onself-interactionsandYukawaterms €F€,multi p liedbyderivativesof f ( Š 1A ).Remarkably,wecanperformthedualitytransformationtoachirals calarmultipletevenintheinteractingtheory.The“rst orderactionequivalentto S is: S= 2 d4xd4 [ f ( X ) Š Š 1( + ) X ].(4.4 .38) Varying ,we “nd X = Š 1G (thenormalizationcanbechosenarbitrarily),andreobtaintheinteractingaction(4.4.37).Varying X ,we “ndthedualactionintermsof :

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4.4.Superforms191S= 2 d4xd4 IK (Š 1( + ))(4. 4.39) where IK isthe Le gendretransform of f : IK (Š 1( + ))= f ( X( Š 1( + ))) Š Š 1( + ) X (Š 1( + )), f ( X ) X Š 1( + ).(4.4 .40) Thedualaction(4.4.39)isrecognizableastheactionforanonlinear -model(seesec. 4.1.b,e.g.(4.1.23)). Wecanalsop erformthereversedualitytransfo rmation,thatis,startwithatheorydescribedbyachiralscalarsuper“eldand“ndanequivalenttheorydescribedbya tensormultiplet.Althoughwecan“ndthemodeldualtoanarbitrarytensormultiplet model,thereverseisnottrue:Forachiralscalarmodel,possiblywithinteractionsto otherchiraland/orgaugemultiplets,wecan“ndthedualtensormodelonlyiftheoriginalactiondependsonlyon + ,orequivalen tly,de “ning eŠ 1,on .Thus, startingwithanaction S= 2 d4xd4 IK (Š 1( + ))(4. 4.41) wecanwrite the“rstorderaction S= 2 d4xd4 [ IK ( X )+ Š 1GX ](4. 4.42) Varying G yields X = Š 1( + )and(4.4 .41),whereasvarying X leadsto(4.4.37), wherenow f isthe(inverse)Legendretransformof IK : f (Š 1G)= IK ( X( Š 1G ))+ Š 1GX (Š 1G), IK ( X ) X = Š Š 1G .(4. 4.43) Wecannow “ndasecondtensormultipletmodeldualtothe free chiralsc alarmultiplet.Web eginwith S= 2 d4xd4 = d4xd4 eŠ 1( + )(4.4.44)

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1924.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSWewritethisin“ rstorderformas S imp= 2 d4xd4 [ eXŠ GX ],(4.4 .45) and “ndthedualaction Simp= Š 2 d4xd4 GlnG ,(4. 4.46) wherenow G hasanonvanishingclassicalvacuumexpectationvalue.Thisdualityholds eveninthepresenceofsupergravity,wheretheequivalenceistothesuperconformal formofthescala rmulti plet( ),asopposedtothe( + )2formobtainedfrom (4.4.36);ingeneralcurvedsuperspace,thesetwoLagrangiansaredierent.Themodel describedbytheaction Simp(4.4.46)iscalledtheimprovedtensormultiplet,because, unliketheunimprovedaction(4.4.34), Simpisconformallyinvariant.(Bothareglobally scaleinvariant,buttheactionforananti symmetric tensorbyitselfisnotinvariant underconformalboosts.) Itisinterestingtostudywhathappenstotheinteractionsofachiralmultiplet afteradualitytransformation.Hereweconsiderinteractionswithagaugevectormultiplet(forotherexamples,seesecs.4.5 e,4.6,5.5).Foranactionoftheform Sgauge= d4xd4 IK ( + + V)+ d4xd2 W2(4.4.47) where V isanabeliangaugesuper“eld, W isits“eldstrength,and IK ( + + V) IK ( ln( eV )),wecanwritethe“rstorderaction S gauge= d4xd4 [ IK ( X + V )+ GX ]+ d4xd2 W2.(4. 4.48) Varying G gives(4.4.47);varying X gives S G= d4xd4 [ f (G) Š GV ]+ d4xd2 W2.(4. 4.49) Thusthegaugeinteractionsoftheoriginaltheoryaredescribedbythesingleterm GV inthedualtheory(thiscouplingisgaugeinvariantbecause G islinear).Observethat fortheusualkineticterm IK = eV ,the dualtheoryhasthei mprovedLagrangian Š GlnG Š GV (4.4.46)ratherthan Š1 2 G2Š GV (4.4.34).Itisstraightforwardtoverify

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4.4.Superforms193thatthelattertheorydescribesamassivevectormultipletratherthanascalarcoupled t oavector.Anoth erwaytodescribeamassivevectormultiplet,butwithoutvector “elds,isintermsofthechiralspinoralonebyaddingamassterm(whichbreaksthe gaugeinvariance(4.4.30))to Sk(4.4.34): Sm= Š1 2 m2 d4xd2 ()2+ h c ..(4.4 .50) Sk+ Smdescribesamassivevector mult iplet.Thecomponentantisymmetrictensor describesamassivespin1“eld, and describeamassiveDi racspinor,Aisamassi ve sc alar,andBisauxiliary. d.Gauge3-formmultiplets d.1.Real 3-form Webeginb yconside ringa real 3-form.Itha sthefo llowingi ndependentcoecientsuper“elds , , ,€, , c, ,€ c, ,( ), ,(€€ ), a,(4. 4.51) wherew ehaveusedthesymmetriesoftowritei tinte rmsofLorentzirreduciblecoecients. b c= C€€ ,( )+ C ,(€€ ), a b c= Š a b c d d= Š i ( C€€C€Š CC€€€),(4.4 .52) Theindependent“eldstrengthsare F , , F , ,€, F ,€ ,€, F , d, F ,€ d, F ,( ), F ,(€€ ), F ,€ ,( ),

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1944.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSF b, F .(4. 4.53) Thelast“ve“eldstrengthsareLorentzirreduciblecoecients,e.g.(see3.1.22)), F a b c d= F a b c d= i ( CCC€€C€€Š CCC€€C€€) F .(4. 4.54) Weimposethefo llowingco nstraintsonthe“eldstrengths: F , = F , ,€= F ,€ ,€=0, F , d= F ,€ d= F ,(€€ )= F ,€ ,( )=0, F ,( )=2 C ( C ) ,(4.4 .55) where i sa n undeterminedgaugeinvariantsuper“eld.Solvingtheconstraintsgives , = ,€= , c= ,(€€ )=0, ,€ c= iCC€€V ,( )= Š C ( D )V €=[ D€, D] V V = V ,(4. 4.56) uptoapuregaugetr ansformationofABC.Giventhesol ution,we“nd = D2V .(4. 4.57) Theprepotential V has gaugetransformations V = Š1 2 ( D+ D€ €), D €=0.(4. 4.58) Thephysicalcomponent“eldsofthismultipletare = | = D2V | = D | = D D2V h =( D2+ D2 ) | = { D2, D2} V | f = Š i ( D2 Š D2 ) | =1 2 a a| =1 2 €[ D€, D] V | .(4. 4.59)

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4.4.Superforms195Thequantity f isthe“eldstrengt hofthecomp onentgaugethree-form l€=€| .The componentthree-formtransformsas(cf.(4.4.33)for f a) l€= i1 2 ( €D( )Š € D(€ € )) | ,(4. 4.60) sothatits“el dstrength f isinvariant. The“eldstrengthisachiral“eldofdimensionone(determinedby ),and hencethekineticactionis S = d4xd4 .(4.4.61) Itgivesconventionalkinetictermsforthecomponents and ;thescal ar“eld h isan auxiliary“eldandthegauge“eld l€enters theactionthroughthesquareofits“eld strength f Suc ha “e lddoesnotpropagatephysicalstatesinfourdimensions. Theonlydierencebetweenthismultiplet,describedby,andtheusualchiral scalarmultip letisthereplacementoftheimagina rypart(thepseudoscalar“eld)of the F auxiliary“eldbythe“eldstrengthofthecomponentgaugethree-form.Massand interactiontermsforcanalsobeusedfor.However,atthecomponentlevel,after e liminationoftheauxiliary“eldsthetheoriesdier:Wenolongerobtainalgebraic equations,since f isthederivativeofanother“eld l€.Another dierenceisthatthe superthree-form gaugemultiplet ca nnot be co upl ed toYang-Millsmultiplets. d.2.Comple x3-form Acomplexsupert hree-formmult ipletcanbetreatedinthesameway.Ithasmore i ndependentcoecientsuper“elds: , , ,€, ,€ ,€,€ ,€ ,€, , c, ,€ c,€ ,€ c, ,( ), ,(€€ ),€ ,( ),€ ,(€€ ), a.(4. 4.62) (Forexample, (€€ ) = € ( ).)Correspondi ngly,therearemoreindependent“eld

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1964.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSstrengths.Theseare F , , F , ,€, F ,€ ,€, F ,€ ,€ ,€, F€ ,€ ,€ ,€, F , d, F ,€ d, F ,€ ,€ d, F€ ,€ ,€ d, F ,( ), F ,(€€ ), F ,€ ,( ), F ,€ ,(€€ ), F€ ,€ ,( ), F€ ,€ ,(€€ ), F b, F€ b, F ,(4. 4.63) Theconstraintshowever,setmore“eldstrengthstozero.Thenonzeroonesare F ,( ), F b, F ,(4. 4.64) andwestillimposetheconstraint: F ,( )=2 C ( C ) .(4.4 .65) Theonlyformcoecientsthatarenotpuregaugearegivenby ,( )= Š C ( D ) D€ €, € ,(€€ )= Š C€ (€ D€ ) D€ €= C€ (€ D2 € ), ,€ c= iCC€€ D€ €, a=[ D€, D] D€ €,(4. 4.66) (uptoarbitrarygaugetransf ormationterms).Theseexpre ssionsallowustocompute; we “ndthatitisexpressedint ermsoftheprepotentialasfollows: = D2D, D€=0.(4 .4.67) Thegaugetransformationsoftheprepotentialare =+ DL( ), D€=0.(4. 4.68) Thecomponentscontainedinthe“eldstrengthare A = | = D2D| ,

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4.4.Superforms197= D | = D D2D| f = D2 | =i 2 €[ D€, D] D| ,(4. 4.69) where f isthe“eldstrengthofa complex 3-form. Thismultipletcanbedescribedintermsoftworealsuper3-formmultiplets: =1+ i 2.Theco nstraintsimp osedabovearetheonesgiveninsec.4.4.d.1for1and2,plusthea ddition alconstraint F€ ,€ ,(€€ )=0.Thisi ssimplyt heconstraint 1+ i 2= D2( V1+ iV2)=0,whic himp lies V1+ iV2= D€ €. The“eldstrengthischiralandofdimensionone.Thereforealloftheactionformulaefortheusu alchiralscalarcanbeusedfor.Asfortherealgaugethree-form mu ltiplet,theequationsofmotionfortheauxiliary“eldsarenolongerpurelyalgebraic. Again,thismultipletcannotbecoupledtoYang-Millsmultiplets. e.4-formmultiplet The“nalsuperformweconsiderhas no physicaldegrees offreedom.Itis describedbyarealsuper4-formABCD.The “eldstrengthsupertensorisasuper5-form FABCDE.Therefo rethe“eldstrengthwithall“vevectorindicesvanishesbyantisymmetry. Asconstraintsweimposetheequations FABCDE=0.Thisi mpliesthatallof ABCDispuregauge.Sinceall“eldstrengthsvanish,nogaugeinvariantactionispossibleattheclassicallevel.However,thismul tiplet(andthecorre spondingcomponent form)hassomeunusu alpropertiesatthequantumlevel,becauseitsgauge“xingtermis notzero.

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1984.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS4.5.Othergaugemultiplets a.GaugeWess-Zuminomodel In(3.5.3)wenotedthatachiralsuper“eldcanbeexpressedintermsofan unconstrainedsuper“eld = D2.(4.5 .1) The“eldprovidesanalternatedescriptionofthescalarmultiplet.Theactionswe consideredinsecs.4.1-2canbeexpressedintermsof.Forexample,theWess-Zumino action(4.1.1-2)becomes S = d4xd4 [( D2 )( D2)+1 2 m ( D2+ D2 ) + 3! (( D2)2+ ( D2 )2)],(4.5.2a) wherew ehaveused d4xd2 ( D2)2= d4xd4 D2,(4.5 .2b) etc. Thesolution(4.5.1)ofthechiralityconstraintintroducestheabeliangaugeinvariance = D€ €(4.5.3) where isanunconstrainedsuper“eld.Thegaugeinvariantsuper“eldisthechiral “eldstrengthofthe gaugesuper“eld,andtheactionisobviouslyinvariant.The gaugetransformationcanbeusedtogotoaWZgauge,byalgebraicallyremovingallthe componentsofexceptthosethatappearin.Inthisformulationthecouplingto su pe rY ang-Millscanbeachievedbycovariantizingthederivatives:Ifiscovariantly chiral,then= 2, = €€.U nde rY ang-Millsgaugetransformationstransformsinthes amewayas.

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4.5.Othergaugemultiplets199b.Thenonminimalscalarmultiplet Thismultiplethasanumberofinterestingfeatures:(a)Itisamultipletwhere thespinofauxiliary“elds ex ceeds thatofthephysical“elds ;(b) noneofthecomponent “elds(inaWess-Zuminogauge)ofthismultipl etaregauge“elds,eventhoughthemultipletisdescribedbyagaugesuper“eld;(c)thismultiplet,unlikeotherscalarmultiplets, formsa reduci ble representationofsupersymmetry. Weintro duceageneralspinorsuper“eldwiththegaugetransformation = DL( ), L( )ar bitrary.Anactionthatisinvariantunderthisgaugetransformationis S = Š d4xd4 ,= D€ €,(4. 5.4) The“eldstrengthsatis“es D2=0,soth atitisa complex linearsuper“eld;incontrast,the“eldstrengthofthetensorgau gemultipletisareallinearsuper“eld. Thecomponent“eldsofthemultipletare A = | €= D€ | = D | P€= D€D | F = D2 | €=1 2 D D€D | .(4. 5.5) Thecomponentactionis S = d4x [ A A + €i €Š| F |2+2 | P€|2+ + € €],(4.5 .6) withpropagatingcomplex A and .A lltheother“eldsareauxiliary. Intermsofsuper“eldswecanseethattheaction(4.5.4)describesascalarmultiplet.Theconstraintand“eldequationsfor are: D2 =0, D€ =0.(4 .5.7) Thesearethesameasthosefortheon-shellchiralscalarmultiplet,butwithconstraint and“eldequationinterchanged.

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2004.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSTo se et he reducibilityofthismultipletweusethesuperprojectorsofsec.3.11. Theactioncanbewritten S = d4xd4 €i €[21,0+(2,1 2 ++2,1 2 Š)], 1,0= Š1 2 Š 1D D2D, 2,1 2 = Š 1 D2D1 2 ( D D€ €).(4.5 .8) Thusthemultipletconsistsofthreeirreduciblesubmultiplets:oneofsuperspin0,and twoofsuperspin1 2 Incontrasttothechiralscalarmultiplet ,itisnotpo ssibletointroducearbitrary massandnonderivativeself-interactionterms.However,wecanwritedowntheaction S = d4xd4 f (, ),(4.5.9) where f ( z z )= f ( z z ).Thus,forexample,itispossibletoformulatesupersymmetric nonlinear -modelsintermsofthenonminimalscalarmultiplet.Furthermore,thenonminimalmultipletcanbecoupledtoYang-Millsmultipletsbycovariantizingthederivatives:= *** Justasforthetensormultiplet(sec.4.4.c),wecanexhibitthedualityofthenonminimalscalarandchiralmultipletsbywritinga“rstorderaction.Mostofthediscussionofsec.4.4.c.2hasananalogforthenonminimalscalarmultiplet,except,sincethe mult ipletisdescribedbyalinearsuper“eld,theLegendretransformistwodimensional andhencethereisnorestrictionontheformofthenonlinear -modelthatcanbe described.Thetwo“rstorderactionsequivalentto(4.5.9)are(see(4.4.38,42)): S= d8z [ f ( X X ) Š X Š X ], D€= 0, (4.5.10a) and

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4.5.Othergaugemultiplets201S= d8z [ IK ( X X )+ X + X ], D2=0,(4.5 .10b) where X isacomplex unconstrainedsuper“eldand IK istheLeg endretransformof f Justasforthetensormultip let,thisdualitytransformationcanbeperformedevenin thepresenceofinteractionswithothermultiplets(e.g.,supergravity). c.Morevariantmultiplets Aswehaves een,severalinequivalentsuper“eldformulationscandescribethe samesetofphysicalstates.The(0,1 2 )multi pletcanbedescribedbyachiralscalar,a gaugetwo-form,real(orcomplex)gaugethree-forms,oragaugespinor.Thechiral scalarprovidesthesimplestrepresentation.Allbutoneoftheotherrepresentationsare obtainedbyreplacingeitherthephysicalorauxiliary“eldbycomponent2-formsor 3-formsrespectively.Wecallthesevariantrepresentationsofthescalarmultiplet.In general,variantrepresentationsareveryrestr ictedineithertheirself-interactionsorcouplingstoothermultiplets.Inthissubsectionwediscussvariantvectorandtensormultiplets. c.1.Vectormultiplet We ha ve de sc ribedsuperYang-Millstheoriesintermsofahermitiangaugeprepotential V .Itcontainsaco mponentvectorasitshighestspincomponent: A a=1 2 [ D€, D] V | .Thereis,howev er,asmallersuper“eldthatcontainsacomponent v ector:Achiral dotted spinor€( D€€=0),hasa sitshighests pincomponent A aŠ i ( D€ + D€) | .(4. 5.11) Thesuper“eld€isreducible;itcanbebi sected(see(3.11.7)1 2 (1 K K )€=1 2 (€Š+ Š 1 D2i € ).(4.5 .12) Sincewewanttodescr ibeagaugetheory,wegaugeawayoneoftherepresentations insteadofconstrainin git.Thetra nsformation

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2024.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS €= D€( D+ D€ €)= D2 €Š i €=(1+ K K ) D2 €, D €=0,(4. 5.13) canbeusedtogaugeaway(1+ K K )€,andleaves(1 Š K K )€inert.Thegaugeparameter D+ D€€describesthetensormultipletofsec.4.4. The“eldstrengthfor isthelowestdimensionlocalgaugeinvariantsuper“eld: W= D2(1 Š K K ) = D2 + i €€.(4. 5.14) The“eldstrength Wisthefamiliarchiral“eldstrengthofthegaugemultiplet describedby V butnowwith= ,€=€, a= Š i ( D€ + D€),and A a= a| .I tscomponentsarethesame,exceptfortheauxiliary“eldD: W| f1 2 D( W )| =1 2 ( €A )€, D 1 2 iDW=1 2 €( D€ Š D€) | =1 2 €B€.(4. 5.15) We thusseetheauxiliarypseudoscalarhasbeenreplacedbythe“eldstrengthofagauge three-form.Theactionisstill(4.2.14),and incomponentsdiersfromtheusualvector mult ipletonlybythe replacementD1 2 €B€. Thisvariantformofthevectormultipletcanalsobeobtainedfromthecovariant approachofsec.4.2:Intheabeliancase,wecansolvetheconstraint F= D( )=0 by= .Justast heusualsolution= Š i1 2 DV directlyintermsofthereal scalarprepotential“xedsomeofthe K invariance(correspondingtoagaugecondition D D2= Š D€D2 €,whichim p lies K =1 2 (+ )),thevariantsolution“xessomeof the K invariancewiththegaugecondition D=0(which,t ogetherwiththeconstraint, impliesthatisantichiral),reducingitto K = D+ D€ €. Thecovariantderivativescanbeusedtocouplethisabelianmultiplettomatter. However,= isnotasolutiontothenonabelianconstraints,nortotheabelian onesingeneralcurvedsuperspace.Thus,likeothervariantmultiplets,itislimitedin thetypesofintera ctionsitcanhave.

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4.5.Othergaugemultiplets203c.2.Tensormultiplet Thevariantrepresentationforthetensormultipletisdescribedbythesamechiral spinorsuper“eldastheusualone(4.4.27),butthegaugetransformationischanged. Inplaceoftherealscalarparameter L (4.4.30),weusethechiraldottedspinor€. Explicitly,themodi“edgaugevariationis(cf.(4.5.14)) = D2 + i €€.(4. 5.16) Thisleadstotheusualtransformationsfor t( )andleaves A and invariant(see (4.4.31)).Butthevariationofthecomponent“eld B = i1 2 ( D€ €Š D) | is B = Š1 2 €v€.(4. 5.17) Thereforethiscomponent“eldisagaugefour-form. Theactionforistheusualoneproportionalto G2,andthefourformdoesnot appearin G2andintheaction.However,atthequantumlevel,thefour-formwould reappearingauge“xingterms,andchiraldottedspinorswouldappearasghosts. d.Super“eldLagrangemultipliers Wehavegi venanumberof examplesofsupersymmetr ictheori esthatdescribe thescalarmultipleton shell(samephysicalstates)butareinequivalentoshell.They dierprimarilyinthetypesofinteractionstheycanhave.Sofar,wehavefoundthat thesimplestformulationofthescalarmultiplet,achiralscalarsuper“eld(or,equivalentlyevenoshell, D2onageneralscalar),hasthemostgeneralinteractions.However,inextendedsupersymmetrynoneoftheknown N =2theori esequivalenton-shell tothe N =2scal armultipletcanhavealltheinteractionsknownfromon-shellformulations.Wenowintro duceaformofthe N =1scal armultipletthati sas ubmultipletof ano-shellformulationofthe N =2scal armultiplet.Itsmostdistinctivefeatureisa super“eldthatappearsonlyasaLagrangemultiplier.Thisformulationhassomedrawbacksincommo nwiththetensormu ltiplet(anothertheoryequivalenttothescalarmultipletonshell),towhichitis closelyrelated:(1)Itdoesnothaverenormalizableselfinteractions(i.e,tho secorrespondingtotermsd4xd2 P()),(2)itisrestrictedinits couplingstosupergravity,and(3)itisnotano-shellrepresentationofthe(chiral) U (1) symmetrywhichthescalarmultiplethasonshell(correspondingto= ei ).Onthe

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2044.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSotherha nd,unlikethetensormultiplet(andmostv ariantformsofthescalarmultiplet), itdoescoupletoYang-Mills.However,becauseof(3),itcanonlybea real representationofanyinternalsymmetrygroup,andcouplestoYang-Millsaccordingly(e.g.,itcan coupletoa U (1)vectormultipletonlyasadoubletofoppositecharges). Theformulationisdescribedbyageneralspinorgaugesuper“eldwithatermin theactionlikethatofthechiralspinorgaugesuper“eldofthetensormultiplet,anda realscalarsuper“eldLagrangemultiplierwithatermintheactionthatconstrainsto zerothesubmultipletsintheformertermthatdontoccurinthetensormultiplet. Explicitly,theactionis S = Š d4xd4 (1 2 F2+ YG ), F =1 2 ( D+ D€ €), G = i1 2 ( DŠ D€ €);(4.5 .18) withgaugeinvariance = DL( )(4.5.19) intermsofageneralsuper“eldgaugeparameter.TheBianchiidentitiesand“eldequationsare: Bianchiidentities : D2( F Š iG )= 0, (4.5.20a) fieldequations : D€( F + iY )= G =0.(4. 5.20b) IfwemakeadualitytransformationbyswitchingtheBianchiidentitieswiththe“eld equations,weobtaintheusualformulationofthescalarmultiplet,withtheidenti“cations F =1 2 (+ ), Y =1 2 i ( Š ), G =0.(4. 5.21) Intermsofirreduciblerepresentationso fsupersy mmetry,thistheorycontains superspins1 2 +1 2 +0inand1 2 +0in Y .There presentationsin Y setthecorrespondingonesintozeroonshell,leavingtheremainingoneasatensormultiplet. However,unlikethetensormultiplet,thephysicalspinzerostatesareallrepresentedby scalars:Thevectorobtainedbyprojectionfrom[ D€, D] F isanunconstrained

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4.5.Othergaugemultiplets205auxiliary“eld,appearingat 2 orderin,whereasthecorre spondingvectorinthetensormultipletisthetransverse“eldstrengthofthetensorappearingat orderi nthe chiralsp inor.Thist heoryh asthesamecomponent-“eldcontentasthenonminimal sc alarmultipletplusanauxiliaryrealscalarsuper“eld. CouplingtoYang-Millsisstraightforward;however,sincebothand appearin F andin G mu sttransformundera real representationoftheYang-Millsgroup.We covariantizebyreplacingthespinorderivativesinthede“nitionsof F and G byYangMillscovariantspinorderivatives.Invaria nceoftheactionundertheYang-Millscovariantizationof(4.5.19)thenrequires 0= = L( )= Š i1 2 FL( ),(4. 5.22) implyi ngthesamerepresentation-preservingconstraint F=0asforthech iralscalar formulation.ThetotalsetofBianchiidenti tiesand“eldequationsisthesameasforthe chiralsc alar. Justasthereisanimprovedformofthete nsormult iplet,withsuperconformal invariance,thereisanimprovedformofthiss calarmultiplet.Inanalogytothetensor mult iplet,itisobtainedbyreplacing1 2 F2in(4.5.18)with FlnF (cf.(4.4.46)).Furthermore,the“rst-orderformulationofthismultipletturnsouttobeequivalenttothe“rstorderformulationofthe nonminimal scalarmultiplet.Westartwith(cf.(4.4.45)) S = d4xd4 [ eXŠ XF Š YG ].(4.5 .23) The“rstorderformofthenonminimalscalarmultipletisusuallywrittenas(4.5.10b): S = d4xd4 [ X XŠ ( XD + h c .)].(4 .5.24) Uponeliminationofthecomplexscalar X,thisgivesth es econd-orderformofthenonminimalscalarmultiplet(4.5.4).Uponeliminationofthespinor ,weobtai nthe constraint D€X=0,whose solution X=inte rmsofachiralscalargivestheminimal scalarmultiplet(provingtheirduality).Theequivalenceoftheactions(4.5.23)and (4.5.24)followsfromthechangeofvariables = X Š 1, X= e1 2 ( X + iY ),(4. 5.25)

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2064.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS(someintegrationby partsisnecessarytoshowtheequivalence). e.Thegravitinomattermultiplet Thusfarwehaveconsideredsupermultipletswithphysical“eldsofspinoneor less.Weconcludeourdiscussionofglobal N =1mult ipletsbyconsideringonewitha spin1and aspin3 2 (gravitino)component “eld.Itispossiblet odiscu ssitwithout introducingsupergravityonlyifthemultipletdescribesafreetheory.Thegravitino mult ipletisofinterestb ecausemanyofthefeaturesencounteredinthesuper“eldformulationofsupergravity,sucha si rreduciblesubmultiplets,compensators,andinequivalent o-shellformulations,arealreadypresent. e.1.O-shell“eldstrengthandprepotential Fo llowingthediscussionofsec.3.12.awedes cribethismultipleton-shellwithcomponent “eldstrengths (v ector“eldstrength)and (theRarita-Schwinger“eld strength ),totallysymmetricintheirindices.Wedenotetheo-shellsuper“eldstrength correspondingto by W.Itisachiral“e ldstrengthofsuperspin1,nobisectionis po ssible( s +1 2 N =3 2 isnotaninteger),andthereforewewrite(see(3.13.1)) W=1 2 D2D( ),(4. 5.26) intermsofageneralspinorsuper“eld.From dimensionalanalysis(thegravitino“eld hascan onicaldimension3 2 ),thedimensionof W is2. Thegaugetransformationsthatleave W invariantare =+ D, D€=0, = ,(4.5 .27) Toan alyzethetransformationsofthecomponents,wede“ne t= W| =1 2 D2D( )| = D( W )| =1 2 D( D2D )| = i1 2 ( €1 2 [ D€, D] )| = DW| =1 2 D D2D( )| ,

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4.5.Othergaugemultiplets207f= D2W| =1 2 D2 D2D( )| = i1 2 ( €D2 D€ )| .(4. 5.28) Weidentify thegravitinoand(comple x)v ector“eldstrengths, and f.Theco rrespondingcomponentgauge“eldsappearinandaregivenby a = i1 2 [ D€, D]| A a= iD2 D€| ,(4. 5.29) Theirgaugetransformationsare a =[1 2 a( D Š )+ iC D€D2] | A a= Š aD2 | .(4. 5.30) Thegravitino“eld,inadditiontoundergoingaRarita-Schwingergaugetransformation describedbythe“rstterm,alsoistranslatedby iC €, €= D€D2 | .(When coupled to N =1superg ravitytogive N =2superg ravity,thetransformation(4.5.30)ispartof the N =2superconfo rmalgroup:The“rsttermbecomesthesecondlocal Q -supersymmetrytransformation,thesecondterm,thesecond S -supersymmetrytransformation.) Werefertothism ultipletastheconforma lgravitino mult iplet. Themultipletofcomponent“eldsof Wisirreducibleandgaugeinvariantand shouldappearinthegaugeinvariantaction.However,intheabsenceofdimensional constantswecannotwriteafreeactionofthecorrectdimensionintermsof W .Wecan writeanonlocalgaugeinvariantactionintermsof,e.g., S =1 2 d4xd4 €i €1,1+ h c .,(4.5 .31) (1,1isthecorrectprojectorontothephysicalgaugeinvariantrepresentation)butit leadstoanonlocalcomponentaction. To “ndalocalaction,weaddmorerepresen tations.Clearlyremovingfromthe actionrestoreslocalitybutintroduces all therepresentationsinanddestroysthe gaugeinvariance.Wecan,however,restrictth erepresentationswh ichappear.Webegin withthegeneralexpression S =1 2 d4xd4 €i €(i cii)+ h c .,(4.5 .32) withthesumrunningoverallprojectors(3.11.38,39),andchoose citoobtainalocal

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2084.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSaction.Werequirethatthesuperprojector1,1bepres ent,and“ndtwosolutionscontainingtheleastnumberofadditionalsuperprojectors.Oneusesthesuperprojectors Š 1,1Š 20,1 2 ŠŠ 2,1 2 ,+theotheruses Š 1,1Š 20,1 2 Š+1,0.(Theoverall signischosentogivethephysicalvectorthecorrect kineticterm.)Theresultingactionsare (1) S(1)= d4xd4 [ Š ( D€)( D €) Š1 2 ( D2+ h c .)+1 4 ( D+ D€ €)2] (2) S(2)= d4xd4 [ Š ( D€)( D €) Š1 2 ( D2+ h c .)].(4 .5.33) However,thegaugegroupisnolongerdescribedby(4.5.27);theinvariancegroupsassociatedwiththeactionsabovearesmallerthantheinvariancegroupof W;theyare (1) = i D2DK1+ DK2, Ki= Ki, (2) =1 + D( D2 + D€ 2€), D€i =0,(4. 5.34) forthetwoactions. Ascomparedto(4.5.27),i nthe “rstcasetheinvariancegrouphasbeenreduced b ecauseisrestrictedto thesp ecialform i D2DK1, K1= K1,andisres trictedto bereal.Inthe secondcaseremainsunrestrictedbutisrestrictedtotheform D2 + D€ 2€.Inbothc asesthe“nalgaugegrouphasfewerparametersthanthe originalone.However,formanypurposes(e.g.,quantization),weneedtousetheoriginalgaugegroup;todothis,weintroducec ompensatingmultiplets(seesec.3.10). e.2.Compensators Forthegravit inomulti plet,twoinequivalentsetsofcompensatorscanbeintroduced.Wedothisby nonlocal “eldrede“nitionsofthebasicgaugesuper“eld.Thus,for thetwoloca lactionswemaketh ered e“nitions (1) + Š 1(1 2 D2W+ D2DG ), W= i D2DV V = V G =1 2 ( D+ D€ €), D€=0,

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4.5.Othergaugemultiplets209(2) + Š 1(1 2 D2W+ D D2 ), D€=0.(4 .5.35) Theyinducethefollowingchangesintheactions (1) S(1) S(1)Š d6zW2+ d8z [ Š ( W+ W€ €)+ G2Š G ( D+ D€ €)], (2) S(2) S(2)Š d6zW2+ d8z [ Š ( W+ W€ €) Š 2 Š ( D+ D€ €)].(4.5.36) Althoughtherede“nitionsarenonlocal,theactionsremainlocal.(Actually,incase(1) wecanalsou sesimply + i1 2 DV +.) Intheabove“eldrede“nitionsweintroducedavectormultiplet V andeithera tensormultipletorachiralscalarmultiplet.Thesechoicesareare”ectionofthe representationsthatwereintroducedbytheadditionalprojections:Avectormultiplet 0,1 2 Š,atensormu ltiplet2,1 2 +,andac hiralscalarmultiplet1,0.Inthe presenceofthecompensatingmultipletsthegaugevariationofisgivenby(4.5.27).The compensatingmu ltipletstransformasfollows: (1) V = i ( Š ), = Š + i D2DK3, (2) V = i ( Š ), = Š D2 ,(4.5 .37) Sincetheyarecompensators,theycanbeal gebraicallygaugedtozero.Intheresulting gauge,thetransformations(4.5.27)ofarerestrictedbackto(4.5.34). Thetwoinequivalentformulationsofthegravitinomultiplet,oneusingatensor mult ipletcompensatorandtheotherusingachiralscalarcompensator,leadtodierent auxiliary“eldstructuresatthecomponentlevel.IntheWess-Zuminogaugeforcase(1) thecomponentso fthegravit inomulti pletare

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2104.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS= D€D €+ D2+1 2 D( D+ D€ €)+ WŠ DG P = Š i ( D D2Š D€D2 €+ DW), V a= i ( D2 D€+ D2D €) Š a[ G Š1 2 ( D+ D€ €)], A a=( D2 D€Š D2D €), t = D( [ )+ )], t=2 W+ ( €B )€, = D2D[ G Š1 2 ( D+ D€ €)], B a=1 2 [ D, D€] V + i ( D €+ D€), a = i1 2 [ D, D€]+ i [ D D€+ D2 €+ W€].(4.5 .38) Forcase( 2)wehaveinstead = D€D €+ D2+ WŠ D, P = Š i ( D D2Š D€D2 €+ DW), J = D2( D+2 ), A a= Š i D2D €+ a, t=2 W+ ( €B )€, = D2 D2+ i €D D€Š i € D€ ,

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4.5.Othergaugemultiplets211B a=1 2 [ D, D€] V + i ( D €+ D€), a = i1 2 [ D, D€]+ i [ D D€+ D2 €+ W€].(4.5 .39) Incase(1),thegauge“eld t ofthetensormultipletreplacesthecomplexscalarcomponent “eld J ,w hi ch co rrespondstotheauxiliary“eldofthechiralscalarmultipletof case(2 ).Inthecomponentaction t onlyappearsas t €A €Š t€€€A €.This te rmisinvariantunderseparategaugetransformationsof t and A €.Alsoits hould benote dthatt he“eld A aisreal.Incase(2) A€ =( A€)hasno gaugetransformationsbecausethephysicalscalarsofthechir alscalarmultiplethavebecomethelongitudinalpartsof A€andcancelthetransformationin(4.5.30).Inbothcases,thephysical v ectorofthecompensatingvectormultiplethasbecomethephysicalvectorofthegravitinomultiplet,whilethephysicalspinorofthevectormultipletbecomesthespin1 2 part ofthegravitinoandcancelsthespinortranslationin(4.5.30). e.3.Duality Sincethetwoformulationsabovedierinthat(1)hasatensorcompensatorwhere (2)hasachiralcompensator,usingtheapproachofsec.4.4.c.2,wecanwrite“rstorder actionsthatdemonstratethedualitybetweenthetwoformulations.Forexample,we canstart with(1)andwrite S (1)= S(1)[, V ]+ d8z [ X2Š X ( D+ D€ €) Š 2 X (+ )].(4.5.40a) Varyingthe chiral “eldleadsto X = G andformulation(1),whereaseliminating X resultsinformulation(2).Similarly,wecanstartwith(2)andwrite S (2)= S(2)[, V ]+ d8z [ Š X2Š X ( D+ D€ €)+2 XG ].(4.5 .40b) Varyingt helinearsuper“eld G we “nd X =+ andform ulation( 2),whereaseliminating X leadsdirectlyto(1). Thereareotherinequivalentformulationswherewereplace V bythevariantvectormultipleta nd/orreplacebyeithertherealorcomplexthree-formmultiplets.This

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2124.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSsi mplyreplacessomeofthescalarauxiliary“eldswithgaugethree-form“eldstrengths. e.4.Ge ometricformulations Finally,wegivegeometricalformulationsofthetheories.Todescribeamultiplet thatgaugesasymmetrywithaspinorialgaugeparameter,weintroduceasuper1-form A withanadditionalspinorgroupindex.Theanalysisissimpli“ediftheirreducible mult ipletisconsidered“rst.Theirreducibletheorywasdescribedby Win(4.5.26). Todescribethismu ltipletgeometrically,weintroducemoregauge“elds(inparticulara complexsuper1-formA( = A))andenlargethegaugegroup.Whenwegettosupergravitywewill“ndthatthisprocesscanalsobecarriedout.TheretheirreduciblemultipletistheWeylmultiplet andtheenlargedgroupisthe conformalgroup.The“nal formofthe(3 2 ,1)multi pletwithmoreirreducib lemultipletsandcompensatorsisanalogoustoPoincar esupergravity. Wew illuset hewordsPoincar eandWe ylforthe (3 2 ,1)multi plettoemphasize thisanalogy. Thecompletesetofgauge“eldsandgaugetransformationsthatdescribetheWeyl (3 2 ,1)multi pletis: A = DAKŠ A L A€= DA K€Š A€ L A= DAL A= DA L .(4. 5.41) The K -termsaretheusualgaugetransformationsassociatedwithasuperformandthe L -termsaretheconformaltransformation.Recallthatwefoundthatthegaugetransformationsoftheirreducib lemultipletcontainan S -supersymmetryterm. L isthe super“eldparameterthatcontainsthesecomponentparameters.Thevectorcomponent ofthecomplexsuper1-formAisthecomponentgauge“eldwhose“eldstrength appearsin(4.5.28).The“eldstrengthsforAarethoseforanordinary(complex)vectormultiplet, butthos eforA anditsconjugate A€mustbe L -covaria nti zed: FAB = D[ AB ) Š TAB DD +[ AB ) FAB€= D[ A B )€Š TAB D D€+ [ AB )€.(4. 5.42) Wecannowimposeth econstraints:

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4.5.Othergaugemultiplets213F = F ,€ = F€ ,€= F a =0, F a€ + F a ,€=0, F = F ,€=0,( F€ ,€ =0).(4 .5.43) Evenwiththe L invariancethegeometricaldescript ionheredoesnotqu itere ducetothe irreduciblemultiplet W.Howev er,theseconstraintsreducethesuper1-formstothe irreduciblemultipletplusthecompensatingvectormultiplet,whicharethetwoirreducib lemultipletscommontobothformsofthePoincar e(3 2 ,1)multi plet,andthusare sucientfortheirgeneralanalysis. Theexplicitsolutionoftheseconstraintsisintermsofprepotentials ,, (complex),and V (real): = D Š ,€ = D€, a = Š i [ D€D + D D€+ (€Š D€)]; = D,€= Š D D€( Š ) Š D2( €Š €)+ D€ Š W€, a= Š iD2 D€( Š )+ a;(4.5 .44) where W= i D2DV .T heprepotentialstransformunder Kand L ,a sw ellasunder newpar ameters(complex)and(c om plex)underwhichthesareinvariant(thisis analogoustothe-groupparametersinsuper-Yang-Mills): = K+ D, = KŠ D€=0; = L Š D2, V = i ( Š ).(4. 5.45) Aswiththevectormultiplet,wecang otoachiralrepres enta tionwhere andonly appearasthecombination= Š ,with = ( Š )=+ D.(4.5 .46)

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2144.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSAtthispo int,bycomparisonwith(4.5.27),wecanidentifyand V withthecorrespondingquantitiesthere.TorecoverthefullPoincar etheory,wemu stbreakthe L invarian ce.Tobreakthe L invariance,weintroducea tensorcompensator Gor ,to obtainthetensor-multipletorscalar-multiplet,respectively.Thesetensor(scalars)are not prepotentials,andtransformcovariantlyunder all ofthegaugetransformations de“nedthusfar.Bycovariant,wemeanthatthesetransform without deriva tives DA. G = Š ( L + L ),(4.5 .47) = Š L .(4. 5.48) Wenowimp osethe L -covariantizedformoftheusualconstraints( D2G =0and D€=0)whichd escribetensorandchir alscalarmultiplets;1 2 D€[ D€ G +(€+ €)]+ h c .=0,(4 .5.49) D€ + €=0.(4. 5.50) Theinvarianceoftheseconstraintsfollowsdirectly(4.5.27,37,47,48).(Thehermitian conjugatetermaboveisnecessarytoavoidconstrainingitself.)Theseconstraints canbesolvedintermsofprepotentials: G = G Š (+ ) Š1 2 ( D+ D€ €),(4.5 .51) = Š ;(4.5 .52) where G andaregivenin(4.5.35)andtransformasin(4.5.37).Toobtaincase(1)as describedabove,weintroducethetensorcompensator G ,choos ethe L -gauge G =0,and solvefor+ inte rmsof G .Thequantity Š i s undetermined,butcanbegauged awaybyusingthere maininginvarianceparametrized L Š L .(R ecallgauging G to zero onlyusesthefreedomin L + L .)Toobtaincase(2),weintroducethechiralscalarcompens ator and gaugeittozerowhichgives= .Thus,gaugingeithertensorcompens ator or G to zeroforcesthestocontainthecorrectandcompletePoincar emultiplets.Alternatively,wecouldgaugeto zero,sothatthetensor(scalar)submultiplet iscontaine donlyin G ( ).Weshouldalsomentionthatotherchoicescouldbemadefor tensorcompensators.Anyofthevariantscal arornonminimalscalarmultipletscanbe

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4.5.Othergaugemultiplets215usedbygeneralizingthediscussionabove.Thesewillleadtoanumberofinequivalent o-shellformulationsofthePoincar e(3 2 ,1)theory. The“eldequations,obtainedfromtheaction(4.5.36),takethecovariantform D€X +(€+ €)=0, X = Gor + .(4.5 .53) (Incase(2),using(4.5.50),thesesimplifyto D€ +€=0.)

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2164.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS4.6.N-extende dmulti plets Sofarinthischapterwehavedescribedthemultipletsof N =1gl obalsupersymmetry.Forinteractingtheoriestherearetwosuchmultiplets,withspins(1 2 ,0),and (1,1 2 ),althoughtheirsuper“elddesc riptionmaytakemanyforms.For N -extended supersymmetry,global mult ipletsexistfor N 4.Theyarenaturallydescribedinterms ofexte ndedsuper“elds.Itispossible,however,todiscussthesemultiplets,andtheir interactions,intermsof N =1super“el dsdescribingtheir N =1s ubmultiplets.In manycasesofinterestthisisthemostcompletedescriptionthatwehaveatthepresent time. a.N=2multiplets Asdisc ussedinsec.3.3,thereexisttwoglobal N =2mult iplets:avectormultipletwithspins(1,1 2 ,1 2 ,0,0),andasc alarmultipletwithspins(1 2 ,1 2 ,0,0,0,0).There existson lyoneglobal N =4mult iplet:the N =4v ectormu ltiplet,with SU (4) representation spins(1 1,4 1 2 ,6 0).(Theonly N =3mult ipletisthesameas thatof N =4.)Webeginbyd iscu ssingthe N =2situation. a.1.Vect ormultiplet The N =2v ectormult ipletconsistsofan N =1 Ya ng -M illsmultipletcoupledtoa scalarmultip letinthesame (adjoint)represe ntationofthe internalsymmetrygroup. Theactionis S = 1 g2 tr ( d4xd4 + d4xd2 W2)(4. 6.1) inthevectorrepresentation.Inadditiontotheusualgaugeinvariance,itisinvariant underthefollowingglobaltransformationswithparameters : = Š W Š i [ 2( ) +( ) iW] = Š W Š [(1 2 [ €, ] ) €+( i 2 ) ], eŠ e= Š i + W€ € .(4. 6.2)

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4.6.N-extendedmultiplets217(Forthe transformationweuse(4.2.52),andalsoagaugetransformationwith K = Š i ( €D2 D€ Š D2D )+€1 2 [ D€, D] .)Duetotheidentity =[ ,( eŠ e)](4.2.77),thesecondtransformationcanbewrittenas = Š( i Š W€ € )= Š i + W€1 2 [ €, ] .(4. 6.3) Bothparametersare x -independentsuper“eldsandcommutewiththegroupgenerators (e.g., = D ).Theparameter ischiralandmixesthetwo N =1mult iplets, whereas istherealpar ameterofthe N =1supersy mmetrytransformations(3.6.13). Since hasthe( x -independent)gaugeinvariance = i ( Š ),theglobals uperparametersthemselves forman abelian N =2v ectormult iplet.Referringtothecomponentsof thisparametermultiplet( )bythe namesofthecorrespondingcomponentsinthe “eldmu ltiplet(, V ),we“ndthefollowi ng:Thephysicalbosonic“eldsgivetranslations(fromthevector a1 2 [ D€, D] | )and centralcharges(fromthescalars z | ); thephysicalfermionic“eldsg ivesupersymmetrytransformations( 1 i D2D | 2 D | );andtheauxiliary“eldsgiveinternalsymmetry U (2)/ SO (2)transformations( r 1 2 D D2D | q D2 ).(Thefull U (2)symmetryhas,inadditionto( r q q ) transformations,phaserotations = iu V =0). Thealgebraofthe N =2gl obaltransformationsclosesoshell;e.g.,thecommutatoroftwo transformationsgivesa transformation: [ 1, 2]= 12, 12= i [12]= i ( 12Š 21).(4.6 .4) Thetransformationstakeasomewhatdie rentforminthechiralrepresentation: = Š W Š i 2( ) eŠ V eV= i ( Š )+( W+ W€ €) ,(4. 6.5a) andhence = [ i ( Š )+( W+ W€ €) ].(4.6 .5b) Nowthe i ( Š )partofthe transformationdoescontri bute,butonlyasa“eld-dependent gaugetransformation= W .

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2184.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSWecanaddan N =2Fayet-I liopoulosterm(param etrizedbyconstants = 1+ i 2, 3= 3) d4xd43V +( Š i d4xd2 + h c .)(4.6 .6) to th ea bo ve ac ti on in th ea be lia n( free)case.Thisisinvariantunder(4.6.5)ifwe restricttheglobalparametersby Š D2 = D2 3 D2 = i (2 D2D2 + u ),(4.6 .7) where u istherealconstantparameterofthephase SO (2)par tof U (2).Theconstraint ontheparametersimpliesthatthe U (2)isbroke ndownto SO (2) U (1). Thismodelhassomeinterestingquantumproperties.Ithasgaugeinvariantdivergencesatone-loop,butexplicitcalculationsshowtheirabsenceatthetwo-andthreelooplevel.Insec.7.7wepresentanargume nttoes tablishtheirabsenceatallhigher loops. a.2.Hypermultiplet a.2.i.Freetheory The N =2scal armultipletcanbedescribedbyach iralscalarisospinorsuper“eld a(theahypermulti plet)withthefreeaction S = d4xd4 aa+1 2 ( d4xd2 amabb+ h c .),(4. 6.8) wherethesymmetricmatrix m satis“esth eco ndition macCcb= Cac mcb.(4. 6.9) (Theexplicitformis mab= iMCabb c, M = M ,with a a=0and b a= a b.Wit hout lossofgenerality, mabcanbechosenproportionalto ab.)Thefreeactionisinvariant undertheglobalsymmetries a= Š ( D2 Cab bŠ Z a) Š i D2[( D ) Da+( D2 )a],(4.6 .10) where Z isacentralcharge: Z a= Cabmbcc, Z a= Cab mbc c.(4. 6.11)

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4.6.N-extendedmultiplets219Onshell,wealsohave Z a= Š Cab D2 b.(4. 6.12) Wecanuse eitheroftheforms(4.6.11,12)inthetransformation(4.6.10),becauseofthe localinvariance a= CabSb, Sb S b ,(4. 6.13) forarbitrary x -dependentchiral .(This isaninvariancebecausethevariationofthe actionisproportionalto S SaCabSb=0.)Ifweus etheform(4 .6.12),thevariations donotdependontheparameters mab.Aninter estingfeatureofthealgebra(4.6.10)is thatitdoesnotcloseo-shellifweuserealization(4.6.11)for Z .Onthe otherh a nd,if weuserea lization(4.6.12)instead,thesymmetries(4.6.10)containpartofthe“eld equations,andhencebecomenonlinearand coupling-dependentwheninteractionsare introdu ced.Theseeectsareasignalthatinthedecompositionofthe N =2super“eld thatdescribesthetheoryinto N =1 su pe r“elds,someauxiliary N =1super“ eldshave b eendiscarded.Wediscussfurtheraspectsofthisproblembelow. Withoutthemassterm,theinternalsymmetr iesofthefreescalarmultipletarethe exp licit SU (2)thatactso ntheisos pinorindexofaandthe U (2)madeupofthe r and q transformationsin and ,andofthe uniformphaserotations a= iu a.Themass termbreakstheexplicit SU (2)tothe U (1)s ubgroupthatcommuteswith mac. a.2.ii.Interactions The N =2scal armultipletcaninteractwithan N =2v ectormult iplet,anditcan haveself-inter actionsdescribinganonlinear model.Aclassofsupersymmetric -modelscanbefoundbyco up linganabelian N =2v ectormult iplet(withnokineticterm butwithaFayet-I liopoulosterm)to nN =2scal armultipletsdescribedbythe n -vector.Thesupersymmetrytransformationsoft hevectormult ipletarethesameasthose givenabovein(4.6.2)or(4.6.5)fortheabeliancase.(Theyareindependentofthe“elds inthescalarmultiplets.)However,thetransformationsofthescalarmultiplets(eachof whichisdescribedbyapairofchiralsuper“eldsa)are gaugecovariantized: a= Š D2[ c(e V)c bCab]Š i D2[( D ) a+( D2 )a] Š i1 2 u a.

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2204.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDS(4.6.14) Thematrixisan SU (2)generatorthatbreakstheexplicit SU (2)ofthescalarmultipletdownto U (1).Because SU (2)preservesthealternatingtensor Cab,(e V)a b(e V)c dCac= Cbd.Theacti onthatisleftinvariantbythesetransformationsis: S = d4xd4[ a(e V)a bb+ 3V]+ d4xd2 i [1 2 aCabb ccŠ ]+ h c .(4. 6.15) provided(4.6.7)aresatis“ed.Thetheoryisalsoinvariantunderlocalabeliangauge transformations: a= i a b b, V = i ( Š ), =0;(4 .6.16) aswellasglobal SU ( n )rot ationsofa.Forexp licitcomputation,itisusefultochoose asp eci“c:Wechoose=3.Wewritea (+,Š) (+ i,Š i)where i =1 ... n isthe SU ( n )i ndex,+, Š arethe SU (2)iso spinindices,and+tr an sformsunderthe SU ( n )representatio nconju gatetoŠ.Thetransf ormations(4.6.14)andtheaction (4.6.15)become(using(4.6.7)) = D2( Š+eŠ+ V) Š1 2 3 ( D2 )Š i D2( D ) (4.6.17a) S = d4xd4[ + ieV+ i+Š ieŠ V Š i+ 3V], + d4xd2 i [Š i+ iŠ ]+ h c .(4. 6.17b) We nowproceedaswedidinthecaseofthe CP ( n )models(see( 4.3.9)):Weeliminate thevectormultipletbyits(algebraic)equationsofmotion.Inthiscase,actsasa Lagrangemultipliertoimposetheconstraint: Š i+ i= .(4. 6.18) Choosingagauge(e.g.,+ 1=Š 1),wecaneasilysolvethisconstraint;forexample,we canparametrizethesolutionas:

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4.6.N-extendedmultiplets221+ i=(1+ u+ uŠ)Š1 2 1 2 (1,u+), Š i=(1+ u+ uŠ)Š1 2 1 2 (1,uŠ).(4. 6.19) The V equationofmotiongives: + ieV+ iŠ Š ieŠ V Š i+ 3=0 ,( 4.6.20a) or Me V=1 2 [( 3 2+4 M+MŠ)1 2 Š+ 3];(4.6 .20b) where M= = | |2= | || 1+ u+ uŠ|Š 1(1+ | u|2).(4.6 .20c) Substituting,we“ndtheaction S = d4xd4{( 3 2+4 M+MŠ)1 2 + | 3| ln[( 3 2+4 M+MŠ)1 2 Š| 3|] }.(4. 6.21) Intermsoftheunconstrainedchiralsuper“elds u,t he transformations(4.6.17a)become u= D2[ eŠ+ V( )1 2 (1+ u+ uŠ)1 2 (1+ u+ uŠ)Š1 2 ( uŠ+Š u)]Š i D2[( D ) Du],( 4.6.22a) wheretheauxiliarygauge“eld V isexpr essedintermsof uby(4.6 .20).Thesupersymmetrytransformations(4.6.22a)includeacompensatinggaugetransformationwith parameter i = Š D2[ ( coshV )( )1 2 (1+ u+ uŠ)1 2 (1+ u+ uŠ)Š1 2 ](4.6.22b) thatmustbeaddedto(4.6.17a)tomaintainthegaugechoicewemadein(4.6.19). Asfort hefree N =2scal armultiplet,wecanaddaninvariantmassterm(which introdu cesanonvanishingcentralcharge).Themasstermnecessarilybreaks SU ( n )and hastheform Im= i1 2 d4xd2 aCabb cM c+ h c .,(4.6 .23) where M isanytra celess n n matrix( M sdie ringby SU ( n )trans formationsare

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2224.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSequivalent).Thesupersymmetrytransformati onsthatleavethisterminvariantarethe sameasbefore,includingthe Z termof(4.6.10).Therealizationof Z givenin(4.6.11) ispreferable,sinceitislinear,whereastherealization(4.6.12)mustbegaugecovariantized. Thesenonlinear -models liveonK¨ ahlermanifoldswith threei ndependentcomplexcoordinatesystemsrelatedby nonholomorphic coordinatetransf ormations(they havethreeindependentcomplexstructures(seetheendofsec.4.1);theconstants 3, parametrizethelinearcombinationofcomplexstructureschosenbytheparticularcoordinatesystem).ThusthesemanifoldsarehyperK¨ ahler.Justaswefoundthat foreveryK¨ ahlermanifoldthereisan N =1no n linear -model(andconversely),onecan showthatforeveryhyperK¨ ahlermanifoldthereisan N =2no n linear -model,andconversely, N =2no n linear -modelsarede“nedonlyonhyperK¨ ahlermanifolds.An immediateconsequenceofthisrelationisastr ongrestrictiononpossibleo-shellformulationsofthe N =2scal armultiplet: Noformulationcanexistthatconta insasphysicals ubmultipletstwo N =1sc alar mult iplets(e.g.,suchaswehaveconsid ered),thatcanbeusedtodescribe N =2 nonlinear -models,andthathassupers ymmetrytransformations independent of theformoftheaction. Ifsuchaformulationexisted,thenthesumoftwo N =2 in va riantactionswouldnecessar ilybeinvariant;however,thesumoftheK¨ ahlerpotentialsoftwo hyper K¨ ahlermanifoldsis not ingenera ltheK¨ ahlerpotentialofahyperK¨ ahlermanifold.Wewillsee belowthatwe can giveano-shellformulationofthe N =2scal armultipletthatavoids thisproblem. Wecang eneralizetheaction(4.6.15)inthesamewaythatwegeneralizedthe CP ( n )models(se e(4. 3.11)): S = d4xd4[ a(e V)a bb+ 3trV]+ d4xd2 i[1 2 a Cabb ccŠ tr ]+ h c ., (4.6.24a) where V = VATA,=ATA,and TAarethegeneratorsofsomegroup.The N =2 transformationsthatleave(4.6.24a)invariantaretheobviousnonabeliangeneralizations

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4.6.N-extendedmultiplets223of(4.6.14).Theequationsthatresultfromvarying(4.6.23a)withrespecttoA, VAare, choosing=3asabove, ŠTA+Š trTA=0, +eVTA+Š ŠTAeŠ VŠ+ 3trTA=0.(4. 6.24b) Asinthe N =1case,the sedonot,ingeneral,have anexplicitsolution. a.3.Tens ormultiplet Justasthe N =1scal armultipletcanbedescribedbydierentsuper“elds,wecan describethe N =2scal armultipletbysuper“eldsoth erthanthechiralisodoubleta. Wenowdiscussthe N =2tensorform ulationofthesc alarmultiplet.Thisisdualtothe previousdescriptioninthesamewaythatthe N =1tensor andscalarmul tipletsare dual(seesec.4.4.c).Wewritethetensorfor mofthescal armultipletintermsofone chiralsc alar“eld andachiralspinorgauge“eld withlinear“eldstrength G =1 2 ( D+ D€€), D2G = D2G =0.The N =2supers ymmetrytransformationsof thistheoryare = Š 2 D Š i D2[( D ) D+( D2 ) ], = Š D2( G ) Š i D2[( D ) D +2( D2 ) ].(4.6 .25) Incontrasttotheahypermulti pletrea lizationofthe N =2scal armultiplet,these transformationsclose o-shell; theyhavethesamealgebraasthetransformationsofthe N =2v ectormultiplet(4.6.4)(uptoagaugetransformationof ).However,although thesuper“eldsdescribeascalarmultiplet,thecentralchargetransformations z = | leavethe “elds in ert;thisgivesoneguidetounderstandingthedualitytothehypermult iplet. Thesimplestactioninvariantunderthetransformations(4.6.25)isthesumofthe usualfreechiralandtensoractions((4.1.1)and(4.4.34)): Skin= d4xd4 [ Š1 2 G2+ ].(4.6 .26) To “ndotheractions,weconsiderageneralansatz,andrequireinvarianceunderthe

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2244.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDStransformations(4.6.25).Actually,wecanconsideraslightlymoregeneralcasethat stillhasfullo-shell N =2invariance byrest rictingthechiralparameter by D2 =0, andtherealparameter by D2D2 =0.Thismea nsthatwedonotimpose SU (2) invariance.Anaction S = d4xd4 f ( G , )(4. 6.27) is in va riantunder(4.6.25)(with D2 =0)if f satis“es 2f G2 + 2f fGG+ f =0.(4. 6.28) ( f containsnoderivativesof G , .)Itdescribe sageneral N =2tensormu ltiplet interactingmodel.Wealsocanco nsidermorethanonemultiplet Gi, i, i,eachtransformingas(4.6.25);thenthemostgenera linvaria ntacti onis(4.6.27)wherethe Lagrangian f satis“es fGiGj+ fi j=0.(4. 6.29) (Actually,wecangeneralize(4.6.27)slightlybyaddingaterm d4xd2 hii+ h c wherethe hisarearbitraryconstants.) a.4.Duality To gaininsightintothephysicsofthesemodelswe“ndthedualtheoriesdescribed bytheahypermulti plet.Weconsiderthefollowing“rstorderaction(cf.(4.4.38)): S= d4xd4 [ f ( Vi, i, i) Š Vi(i+ i)].(4.6.30) Eliminating, gives(4 .6.27),whileeliminating V resultsinthedualtheory.We“nd the N =2transf ormationsoftheresultinga ihypermulti pletsfromthetransformations thatleavethe“rstorderaction(4.6.43)invariant.Since d4xd4 f ( V , )isinvarian t under(4.6.25)with G Vexcept forterms D2V or D2V ( V diersfrom G onlybecauseitdoesnotsatisfytheBianchiidentities D2G = D2G =0),weca ncan cel thesetermsbychoosingthevariationofappropriately.The“rstorderaction(4.6.43) is in va riantunder

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4.6.N-extendedmultiplets225 Vi= DiD + D€ i D€ + Vi,( 4.6.31a) i= Š D2( Vi)+ i,(4. 6.31b) i= Š D2[ ( fi+ Vj( f jViŠ f iVj))]+ i;(4. 6.31c) where istheusual N =1supersy mmetry(3.6.13)(with wV=0, w= Š 2, w=0). (T op rovetheinvarianceof(4.6.30)under(4.6.31),weneed(4.6.29)anditsconsequen ces,inparticular, fViV[ jk ]=0and f i[ jVk ]=0b ecauseoftheantisymmetrization, andtherefore,using thechain rulewe“nd D€f [ jVi ]=0.)Perfo rmingthedualitytransformations,wecanrewritethetransformati ons(4.6.31)andthecondition(4.6.29)in termsofthedua lvariables, andtheLegendretransformedLagrangian IK (+ , ).We“nd(dro ppingtheuninteresting terms) i= D2( IKi), (4.6.32a) i= Š D2[ ( IKi+ IKj(( IKk i)Š 1IKjkŠ ( IKk j)Š 1IKi k))],(4.6.32b) forthetran sformations,and IKi j=( IKi j)Š 1+ IKi m( IKm n)Š 1IKn j(4.6.33) fortheconditionthat theLagrangianmustsatisfytoguaranteeinvariance.Notethatin contrast withthe o-shell transformations(4.6.25),the on-shell transformations (4.6.31,32)dependexplicitlyontheformoftheaction.Furthermore,thecondition (4.6.29)need edforinvarianceoftheo-shellversionofthemodelis linear, andhence thesumoftwoinvariantactionsisautomaticallyinvariant,whereasthecondition (4.6.33)is nonlinear. TheLegendretransformationallowsthistooccur,andallowsusto complywith therestrictionono-shellformulationsthatwediscussedabove. Althoughitisalwayspossibletogofromtheo-shellformulation(intermsofthe tensormultiplet)totheon-shellformulation(intermsofthehypermultiplet),thereverse transformationisgenerallynotsostraightforward.Theimprovedform(see(4.4.45-5)) ofthefreemultipletcanbefoundbyexploitingananalogywiththenonlinear -models discussedabove(Actually,thetensormult ipletformoftheinteractingmodelscanbe foundinthisway).Alternatively,somesimplemodelscanbefoundbyusingthecentralchargeinvarianceofthetensormultiplet(seebelow).

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2264.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSByanalogywith(4.6.17),wecanwritedownthefollowing“rstorder N =2invariantactionbyintroducinganauxiliary N =2v ectormult iplet V ,: S= d4xd4[ +eV++ŠeŠ V ŠŠ GV]+ d4xd2 i [Š+Š ]+ h c .(4. 6.34) Thisactionisinvariantunderthetransformations(4.6.5,14,25).Varying G and ,we “ndthefreehypermu ltiplet(seediscussioninsec.4.6.a.2);varying V and,w e “nd thatdrop outoftheactionentirely,andtheimproved( N =2)tensorm ultiplet results: Simp= d4xd4[( G2+4 )1 2 Š Gln ( G +( G2+4 )1 2 )].(4. 6.35) Thiscomplicatednonlinearactioncorrespondstoafreehypermultiplet!Itis,however, an o -shellformulation,invariantunderthetransformations(4.6.25).Itgeneralizes directlytogiveano-shellformulationofthenonlinear -modelswediscussedabove. Analternativederivationoftheimprovedtensormultipletdoesnotrequirean N =2v ectormultiplet,butusesthecentralchargeinvarianceofthetensormultiplet. Webeginwit hthefreehyperm ultipletaction (4.6.8)(withoutlossofgenerality,wetake mab= imCab( 3)b c).WewishtoLegendr etransfo rmoneofthechiral“elds a=(+,Š),andkeeptheother“eldasthechiral“eld ofthetensormultiplet. However,though is in ertundercentralchargetransformations,arenot;wethereforede“netheinvar iantcombination i +Š,andinte rmsofitwritethe“rstorder action S= d4xd4 [ eŠ V+ eVŠ GV ]+1 2 m [ d4xd2 + h c .].(4. 6.36) Varying G ,wer ecoverthehypermultipletaction(4.6.8)with V = ln ( ++);varying V wer ecovertheimprovedtensormult ipletaction(4.6.35)witha linear termthatacts asamassterm.Thealgebraoftransformationsthatactonthemassivescalarmultiplet has a centralcharge;however,thedescri ptionofthe mult ipletgivenbythe N =2tensor mult ipletonlyinvolves“eldsthatare inert underthecentralcharge.

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4.6.N-extendedmultiplets227Finally,wenotethatthegaugeinteractionsofthe N =2tensormu ltipletareanalogoustothe N =1case(s eesec.4.4.c). a.5.N=2super“eldLa grangemultiplier Anotherformulationofthe N =2scal armultipletwitho-shell N =2supersymmetryisthe N =2L agrangemultipliermultiplet.Itisthe N =2genera lizationofthe mult ipletdiscussedinsec.4.5.d,andcontainsthat N =1mult ipletasasubm ultiplet. Unliketheo-shell N =2supersy mmetricscalarmultipletdiscussedabove(the N =2 tensormultipletofsecs.4.6.a.3,4),thismultipletcanbecoupledtothe( N =2)nonabelianvectormultiplet,thoughonlyinrealrepresentations.Byusingtheadjointrepresentation,thisallowsconstructionof N =4 Ya ng -M illswitho-shell N =2supersymmetry,asdiscussedbelowinsec.4.6.b.2. The N =2L agrangemultipliermultipletisdescribedbythefollowing N =1 super“elds:(1)1 and Y ,des crib ingan N =1L agrangemultipliermultipletasin (4.5.18),withthegaugeinvarianceof(4.5.19),and“eldstrength1= D€ 1€(forwhich F and G of(4.5.18)aretherealandimaginaryparts);(2)asecondspinor2 ,withthe sa medimensionandgaugeinvariance,butwhichisauxiliary;(3)acomplexLagrange mult iplier,whichco nstrainsallof2tovanish(insteadofjusttheimaginarypart,as does Y for1),andhasa“eldstrength D€with gaugeinvariance =(forchiral);(4)aminimalscalarmultiplet,describedbyacomplexgauge“eld1withchiral “eldstrength1(seesec.4.5.a);and(5)twomo reminimalscalarmultiplets2and3, but auxiliary.Wethushavean N =1L agrangemultipliermultiplet,aminimalscalar mu ltiplet,andassortedauxiliarysuper“elds. Theactionis S = Š d4xd4 [1 8 (1+ 1)2+i 2 Y (1Š 1)] + d4xd4 [ 11+(2+ 2)+(23+ 2 3)].(4. 6.37) Themostinterestingpropertiesofthistheoryappearwhenitiscoupledto N =2superYa ng -M ills.Wedothisby N =2 gaugecovariantizingthe N =2L agrangemultiplier mult iplet“eldstrengths.(Inthe absence ofYang-Millscoupling,thescanbe

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2284.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSconsideredasordinaryscalarmultiplets,r athert han“el dstrengths.)Th iscouplingis discussedins ec.4.6.b.2. b.N=4Yang-Mills Inmanyrespects,the N =2no n linear -models,whenstudiedintwodimensions,areanalogsof N =4 Ya ng -M illstheoryinfourdimensions.Despitepower-countingarguments,theyarecompletely“niteonshell,andtheyarethemaximallysupersy mmetricmodelscontainingonlyscalarmultiplets(thevectorisauxiliaryandcanbe e liminated).The N =4 Ya ng -M illstheoryisthe“rstandbest-studied4-dimensional theorythatisultraviolet“nitetoallordersofperturbationtheory,andthusscaleinvariantatthequantumaswellastheclassicallevel.(Its -functionhasbeencalculatedto vanishthro ughthreeloops;ar gumentsfortotal“nitenessaregiveninsec.7.7.Indepe ndentargumentsusinglight-conesuper“eld shaveb eengivenelsewhe re.)Itisselfconjugateandisthemaxima llyextendedgloballysupersymmetrictheory.Twosuper“eldformulationsofthetheoryhavebeengiven:Oneusesan N =2v ectormult iplet coupledtoaahypermul tipletandhasonly N =1supersy mmetryoshell,andthe otherusesan N =2v ectormult ipletcoupledtoan N =2L agrangemultipliermultiplet andhas N =2 su pe rsymmetryoshell(however,ithasalargenumberofauxiliary super“elds). b.1.Minimalf ormulation Atthecomponentleve lthetheory containsagaugevectorparticle,fourspin1 2 Weylspinor s,andsixspin0particles,allintheadjointrepresentationoftheinternal symmetrygroup.Itcanbedescribedbyonerealscalargaugesuper“eld V andthree chiralsc alarsuper“eldsi,andisthesameasan N =2v ectormult ipletcoupledtoan N =2scal armultiplet.Ifweuseamatrixrepresentationforthei,the(c hiralrepresentation)conjugatecanbewrittenasi= eŠ V ieV.The N =1supers ymmetricaction (inthechiralrepresentation)isgivenby S = 1 g2 tr ( d4xd4 eŠ V ieVi+ d4xd2 W2+1 3! d4xd2 iCijki[j,k]+1 3! d4xd2 iCijk i[ j, k]).(4.6.38)

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4.6.N-extendedmultiplets229InadditiontothemanifestSU(3)symmetryonthe i j k i ndicesofand ,ithasthe fo llowingglobalsymmetries: i= Š ( Wi+ Cijk 2 jk) Š i 2[( ) i+2 3 ( 2 )i], = [ i ( iiŠ ii)+( W+ W€ €) ];(4.6 .39) inthechiralrepresentation,an dinthev ectorrepresentation i= Š ( Wi+ Cijk 2 j kŠ i [ jj,i]) Š i [ 2( ) i+( ) iWi+2 3 2( 2 )i]; = Š( i i i+ W€ € ).(4.6 .40) The iarethegeneralizationofthosegivenforthe N =2mult ipletsabov e,butnow theyforman SU (3)iso spinor,asdoesi.Theidenti “cationofthe componentsof and isthesame:Thephysicalbosonic“eldsarethetranslationsandthecentralcharge parameters(3complex=6real,asfollowsfromdimensionalreductionfromD=10:see s ec. 10.6),thespinorsarethesupersymmetryparameters,andtheauxiliary“eldsare internalsymmetryparametersof SU (4)/ SU (3).Thealgebradoesnotcloseo-shell. Uponreductiontoits N =2s ubmultiplets,(4.6.39)(or(4. 6.40))reducesto(4.6.5)(or (4.6.2))and(4.6.14)(butwithdierentR-weights). Thecorrespondingcomponentactionhasaconventionalappearance,withgauge, Yukawa,andquarticscalarc ouplingsallgovernedbythesamecouplingconstant.In sec.6.4wediscusssomeofthequantumpropertiesofthistheory. b.2.La grangemultiplierformulation Wenowbrie” ydes cribeanother N =1super“eldf ormulationof N =4superYa ng -M ills;itemploysthe(unimproved)typeof N =1scal armultipletofsec.4.5.d. Althoughevenlessofthe SU (4)symmetryismanifest,thisformulationiso-shell N =2supersy mmetric:Itfollowsfromthe N =2super“eldformula tionofthetheory, asdescribedbyt hecouplingof N =2 su pe rY ang-Millstoan N =2(L agrangemultip lier)scalarmultiplet.Thisformulationhasanumberofothernovelfeatures:(1)

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2304.CLASSICAL,GLOBAL,SIMPLE(N=1)SUPERFIELDSrenormalizablecouplingsbetweennonminimalandotherscalarmultiplets,(2)thenecessaryappearance(ininteractionterms)oftheminimalscalarmultipletintheformofa gaugemultiplet(sec.4.5.a),and(3)lossof(super)conformalinvarianceoshell(this o ccursbecausethemodelincludesan unimpr oved Lagrangemultipliermultiplet). Theactioncanbewrittenas(inthesuper-Yang-Millsvectorrepresentation)the sumof(4.6.1)and(4.6.37).However,thede“nitionsofthe“eldstrengthsiandiare nowmodi“ed: i= € i€Š i [0,i] i= 2i( i =0,1,2),3= 23Š i [0,];(4 .6.41) where0(withprepotential0)isthechiralsu per“eldofthe N =2 Ya ng -M illsmultiplet.The Ainthesede“nitionsistheYang-Millscovariantderivative.Inadditionto theusual(adjoint,vectorrepresentation)Yang-Millsgaugetransformations,wehave manynewlocalsymmetriesoftheaction: i= € Ki€( i =0,1,2), 3= € K3€+ i [0, ]; (4.6.42a) i = Ki ( )+ i [ 0, Ki ]( i =1,2); =; Y = =0;(4.6 .42b) whereiscovariantlychiral( €= 0),andistheYang-Millsvector-representation prepotential.Underthesetransformationsthe“eldstrengthsi( i =0,. ..,3),i( i =1,2), €, Y ,and Wareinvariant.Intheabelian(orlinearized)case,thesumof (4.6.1)and(4.6.37)asmodi“edby(4.6.42)describesan N =2v ectormult iplet( Wand 0)plusan N =2scal armultipletconsistingofthe N =1L agrangemultipliermultiplet of(4.5.18)(1 and Y ),aminimal N =1scal armultiplet(1),andsomeauxiliary super“elds(2 ,,2,and3).However,intheinteractingcasetheformulationis somewhatunusualinthat3isnotj ust N =1covarian tlychiral( €3 =0)nor areiN =1covarian tlylinear( 2i =0), buttheysatisfythe N =2covariantB ianc hiidentities €3= Š i [0, €], 2i= Š i [0,i].(4.6 .43) Theinteractiontermsoftheauxiliarysuper“elds(introducedthroughthenonlinearities

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4.6.N-extendedmultiplets231ofthe“eldstrengths2and3)can celamongthemselves:Theirtermsintheactioncan berewri ttenas,inthe chiral representation, d4xd4 ( D€ 2€+ h c .)+[ d4xd2 2( D23)+ h c .].(4. 6.44) BycombiningtheBianchiidentities(4.6.43),theusualconstraint €i=0(for i =0,1,2),and €W=0, W+ €W€=0with the“eldequationswhichfollow fromtheaction,weobtaintheon-shellequationsforallofthesuper“elds D€=1Š 1=2=2=3=0, i W= Š i € W€=[0, 0]+[1, 1]+1 4 [(1+ iY ), (1+ iY )], €0= 2 0+ i [1,1 2 (1+ iY )]=0, €1= 2 1+ i [1 2 (1+ iY ),0]=0, €(1+ iY )= 2 (1+ iY )+2 i [0,1]=0.(4 .6.45) Wecanthuside ntifythisfor mulation onshellwiththatgivenaboveinsubsec.4.6.b.1. bytheco rrespondences W W,(0,1,1 2 (1+ iY )) i.(4. 6.46)

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Contentsof5.CLASSICALN=1SUPERGRAVITY 5.1.Reviewofgravity232 a.Potentials232 b.Covariantd erivatives235 c.Actions238 d.Conformalc ompensator240 5.2.Prepotentials244 a.Conformal244 a.1.Linearizedtheory244 a.2.Nonlineartheory247 a.3.Covariantderivatives249 a.4.Covariantactions254 b.Poincar e 255 c.Densitycompensators259 d.Gaugechoices261 e.Summary263 f.Torsionsandc urvatu res264 5.3.Covariantapproachtosupergravity267 a.Choiceofconstraints267 a.1.Compensators267 a.2.Conformalsupergravityconstraints270 a.3.Contortion273 a.4.Poincar esupergravityc onstraints274 b.Solutiontoco nstrai nts 276 b.1.Conven tionalconstraints276 b.2.Representationpreservingconstraints278 b.3.The gaugegroup279 b.4.Ev aluationofand R 281 b.5.Chir alrepresentation284 b.6.Densitycom pensat ors286 b.6.i.Minimal( n = Š1 3 )supergr avit y 287 b. 6.ii.Nonminimal( n = Š1 3 )supergr avit y 287 b. 6.iii.Axial( n =0)supergr avit y 288 b.7.De gauging289

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5.4.SolutiontoBianchiidentities292 5.5.Actions 299 a.Reviewofvectorandchiralrepresentations299 b.Thegeneral measure300 c.Tensorcompensators300 d.Thechiralmeasure301 e.Representationindependentformofthechiralmeasure301 f.Scal armultiplet302 f.1.Superconforma linterac tions303 f.2.Conformallynoni nvariantac tions304 f.3.Chir alself-interactions305 g.Vectormultiplet306 h.Generalmattermodels307 i.Supergravityactions309 i.1.Poincar e 309 i.2.Cosmologicalterm312 i.3.Conformalsupergravity312 j.Fieldequ ations313 5.6.Fromsuperspacetocomponents315 a.Generalconsiderations315 b.Wess-Zumino gaugeforsupergravity317 c.Commutatoralgebra320 d.Localsupersymmetryand componentgauge“elds321 e.Superspace“eldstrengths323 f.Supercovariantsupergravity“eldstrengths325 g.Tensorcalculus326 h.Componen tactio ns 331 5.7.DeSittersupersymmetry335

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5.CLA SSICALN=1SUPERGRAVITY 5.1.Reviewofgravity a.Potentials Ourreviewisintendedtodescribetheapproachtogravitythatismostusefulin understandingsupergravity.Wetreatgravityasthetheoryofamasslessspin-2particle describedbya gauge“eldwithanadditional vector indexasagroupi ndex(sothatit containsspin2).Byanalogywiththetheor yofama sslessspin1partic leitslinearized transformationlawis h a m= a m.(5. 1.1) SincetheonlyglobalsymmetryoftheS-matrixwithavectorgeneratoristranslations, wechoosepartials pacetimederivatives(momentum)asthegeneratorsappearingcontractedwiththegauge“eldsgroupindexinthecovariantderivative e a aŠ ih a m( i m) =( a m+ h a m) m e a m m.(5. 1.2a) Thus,incontrastwithYang-Millstheory,weareabletocombinethederivativeand gr ouptermsintoasingleterm.Thegauge“eld e a misthe vierbein, whichreducestoa Kroneckerdeltain”atspace.It isinvertible:Itsinverse e m aisde “nedby e m ae a n= m n, e a me m b= a b.(5. 1.2b) Finitegaugetransformationsarealsode“nedbyanalogywithYang-Millstheory: e a= ei e aeŠ i mi m.(5. 1.3) Thelinearizedtransformationtakestheformof(5.1.1),whereasthefullin“nitesimal formtakestheformofa Liederiv ative: ( e a m) m= i [ e a]= Š [ n n, e a m m],(5.1.4a) or,inmoreconventionalnotation, e a m= e a n n mŠ n ne a m.(5. 1.4b)

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5.1.Reviewofgravity233Thegaugetransformationofascalarmatter“eldis,againbyanalogywithYang-Mills theory, = ei ei eŠ i ,(5. 1.5) an di ni n“nitesimalform = i [ ]= Š m m .(5. 1.6) Equation(5.1.5)canalsobewrittenasthemorecommongeneralcoordinatetransformation ( x) ( x ), x= eŠ i xei .(5. 1.7) (Thiscanbeveri“edbyaTaylorexpansion.)Forthecaseofconstant ittakesthe fa miliarformofglobaltranslations.Orbital(global)Lorentztransformationsare obtainedbychoosing m= x€+€€x€(whichju stequals x min th ei n“nitesimal case);istraceless.Scaletransfo rmationsareobtainedbychoosing m= x m. Wecouldatt hispointde“ne“eldstrengthsintermsofthecovariantderivatives (5.1.2),buttheinvarianceg roupwehavede“nedistoosmallfortworeasons:(1)The vierbeinisareduciblerepresentationofthe(global)Lorentzgroup,somoreofitshould be gaugedaway;and(2)therearedicultiesinrealizing(global)Lorentztransformationsongen eralrepresentations,aswenowdiscuss. SinceunderglobalLorentztransformations ( x )transfo rmsasascalar“eld,its gradient m willtransformasacovariantvector.Ingeneral,wede“neacovariantvectortobeanyobjectthattransformslike m .Wecande “neacontr avariant v ectorto belongto theadjointrepresentationofourgaugegroup.Indeed,ifwede“ne V V mi mandrequirethat[ V ]= V mi m transformasascalar,i.e., V V mi m= ei VeŠ i ,(5. 1.8) then V mtr an sformscontravariantlyunderglobalLorentztransformations.However, th is proceduredoesnotallowustode“neobjectswhichtransformasspinorsunder globalLorentztransformations,andinfacti tisimpo ssibletode“nea“eld,transforming linearly underthe group,whichals otransfo rmsasaspinorwhenthe sarerestricted torepresentglobalLorentztransformation s.Itispossibletogetaroundthisdiculty byrea lizingthe transformationsnonlinearly,butthisisnotaconvenientsolution.

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2345.CLASSICALN=1SUPERGRAVITYComparing(5.1.3)to(5.1.8),wesee e a mtransformsasfourinde pe ndentco ntrava riant v ectorsundertheglobalLorentzgroup:The transformationsdonotactonthe a i ndices. Tosolvethe seproblemsweenlargethegaugegroupbyadjoiningtothe transformationsagroupof local Lorentztransformations,andde “nespinorsw ithresp ectto this group.Thisisaprocedurefamiliarintreatmentsofnonlinear models.Nonlinearrealizationsofagrouparereplacedbylinearrepresentationsofanenlarged(gauge)group. Thenonlinearitiesreappearonlywhenade“nitegaugechoiceismade.Similarlyhere, byenlarg ingthegaugegroup,weobtainlinearspi norrepresentations.Thenonlinear spinorrepresentationsofthegeneralcoordinategroupreappearonlyifwe“xagaugefor thelocalLorentztransformations.Itwillthusturnoutthatour“nalgaugegroupfor gravitycanbeinterpretedphysicallyasthedirectproductofthetranslation(general coordinate)groupwiththespin(internal)angularmomentumgroup. Wede “netheactionofthelocalLorentzgrouponthevierbeintobe e a m= Š e€ m+ h c ., =0.(5. 1.9) Thesetransformationsactonlyon thefreeindicesintheoperator e a( butnotthehidden i ndicescontractedwith ,sincewe wanttheope ratortotransform covariantly). From nowonwewillindicateindicesonwhicht helocalLorentztr ansformationacts (”at or tangentspace i ndices)byusinglettersfromthebeginningoftheGreekandRoman al phabets( ,... a b ,...),andi ndicesonwhichl ocaltr anslations(generalcoordinate transformations)act (curved or world i ndices)bylettersfromthemiddle ( ,... m n ,...).Transfo rmationsrepresentedbyamatri xmulti plyingthefreeindex of e aarecalledtangentspacetransformations. ThelinearizedformofthelocalLorentztransformationsis h a m= Š €€ + h c ..(5.1 .10) Itisthuspossibletogaugeawaytheantisymmetric-tensorpartofthevierbein(although notthescalarpart)witha nonderivative transformation.Tostayinthisgaugealocal coordinatetransformationmustbeaccompan iedbyarelatedlocalLorentztransformation;theLorentzparameterisdeterminedint ermsofthetranslationparameter.Atthe lin earizedlevelwe“nd,usingthecombinedtransformations(5.1.1)and(5.1.10),

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5.1.Reviewofgravity235€€h( € )€=0 = Š1 2 ( € )€.(5. 1.11) Inthisgaugethe(orbitalLorentzcoordinate)transformationde“nedaboveinduces thesameglobalLorentztransformationactingonthe”atindices.Wecanthusde“nea Lorentz spinor bychoosingitss pinorindextobea”atindex;”at,ortangentspace, i ndicestransformunderlocalLorentztransformationsbutnotlocaltranslations except whenagaugeischosen,e.g.,asin(5.1.11).Furthermore,wecande“ne all covariant objects exceptthevierbein tohaveonly ”atindices.Thecurved-indexvectorsde“ned abovecanberelatedto”at-i ndexonesbymultiplyingwiththevierbeinoritsinverse. b.Covariantderivatives Wenowde “neournewlocalgroupofPoincar etransfo rmations,derivatives covariantunderit,anditsrepresentationonall“elds.Theparameterofourenlarged localgroup isde “nedby = mi m+( iM + €€i M€€).(5.1 .12) Thegenerator M (andtheparameter )istra celessandactsonlyonfree”at indices.Itsactiononsu chi ndicesisde“nedby [ M ]= ,[ M €]=0, [ €€ M€€, ]=0,[ €€ M€€, €]= €€€.(5. 1.13) An yc ov ar ia nt “eldwithonly”atindicestransformsunderthisgaugegroupas: ...= ei ...eŠ i .(5. 1.14) Thecovariantderivativeisde“nedbyintroducingagauge“eldforeachgroupgenerator, sowemustnowaddto e aof(5.1.2)anewgauge“eldfortheLorentzgenerators: D D a= e a+( a M + a ,€€ M€€).(5.1 .15) Itstransformationlawtakesthecovariantform D D a= ei D D aeŠ i .(5. 1.16)

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2365.CLASSICALN=1SUPERGRAVITYInde“ningthiscovariantderivativewehaveintroducedagauge“eld awhich transformswithaderivativeoftheLorentzgaugeparameter( a = a + ... ). However,duetothevierbeinsLorentztransformationlaw(5.1.10),wecande“nethis Lorentzgauge“eldtobeaderivativeofourfundamental“eld(thevierbein),justasfor thenonlinear -model(seesec.3.10),ratherthanhavingitasanindependent“eld. Therearetwowaysto“ndthisexpressionfor a:(1)Comp arethefulltransformation lawsofthevierbeinandtheLorentzgauge“eld,andconstructdirectlyfromthevierbein aLorentz connectionthathasthecorrecttransformationproperties;or(2)constrain someofthe“eldstrengthsinsuchawaythattheLorentzgauge“eldisdeterminedin termsofthevierbein.Becaus ethe “eldstrengthsarecovariantthiswillautomatically leadtocorrectlytransforminggauge“elds a. The“eldstrengths t a b cand r a b( M )arede “nedby: [ D D a, D D b]= t a b cD D c+( r a b M + r a b ,€€ M€€).(5.1 .17) Wehaveex pandedtheright-handsideover D D and M insteadof and M b ecausethen thetorsion t andcurvature r arecovariant.Byexaminingth eresulta ntexpr essionsfor the“eldstrengthsintermsofthegauge“elds,we“nd t a b c= c a b c+[( a €€+ a ,€€ ) Š a b ], r a b =( e a b Š a b ) Š c a b e e + a ,( | b | ),( 5.1.18a) wherethe anholonomycoecientc isde “nedby [ e a, e b]= c a b ce c.(5. 1.18b) Weseethatcon strain ingthetorsiontovanishgivesasuitableLorentzgauge“eld: a = Š1 4 ( c a ,( € )€+ c( € )€ €).(5.1 .19) (Ifinsteadofthetorsionweconstrainedthecurvaturetovanish,theconnection awouldbe pureLorentzgauge,andunrelatedtothevierbein.However,theantisymmetricpartofthevierbeinwouldremainasacompensatorforasecond hidden localL orentz gr ou po ft hetheory,underwhich awouldtra nsformhomogeneouslyandnotasaconnection.Hence D D de“nedby(5.1. 15)wouldbe noncovariant underthenewtransformations,andinstead

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5.1.Reviewofgravity237 D D a= e aŠ [1 2 ( c a € €+ c€ € €) M + h c .] = e aŠ [1 2 ( t a € €+ t€ € €) M + h c .](5.1 .20) wouldbecovariant.S inceEinsteintheory isnowdes cribedintermsofacurvatureconstructedoutof D D ,theor iginal aanditsassociatedLorentzinvariancewouldbeirrelevantto thetheory.Co nstrainingthecurvaturetovanishisgaugeequivalenttonot introducinganyconnectionatall.Suchaformulationofgravityisoftenreferredtoasa telepara llelismtheory.Ofcourse,ifweweretoconstrainboth t and r tovanish, D D wouldbe gaugeequivalentto ,andwe wouldhavenogravity.) Intheabsenceofanyconstraint,wecouldalwaysexpressthecovariantderivative astheconstrainedcovariantderivative( t =0) plus Lorentzcovar ianttermsthatcontain onlythetorsion.Thetorsioncouldthusbe consideredasanindependenttensorwithno relationtogravity.Ourtorsionconstraintisthusaconventionalconstraint,justlike theconventionalconstraint(4.2.60)ofsuper-Yang-Millstheories. Allremaini ngtensors(i.e.,covariantobjectsthatarenotoperators)canbe expressedintermsofthecurvatureanditscovariantderivatives.Thecurvatureitselfis al ge braicallyreducible(undertheLorentzgroup)intothreetensors: r a b = C€€( w Š1 2 ( ) r )+ Cr€€,(5. 1.21) wherethetensorsaretotallysymmetricinundottedindicesandindottedindices(which isequivalenttobeingalgebraicallyLorentz-irreducible).Thetensors r and r€€arethe traceandtracelesspartsofthe Riccitensor, and wisthe Weyltensor. (Notethat ournormalizationoftheRicciscalardiersfromthestandard:Weusethemoreconvenientno rmalization,ingeneralspacetimedimensionD, r a b c d= Š [ a c b ] dr + ... ,rather than r a b a b= r .Thesignisch osensothat r isno nnegativeonshellinunbrokensupersymmetrictheories.)Thesetensorsare,ofcourse,relateddierentiallythroughthe Bianchiidentities(theJacobiidentitiesofthecovariantderivatives).Explicitlyfrom [[ D D a, D D b], D D c]+[[ D D b, D D c], D D a]+[[ D D c, D D a], D D b]=0,(5 .1.22) we “nd D D[ at b c ] dŠ t[ a b | et e | c ] dŠ r[ a b c ] d=0 ,( 5.1.23a)

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2385.CLASSICALN=1SUPERGRAVITYD D[ ar b c ] Š t[ a b | er e | c ] =0.(5. 1.23b) Theselasttwoequations(whichfollowfromthelinearindependenceof D D aand M )are the“rstands econdBianchiidentities,respectively. c.Actions IncontrasttoYang-Millstheory,ingravityonecannottraceoverthegroupwithoutintegratingoverthespacetimecoordinates,sincethetranslationgroupactsonthe coordinatesthemselves.Thus,onlyintegratedquantitiescanforminvariants.Furthermore,gravitydiersevenfromthegroupmanifoldapproachtoYang-Mills,wherethe groupgeneratorsaretreatedastranslationsinthegroupspace,inthatthelocaltranslationgroupisnotunitary:Although mis hermitian,anin“nitesimaltranslationisnot: ( mi m)= i m m= mi m+( i m m).(5.1 .24) Fromthereord eringofthetwofactors,wegetanadditionaltermproportionaltothe divergenceof .Thistermarise sb ecausesomecoordinatetransformationsarenot volume-pr eserving: e.g.,thetransformationgivenby m x misascaletransformation. Consequentlythevolumeelement d4x dx+€+/ / \ \dx+€Š/ / \ \dxŠ€+/ / \ \dxŠ€Šisnotcovariant.To covariantize,wesimplyreplace dx mwithanobjectthatisascalarundercoordinate transformations(aworldscalar): a= dx me m a.Theresult ingvolumeelementis 4= d4x eŠ 1,whereei sthe determinantof e a m. Theinvarianceofascalarintegratedwiththecovariantvolumeelementcanalso bes eenfromthetransformationlawofe,whichwewriteinthecompactandconvenient form e Š 1=eŠ 1ei .(5. 1.25) Here = mi mmeansthatthederivativeactsonallobjectstoitsleft.(ForthepresentdiscussionwemayignoreLorentztransformations.)Beforederivingthistransformationlaw,weshowhowitallowseŠ 1toforminvariantintegrals:Foranyscalar L d4x (eŠ 1L )= d4x (eŠ 1ei )( ei LeŠ i ) = d4x eŠ 1ei ( eŠ i Lei )

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5.1.Reviewofgravity239= d4x (eŠ 1L ) ei ,( 5.1.26a) wherew ehaveusedtheid entity (forany X ) [ X ]=[ X ] ei XeŠ i = eŠ i Xei .(5. 1.26b) Finally,using Xei = X +tot alderivati ve,w e “nd d4x (eŠ 1L )= d4x eŠ 1L .(5. 1.27) Toderivethet ransformationlaw(5.1.25),weneedtheidentity detX = detXtr ( XŠ 1 X ),(5.1 .28) whichfollowsfrom detX = etrlnX.T hu sw e “nd,from(5.1.4b), eŠ 1= Š eŠ 1( e m a( e a n n mŠ n ne a m)) = Š eŠ 1( m mŠ e m a n ne a m) = Š eŠ 1 m mŠ m meŠ 1= Š m( meŠ 1)=eŠ 1i .(5. 1.29) To “ndthe“nitetransformation,weiteratethein“nitesimaltransformation(5.1.29)and use ex=n lim(1+x n )n;wethusarriveatthedes iredresult(5.1.25).Anequivalentstatementofourresultisthat1 ei istheJacobiandeterminant ofthecoordinatetransformation ei .(1 ei meansthatderivativesacttotheleftuntilannihilatingthe1.) Wecannow constructinvariantactionsforgravityanditscouplingstomatter. Theonlypossibleactionthatgives h a bas econd-orderkineticoperatoris S = Š3 2 d4x eŠ 1r ,(5. 1.30) where r isthecurvaturescalarde“nedby(5.1.18)and(5.1.21).Theresultant“eldequationsare r = r€€=0.Coup lingtomatterisachievedbycovariantizationofthe de rivatives,asinYang-Millstheory,butnowthevolumeelementisalsocovariantized (witheŠ 1).AsinYang-Mills,wearealsofreetoaddnonminimalcouplingsdepending

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2405.CLASSICALN=1SUPERGRAVITYonthecurvature.(Fortheteleparallelismtheorywecanstillusethisactionif r is de“nedbythecommutatorof D D a.Thisactionl eads,byuseofthe“rstBianchiidentity, toanexpressionthat is purelyquadraticin t a b c.) d.Conformalcompensator In”atspace(i.e.,withoutgravity)certaintheories(e.g.,massless 4,orma ssless QCD)areinvariantunder(global)conformaltransformationsattheclassicallevel.On theother hand,whengravityispresent all theoriesareconformally invariantsinceconformaltransformationsareaspecialcaseof generalcoordinatetransformations.However,thistypeofconformalinvariancehasnophysicalsigni“cance,andispresentsimply b ecausethevierbeinautomaticallycompensa testheconformaltransformationsofother “e lds.ThisisanalogoustoglobalorbitalLorentztransformations: Any nonLorentz covariant”at-spacetheorycanbemadecovariantundertheseorbitaltransformationsin curvedspace,becausetheantisymmetricpartofthevierbeinactsasacompensator (e.g.,d4x ( 0 )2 d4x eŠ 1( e 0 m m )2).Aswesawabove,itisnecessarytointroduceadditional,local,tangent-spaceLorentztransformationstogiveameaningfulde“niti on of Lorentzinvarianceincurvedspace.Theoriesthatareinvariantunderthesetange nt spaceLorentztransformationswillautomaticallybeinvariantundertheusual Lorentztransformationsin”atspace,orwhenagaugeforlocalLorentzandgeneral coordinatetransformationsischosen. Sim ilarly,inthepresenceofgravityitispossibletogiveameaningtoglobalconformalinvariancebyobservingthatincurv edspaceitcorrespondstoanadditional in va rianceunder local scaletransformations e a= ee a, ...= ed ....(5. 1.31) Here ( x )isalocalpar ameterand d isthecanonicaldi mensionofthe“eld ...(usually 1forbosons,3 2 forfermions)whenwrittenwith ”at tangent-spaceindices.(Notethat e a,which hasnofreecurvedindicesanddescribesaboson,hascanonicaldimension1 sinceitcontainsaderivative.)Anytheoryincurvedspacethathaslocalscaleinvariancegivesa”atspacetheorywhichiscon formallyinvariant.Thetransformationin (5.1.31)isanotherexampleofatangentspacetransformation.

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5.1.Reviewofgravity241Thus,bothlocalLorentzandlocalscaleinvariancere”ect”atspaceinvariance propertiesofmattersystems.However,t hereisan importantdistinctionbetween Lorentzandconformaltransfo rmations:Conformalinvarianceisnotageneralproperty ofphysicalsystems,andconsequentlywedonotintroducelocalscalegeneratorsandcorrespondinggauge“eldsintoourcovariantderivatives. Asdescribedabove,theantisymmetricpartofthevierbeincanbegaugedawayby localLorentztransformations.Intheresultinggauge,generalcoordinatetransformationsmustbeaccompaniedbyrelatedlocalLorentztransformationsthatrestorethe gauge.ThelocalLorentzparameterbecomesanonlinearfunctionofthegeneralcoordinateparameter,makingconstructionofLore ntzcovariant actionsmoredicult.Similarly,localscaletransformationscanbeusedtogaugeawaythetraceofthevierbein.In fact, inlocallyscaleinvarianttheories, thedeterminantofthevierbeincanbegaugedto 1bylocalsc aletransformations.Intheresultinggauge,generalcoordinatetransformationsmustbeaccompaniedbylocalscaletransformationswithparameter determined by 1=(eŠ 1)=eŠ 1ei eŠ 4 =(1 ei ) eŠ 4 .(5. 1.32) Thelocalscaleparameterbecomesanonlinearfunctionofthegeneralcoordinateparameter.Inparticular,dimension d “elds ...nowtransformas densities undergeneralcoordinatetransfo rmations,i.e.,withanadditionalfactor(1 ei )d /4.Thefo rmalism b ecomesrathercumbersome.(Wenotethatevenintheoriesthatarenotinvariant underlocalscaletransformations,ecanstillbegaugedto1,atleastinsmallregionsof spacetime,bysomeofthegeneralcoordinatetransformations: e m m.Howev er,this resultsintheconstraint m m=0onfurtherc oordinatet ransformations,anddierenti a lly constrainedgaugeparametersareundesirablewhenatheoryisquantized(seesec. 7.3);suchgaugechoicesarepossibleuponquantization,buteshouldnotbesetto1 beforequan tization.) Wehavealre adyindicatedthateactsasacompensatorforthelocalscaletransformationsof“elds ...asgivenin(5.1.31).Infact,bymakingthe“eldrede“nition ... eŠ d /4...we canmakeall“eldsexcepteinertunderscaletransformations.In termsofthenew“eldslocalscaleinvarianc eofanactio nisequivalentt oi ndependence ofe.However,tomaintainmanifestcoordinateinvariance,itispreferabletokeep

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2425.CLASSICALN=1SUPERGRAVITYexplicitthedependenceone.Ontheotherhand,itisfrequentlyusefultodescribeto whatextentatheorybreakslocalscaleinvariance,bothbecauselocallyscaleinvariant theori esareinterestingintheirownright,andbecauseadecompositionintolocallyscale invariantplusscale-breakingpartscanbeh elpful.Thiscanbedonebyintroducingan additional compensating“eldintothetheory,butonewhichunlikeeisa scalar under generalcoordinateandlocalLorentztransformations.Todistinguishthistypeofcompensat orfromtheetype,wewillhenceforthrefertothemas tensorcompensators and densitycompensators, respectively.Densitycompensators(e.g., e[ a m ]forlocalLorentz, or ef or lo calscale)generallyoccuraspartsofphysical“eldsandarenottensorsunder thelocalsymmetrygroup(e.g.,generalcoord inatetransformations)oftheactionwithoutcompensators.Tensorcompensatorsarecovariant,andtheirpresenceallowsthe introductionofalocalsymmetryevenintheabsenceofacorrespondingglobal,”at spacesymmetry. Forlocalsc aletransformationsweintroduceasc alarcompensatortransformingas = e .(5. 1.33) Startingwith“eldsinvariantunder transformations,wenowmakethereplacements e a Š 1e a, ... Š d....(5. 1.34) Thenew“eldsstilltransformaccordingto(5.1.31).Thereplacement(5.1.34)isjusta -dependentscaletransformation.Hencelocalscaleinvarianceofagivenquantityis equivalent toi ndependencefrom .Forex ample,aftert herede“nition(5.1.34),the usualgravityaction (5.1.30)becomes S = Š3 2 d4x eŠ 1 ( + r ) .(5. 1.35) Thisactionisscaleinvariantbecause compensates thetransformationofeand + r Sinceitisnot -independenttheoriginalEinsteinactionwasnotscaleinvariant.Alternatively,(5.1.35)canbeinterpretedasascaleinvariantactionforthe“eld .Thescale invarianceallows tobegaugedtoone.InthatgaugeonerecoverstheusualEinstein action. Incontrast,theWeyltensor,whichistheonlypartofthecurvaturewhichis homogene ousin after(5.1.34),canforma i ndependent,locallyscaleinvariant(but

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5.1.Reviewofgravity243higher-derivat ive)action: SWeyl d4x eŠ 1( w)2.(5. 1.36) *** Weintro ducedthelinearizedvierbein h a masagauge“eldfortranslations;alternatively,wecanusetheanalysisofchapter3to “ndh a m.Lineari zedgravityisthetheory ofamasslessspin2“eld.Asdiscussedinsec.3.12,itisdescribedbyanirreducible onshell “eldstrength satisfying(s ee(3.12.1)) €=0.(5. 1.37) Usingtheresultsofsec.3.13,th eco rrespondingirreducible o-shell “eldstrengthisthe lineari zedWeyltensor wsatisfyingthebis ectioncondition( s +N 2 =2isan integer) w= K K w=€€€€ w€€€€(5.1.38) whichisequivalentto €€w= €€ w€€€€.(5. 1.39) By(3.13.2)appliedto N =0,thesolu tiontothisequationis w= ( €€V )€€,(5. 1.40) where h a b h€ € V( )(€€ )isatracelesssymmetrictensor.Themaximalgauge invarianceof(5.1.40)is: h a b= ( a b )Š1 2 a b c c.(5. 1.41) Thesearelinearizedcoordinatetransformationsidenticalto(5.1.1), except thatscale transformationsandLorentztransformati onsarenoti ncluded.Theycanbeaddedby introducingcompensators:thetraceandantisymmetricpartsof h a b.

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2445.CLASSICALN=1SUPERGRAVITY5.2.Prepotentials a.Conformal Asforsuper“eldYang-Millsandforgravity,onecandevelopaformulationfor super“eldsupergravityineitheroftwoways:(1)Studyo-shellrepresentationsto determinethelinearizedfo rmulationintermsofunconstrainedsuper“elds (prepotentials), andthenconstructcovariantderivatives,whichprovidethegeneralizationtothe nonlinearcase;or(2)startbypostulatingcovariantderivatives,determinewhatconstraintstheymustsatisfy,andsolvethemin termsofprepotential s.Inthiss ectionwe willdescribetheformerapproach,andinthefollowingsectionthelatter. a.1.Linearizedtheory Fromtheanalysi sins ec.3.3.a.1,weknowthatthe N =1superg ravitymultiplet consistsofmasslessspin2andspin3 2 physicalstates.Thecorrespondingon-shellcomponent “eldstrengthsare and (on-shellW eyltensorandRarita-Schwinger “eldstrength), totallysymmetricintheirindices,a sdiscu ssedinsec.3.12.a.Theselie inanirreducibleon-shellmultipletdescribedbyachiralsuper“eld(0) thatsatis“es theconstraint D(0) =(1) ,(5. 2.1) where(1)istotallysymmetricandisthesuper“eldcontainingtheon-shellWeyltensor (= w)atthe =0leve l.Theconstraintimplies D(0) =0.(5. 2.2) Bytheanalysisofsec.3.13,thecorresponding irreducibleo-shell super“eld strengthisachiralsuper“eld Wsatisfyingthebis ectioncondition( s +N 2 =3 2 +1 2 is aninteger) W= Š K K W= Š Š 1 2 D2€€€ W€€€,(5. 2.3) whichcanberewrittenas €DW= Š € D€ W€€€.(5. 2.4)

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5.2.Prepotentials245By(3.13.2)thesolutiontothisequationis W= Š i1 3! D2D( €H )€,(5. 2.5) where H€isreal.( H mightbeexpressedasaderivativeofamorefundamental“eld; thispossibilityiseliminatedwhenweexaminethe N =1theoryat thenonlinearlevel.) Were markinpassingthat Sconf= d4xd2 W2= d4x [ WD2W +( DW )2] | (5.2.6) containstheactionforlinearizedconformalgravityd4x ( w)2.Thus( 5.2.6)isthe extensionofconformalgravitytoconfo rmalsupergravityatthelinearizedlevel. Acareful examinationof(5.2.4)revealsthatthelargestgaugeinvarianceof W, writteninaformcontainingthefewestderivatives(andthusthecomponenttransformationscontainthefewestpossi blespacetimeder ivatives),is H a= D L€Š D€L.(5. 2.7) Togetinsight intothephysicalcontentof H anditstransformation,weconsidertheir componentsusing D projection. Thecomponentsof H aare h a= H a| h€= DH€| h(2) a= D2H a| h a b= Š1 2 [ D, D€] H€| a ,€= Š iD2 D€H€| A a= Š2 3 D D2DH a|Š1 6 a b c d b[ D, D€] H d| .(5. 2.8) where a b c d= i ( CCC€€C€€Š CCC€€C€€)(3. 1.22).Althoughitisconvenientto de“nethecomponent“elds h a b, a ,€, A aasabove,theseareonlythe linearized ,conformal de“nitionsofthesecomponent“elds.Inthe“nalPoincar etheorya dditi onal H aandcompensatorsuper“elddependentterms,aswellasnonlinearities,arepresent. Thecomponentsof D€L(therestof L neverenters)are: a= D€L| L1 €= D D€L| = D2L| ,

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2465.CLASSICALN=1SUPERGRAVITYL2 a= D2 D€L| = D D2L| =1 2 D( D2L )| = D2 D2L| ,(5. 2.9) andsimilarlyforthecom plexconjugate(butnote a= Š D L€).Thetransformations ofthei ndependentcomponentsof H are: h a=2 Re €, h€=1 2 C €Š L1 €, h(2) a= Š L2 a, h a b= Š ( C€€+ C€€)+ CC€€Re Š aIm b, a ,€= a €Š iC€€, A a=2 3 aIm .(5. 2.10) Wecantherefo regotoaWess-Zuminogaugebyusing Re L1, L2toalgebraicallygauge awayallof H aexcept h a b, a A a.Thesecanbei denti“edasthelinearizedvierbein, thespin3 2 Ra rita-Schwinger“eld,andanaxialvectorauxiliary“eld,respectively. West udytheremainingtransf ormations:Examining h a bwenote that canbe usedtoelim inatetheantisymmetricpartofthevierbein,whichidenti“esitasan i n“nitesimallocalLorentztransformation. Re removesthetrace,andisthereforea localscaletransformation.Finally, Im generatesacoordinatetransformation.Examining a ,€weidenti fythe €termasaRarita-Schwingergaugetransformation(alinearizedlocalsupersymmetr ytransfo rmation).The termisalocal S -supersymmetry transformation:itgaugesawaythetrace € ,€.From A aweidentify Im asanaxial gaugetransformation.(Notethatthelocal S -transformationofthespin3 2 “eldcontains nospacetimederivatives.Avo idingderivativesisimportantforquantization(seesec. 7.3)).

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5.2.Prepotentials247Thus,intheWess-Zuminogauge,the L -gaugegroupthatremainsconsistsoflocal superconformaltransformati ons:thesuper-Poincar es ubgroup(coordinate,Lorentz,and localsupersymmetry),andaxial, S -supersymmetry,andscaletransformations(seesec. 3.2.e).Localconformalinvarianceplaysamoreimportantroleinsupergravitythanin gravity:Whereastheno nconformalpartofthevierbein(itstrace)canbeprojectedout algebraically,theanalogousstatementdoesnotholdfor H (avectorisnotalgebraically reducibleinaLorentzcovariantway).Thesamedistinctionbetweenthereducibilityof thevierbeinand H applieswithregardtolocalLorentzinvariance(whichmustbemaintainedinsupergravitysimplybecausesup ersymmetrictheoriescontainspinors). a.2.Nonlin eartheory To generalizetothenonlinearcaseweexamine(asinsuper-Yang-Mills)theappropriate transformationsofthesimplestmultiplet,thechiralscalarsuper“eld.Sincegravitygaugestranslations,supergravitywillgaugesupertranslations.Wethereforelookfor themostgeneraltransformationoftheform = ei eŠ i ,=MiDM;(5. 2.11) (Wechoosetoparametrizewith DMratherthan Minordertokeepmanifestglobal supersymmetry.Thissimplyamountstoarede“nitionoftheparameters.)Wemaintain thechiralityof ( D€ =0),byrequ iringtosatisfy [ D€,] =0,(5. 2.12) whichimplies D€=0, D€ n= i €€,(5. 2.13) andhasthesolution m= Š i D€L,= D2L,€arbitrary ;( 5.2.14a) i.e., m m+D=1 2 { D€,[ D€, LD] } .(5. 2.14b) Notethattheparametersuper“eldMmustbecomplex. Inparticularthismeansthat m =( m), =(€),and€ =().Caremustb etakentodi stingu ishand

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2485.CLASSICALN=1SUPERGRAVITY€fromthehermi tianconjugatedquantities (=(€))and €(=()).We de“ne thetransformationofanantichiralscalarinasimilarfashion: = ei eŠ i = MiDM, m= Š iD L€, €= D2 L€, arbitrary .(5. 2.15) Thequantity Misthecomplex conjugateofM. At thispointitisclear,byanalogywithsuper-Yang-Mills,that H misthec orrect “eldtocovari antizethe mpartofthetransformationofthescalarmultipletkinetic term,sinceitslinearizedtransformationis(from(5.2.7)) H m= i mŠ i m.Wethereforecomplete H mtoasuper v ector HM=( H, H€, H m)andintro duceanexp onential eH, H = HMiDM.A sf orYang-Mills,thenonlineartransformationlawis eH= ei eHeŠ i .(5. 2.16) Wenote that H€canbetriviallygaugedawaybecausetheparameter€isarbitrary;cons equently Hisalsogaugedawayby .Toprese rvethisgaugechoice,the L gaugetransformations(5.2.14,15)mustbeaccompaniednowbycompensating€and tr an sformations.Forin“nitesimalwehave (eH mi m)= Š ( m m+ € D€+ D)(eH mi m) +(eH mi m)( m m+D+€ D€).(5.2 .17) (Thisequationistobeinterpretedasanoperatorequationactingonanarbitrarysuperfunction totheright.)Thisimpliesthatwemustcancel Dand D€termsontherighthandsideandhence = eHeŠ H= eH D2LeŠ H,€= eŠ H €eH= eŠ HD2 L€eH.(5. 2.18) However,wewillnotrestrictourselvesto thisgaugeinthesubseq uentdiscussion.

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5.2.Prepotentials249a.3.Covariantderivatives Ournexttaskistoconstructcovariantderivatives A=( €, a).Byanalogy withYang-Millstheorywewouldrequire( A)= ei ( A),i.e., A = ei AeŠ i or A= i [, A].However,sincew eexp ectlocalLorentztransformationstobepresent, wecang eneralizeto ( A)= LA Bei B, LA B=( L L€€, L L€€),(5.2 .19) Wede “ne,byanalogywithEinsteinstheory(5.1.15),covariantderivativesthattakethe form: A= EA+A M +A€€ M€€,(5. 2.20) wherethe M sareLorentzrotationoperators.Theiractionisde“nedin(5.1.13).Again inanalogywithordinarygravity,weadheretoalate-earlyindexconventiontodistingu is hb etweenquantitieswithcurvedindices(thattransformonlyunderthe-gauge gr o up)andquantitieswith”atindices(thattransformonlyundertheactionofthe Lorentzgenerators M and M€€).Theform(5.2.19)assumesthattheLorentztransformations LA Bactinthe usualmanner:Spinorsandvectorsdonotmix,andboth rotatewiththesameparameter.Foranin“nitesimalLorentztransformation, LA B= A B+ A B,thecovarian tderivat ivestransformas A=[ i A]+ A BB,(5. 2.21) where A B=( €€, a b)and(from( 5.2.19)) a b= €€+ €€( isthechiralrepresentationconjugateof €€:seebelow).Thisi mpliesthefollowingtransformationlawsforthe connections: A€€=[ i ,A€€] Š EA€€+ A DD€€+ €€A€€Š A€€€€.(5. 2.22) Thereisacertainamountofarbitrarinessinde“ningconnectionsthattransform properly:OnecanalwaysaddtoA anytensor KA thattransformscovariantly.As willbediscussedinthenextsection,thisarbitrarinessisphysicallyirrelevant.Astandardwayto“ndconnectionsistocomputethe anholonomycoecientsCAB Cde“nedby

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2505.CLASSICALN=1SUPERGRAVITY[ EA, EB} = CAB CEC. Suitableconnectionscanbede“nedaslinearcombinationsofthe C s. PushingtheanalogywithYang-Millsfurther,wecantrytoconstructthefollowing chiral-representation covariantd erivat ives: E€ D€, E eŠ HDeH, E mŠ i { E, E€} .(5. 2.23) However,atthispointtheanalogywithYang-Millstheorybreaksdown.Thesederivativesarenotcovariantfortworeasons:(1)Actingonnontrivialrepresentationsofthe Lorentzgroup,theyarenoncovariantbecausetheyhavenoconnections(thisiseasily cured);and(2)moreseriously,evenactingonscalarstheyarenoncovariantbecause€isnotc hiral.Thus, E€=[ i E€] Š ( E€€) E€=[ i E€]+ €€ E€+ E€,(5. 2.24) where €€= Š1 2 E(€€ )= Š1 2 D(€€ ),= Š1 2 E€€= Š1 2 D€€.(5. 2.25) Theterminvolving isharmless:itisjustaLorentzrotation,andwillbeperfectly covariantafterweintroduceLorentzconnections.(Thereisaslightproblem,however. Theindicesin(5.2.19,20)are”atspinorindiceswhereasthosein(5.2.24,25)arecurved indicesinanalogywithourdiscussionofordinarygravity.Thereforeitisnotquitecorrecttoidentifythe in(5.2.25)withtheonein(5.2.20).Wewill“ndasolutionforthis shortly.) Bycontrast,thetermproportionaltoisa supers pace scaletransformation whichisnotpartofouroriginalgaugegroupasde“nedby(5.2.19)and(5.2.20).For thetimebeingwein tro duceintothetheorya(density) compensator thattra nsforms as: =[ i ,] Š .(5. 2.26) Lateron,willbedeterminedintermsof H .Withthiso bj ect,wecanconstructa covariantspinorderivative:

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5.2.Prepotentials251 E€ E€= D€, E€=[ i E€]+ €€ E€.(5. 2.27) Thecomplexconjugateof E€iscovariantnotwi threspecttobutratherwithrespect to ; ho we ve r,justasintheYang-Millscase,wecanuse eHtoconv ertanyobject covariantwithrespectto intoanobj ectcovariantwithrespectto.Weobtain E eŠ H DeH=( eŠ H eH) E E,(5. 2.28) whereisthechiral-representationHermitianconjugateof(asinsuper-Yang-Mills: see(4.2.37)and(4.2.78)).Thecovarianttransformationof Eis E=[ i E]+ E,(5. 2.29) where = eŠ H eH. Thespinorvielbeinsthatwehaveconstructedtransformasin(5.2.21)butwith theimportantrestrictionthatthe parameterofLor entzro tations, A B,mustbed etermined(bythede“nitionin(5.2.25)andthosefollowing(5.2.29)and(5.2.21))intermsof theparameterofsupercoordinatetransformationsM.Inthe discussionofordinary gravity(see(5.1.10,11)),wesawthatananalogoussituationoccurredonlyiftheantisymmetricpartofthevierbeinwasgaugedaway.Wealsohavetherelatedproblemthat thefreei ndexonthevielbeiniscurvedwhereastheindexin(5.2.19)is”at(andconsequentlythe problemofidentifying €€with €€).Thissituationari sesbecausethevielbein E€€asde“nedin(5.2.27)isgivenby €€andthushasnosymmetricpart(i.e., E(€€ )=0).Thesolutionist orestore themissingpartbyin tro ducinganewsuper“eld N€€.InageneralL orentzframethespinorvielbein(5.2.27)ismodi“edto E€= N€€ D€,(5. 2.30) where N€€isanarbitrary SL (2 C )matrixsup er“eld( detN =1).Itac tsasacompensating“eldfortangentspaceLorentztransformations.(Thisisanalogoustogeneralizing fromaframewheretheusualvierbeinissymmetric.)The N -dependenceoftheother equationscaneasilybefoundbysimplyperf ormingthegeneralLorentztransformation whichtakes E€€from €€to N€€.Thequantity N mapsbetweencurvedand”at

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2525.CLASSICALN=1SUPERGRAVITYspinorindices.Thispermitsustosolvetheproblemofidentifying €€with €€.Since webe ganinthegauge N€€= €€,thetwoq uantitiesareequal.Furthermore,aslongas were maininthisgauge,weneednotbecarefultodistinguishcurvedand”atspinor i ndices. Adistinc tionmuststillbemadebetween”atandcurvedvectorindices. Wenowa ttempttoconstructthevectorcovariantderivativebyanalogywith Ya ng -M illstheory: E m= Š i { E, E€} .(5. 2.31) Thetransformationlawfollowsfrom(5.2.27,29): E m=[ i E m]+ E€+ €€ E€Š i ( E€€) E€Š i ( E€ ) E.(5. 2.32) De“ning EM ( E, E€, E m)wecanwrite( 5.2.27,29,32)as EM=[ i EM]+ M N EN.(5. 2.33) However,becauseoftermslike m€= Š i E€€,whicharen otpresentin(5.2.19), EMis notquitecovariant. Thetermswewanttoeliminateare(spinor) derivativesoftheLorentztransformationparameter €€;therefo re,theremedyistointroduce(spinor)Lorentzconnections into(5.2.31).Theseconnectiontermswillrede“ne E msothatittransformscovariantly. To “ndtheconnections,wede“nea(noncovariant)setofanholonomycoecients CMN Pby [ EM, EN} = CMN P EP.(5. 2.34) Fromthetransfo rmationsin(5.2.27,29,32)weobtain CMN P=[ i CMN P]+ E[ MN ) P+ [ M | R CR | N ) PŠ CMN RR P.(5. 2.35) Inparticularwe“nd C€€€=[ i C€€€]+ E(€€ )€+ (€ |€ C€ |€ )€Š C€ ,€€€€, C m ,€ r=[ i C m ,€ r] Š E€ m r+ m n C n ,€ rŠ C m ,€ n n r+ €€ C€ m r+ i m €€,(5. 2.36)

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5.2.Prepotentials253andcorrespondingequationsfortheconjugates C and C m r.From( 5.2.36)wesee that Š1 2 C( € ,€ )€transformsastheneed edspinorconnection: [ Š i1 2 C( € ,€ )€ E]= i ( E€ ) EŠ i1 2 C( € ,€ )€ EŠ i1 2 €€ C( € ,€ )€ E.(5. 2.37) Thereforewede“ne E a a m[ E mŠ i1 2 C( € ,€ )€ EŠ i1 2 C (€ € ) E€],(5.2 .38) where a m N N€€inthegauge N€€= €€.Thev ectorvielbein E atransformscovariantly. Wehavealre adyconstructedoneoftheLorentzconnectionsuper“eldsA (as notedabove ,inthe gauge N = wen eednotdistinguishcurvedand”atspinor i ndices).Wecanconstructtheremainingconnectionsinthestandardway(seesubsec. 5.3.b.1)fromtheanholonomycoecients CAB Cde“nedby EA ( E€, E, E a),where E€= €€ E€, E= EinourparticularLorentzgauge,and E aisgivenin(5.2.38). Alternativel y,wecanuse C directly.Aswesawabove, anappropriatelytransformingspinconnection€ isgivenby1 2 C( € ,€ )€.For€€€wehave achoi ce:Both Š1 4 C€ (€ € )andŠ 1 2 [ C€ ,€€+ C€ ,€ ,€Š C€ ,€ ,€]transfo rmappropriately.Ingeneral,any linearcombinationofthesecanbeus edasaspinconnection.Furthermore € (€ C€ ), d dis Lorentzcovariant,andcanbeaddedto€€€;sees ec.5.3.a.3.Wechoose €€€=1 4 [ Š C€ (€ € )+ € (€ C€ ), d d]. (5.2.39a) Wealre adyhad €€= Š1 2 C (€ € ).(5. 2.39b) Wealsohaveco rrespondingexpressionsforthecomplexconjugates ,€ Thevectorconnectionisde“nedby: a = Š i [ E€ + ,€€€ + E€ +€ + ( | € | )].(5.2 .40)

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2545.CLASSICALN=1SUPERGRAVITYasfollo wsfrom €= Š i {, €} a.4.Covariantactions InsupergravityLagrangianscannotbeinvariantbecauseallourquantities,includin gs calars,transformundertransformations.AtbestaLagrangiancantransformas atot alderiva tive: IL = Š ( Š )MDM(MIL ).(5.2 .41) Thiscanberewrittenas IL = iIL = i [, IL ]+ i (1 ) IL ,(5. 2.42) where = i MD M= i ( Š )M[ D MM+( DMM)].(5.2.43) Equation(5.2.42)isthetransformationlawfora density. Itiseasytocheckthata scalartimesadensityisalsoadensity. Byanalogywithgravity,wetakethevielbeinsuperdeterminant E = sdetEA Masa candidateforadensity.Indeed,from(3.7.17) EŠ 1= Š ( Š )MEŠ 1[( EŠ 1)M A EA M],(5.2 .44) where,from(5.2.21) EA M= i ( EA M)+( EAM)+ A BEB M.(5. 2.45) (HowevertheLorentzrotationtermstriviallydropoutof(5.2.44).)Consequently, EŠ 1= Š i ( Š )MEŠ 1[( EŠ 1)M A EA M] Š ( Š )MEŠ 1[ DMM] = i [, EŠ 1] Š ( Š )M( DMM) EŠ 1= iEŠ 1.(5. 2.46) Therefore,invariantactionscanbeconstructedasintegralsofproductsof EŠ 1and scalarquantitiesoftheappropriatedimension( E itselfisdimensionless): ( EŠ 1IL )= i ( EŠ 1IL ).

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5.2.Prepotentials255Forsupergravitywew anttohavetheusualEinsteinterminthecomponentaction; hencewerequire SSG=1 2 d4xd4 EŠ 1ILSG=1 2 d4x eŠ 1ILcomponent.(5. 2.47) Thisimpliesthat ILSGisadimensionlessscalar.Ingeneral,theonlydimensionless scalarsinthetheoryareconstants,sowemusttake(however,seebelow) SSG1 2 d4xd4 EŠ 1(5.2.48) asthelocallysupersymmetricinvariantactionforPoincar esupergravity. Thevielbeinsuperdeterminantcanbeworkedoutinastraightforwardmanner using( 3.7.15).We“nd E = sdet [ EA M]= sdet [ EA M]=22sdet [ EA M( H )].(5.2.49) However,variationoftheactionwiththe“eldconsideredasanindependentvariable leadstoasingular“eldequation:( E )Š 1=0.Thisisnot surprising:wasintroduced asadevicetosimplifytheconstructionofth ecovariantderivati ves,andits houldbe relatedtothefundamentalprepotential HM. b.Poincar e Wenowconsi derspeci“cformsforthecompensatorintermsof H .Aswediscussedearlier,the-gaugegroupincludessuperconformaltransformations:Thus,the covariantderivativeswehaveconstructedareappropriatefordescribingasuperconformallyinvarianttheory.Thesuperconformalactionisthenonlinearversionof(5.2.6).It isafunctionalof HMonly;dropsoutcompletely.TodescribePoincar esupergravity, wew illhavetobreaktheextrainvariance,i.e.,thecomponentsuperscaleinvariance. To “ndanappropri ateexpressionforwerecallthatittransformsasa(noncovariant) density( 5.2.25,26) =[ i ,] Š .(5. 2.50) Theonlyotherdimensionlessobjectthattransformsasadensitywithrespecttoand not is E (, H )(see(5. 2.46)).Wethereforeexpressintermsof HM(implicitly)by writing

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2565.CLASSICALN=1SUPERGRAVITY4 n= En +1.(5. 2.51) (Thisparticularparametrizationwillproveconvenientwhenwritingtheexplicitaction (5.2.65).)However,thisrelationisnotpreservedbythefull-group:Ifwetransform bothsidesof(5. 2.51),using(5.2.46,50)we“ndtherestriction Š 4 n = Š ( n +1 )(1 i ),(5.2 .52) or,moreexplicitly, (3 n +1) D€€=( n +1)( m mŠ D).(5.2 .53) Thisisanacceptablerestrictiononthegaugegroup:Wecanshowthatitcorrespondstoreducingthe component localsup erconformalgrouptothesuper-Poincar e group.We note thatwhen n = Š1 3 (5.2.53)setsthe chiral quantity m mŠ Dto zero:i.e.,using(5.2.14)therestrictioncanbewrittenas n = Š1 3 : DŠ m m= D2DL=0.(5. 2.54) Ontheotherhand,for n = Š1 3 theconditionrestricts€:(5. 2.53,54)imply n = Š1 3 : D2€=0.(5. 2.55) Wean alyzethecase n = Š1 3 “rst.Since€isunrestricted,wecanstilluseitto gaugeaway H€;wethenn eedonlyreconsiderourdiscussionoftheWess-Zuminogauge for H msubjecttotheres triction(5.2.54).Thisrestrictionimpliesthefollowingrelations amongthecomponentsof Lin(5.2.9): = i a a, = Š i €L1 €, aL2 a=0.(5. 2.56) Thusthelocalsuperconformaltransfo rmationsarereducedtothoseoflocal super-Poincar e: and havebeenremovedasindependentparameters.Further,a differential constrainthasbeenimposedononeoftheparameters,the L2thatweusedto gaugeawayextracomponentsof H m: B = mh(2) mcannolongerbeeliminated.Consequently( cf.thediscussionfollowing(5.2.10)),we“ndthattheminimalsetofcomponent

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5.2.Prepotentials257“eldsdes crib ing N =1Poincar esupergravityare thevierbein(then onlinearextension of h a b),thegravitino a ,anaxialv ector“eld A a,andina ddition thecomplexscalar “eld B (whichcouldbegaugedawayinthesuperconformalcase). For n = Š1 3 ,weca nnotuse€togaugeawayallof H€.Wecan gaugeawayparts ofitbyusingupallthecomponentsof D€€andhence,becauseoftheconstraint (5.2.52),thoseof D2DL.A gainthelocalsuperconfo rmalgrouphasbeenreducedto thesuper-P oincar egro up: and nolongerenterasindependentparameters.Wewill discussthecomponentcontentof n = Š1 3 supergravitylater. For n =0,(5. 2.51)impliesthat E =1andhen cethattheaction(5.2.48)vanishes. Itisclearfrom(5.2.53)thatfor n =0thepar ameterMsatis“es( Š )MDMM=0. Thisispreciselytheconditionthatthesupercoordinatetransformationparametrizedby aresupe rvolumepreserving.However,the n =0caseis uniquebecauseitcontains a(constrai ned)dimensionlessscalar V (anabeliangaugeprepotential)whichcanbe usedtoconstructanaction.Furthermore,theconstraint(5.2.51)isinvariantunderan arbitrarylocalphaserotationof:= eiK5, K5= K5= eŠ H K5eH.Thisinv ariance canbeusedtochooseagaugewhere=.Another consequenceof(5.2.51), E = E =1,isthatweh aveimposedap artialgaugeconditionon H :The hermiticity conditi onthat EŠ 1satis“es(s ee(5.2 .60)below)impliesthat(1 eŠ H)=1, i.e., 1 H=0.Thisisachievedbyc hoosingthe gaugewhere H= Š i D€H€insteadof zero,sothat H = DH€ D€+ D€H€Dwherethe D and D preceding H€actonall objectstotheright.Thecaseof n =0w illbediscu ssedinmoredetailinthefollowing sections. To “ndtheexplic itexpressionforintermsof H ,weuse(5 .2.49)and(5.2.51) andwrite 4 n= eŠ H 4 neH= eŠ H En +1eH.(5. 2.57) (Forsimplicitywehaveassumedthat n isreal;thegeneralizationtocomplex n is straightforwardbutnotinteresti ng).Therefore,wemustcompute E .Altho ughthis couldbedonebybruteforce,amoreelegantprocedureispossible: In(5.2.28)wede“ned

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2585.CLASSICALN=1SUPERGRAVITYE= eŠ H EeH,(5. 2.58) where Eisthehermitianconjugateof E€.Therighthandsidecanbeinterpretedasa coordinatetransformationwithimaginaryparameter iHMinsteadof i M.Byhe rmitian conjugation E€isthecoordinatetransformof E€(thehermitianconjugateof E). Therefore,anycovariantconstructedfrom Eand E€canbeobtainedbyacomplex coordinatetransformationfromthecorrespondingobjectconstructedoutof E€and E. Thisisthecasefor E a,andalsofo rthevielbe insuperdeterminant. Thefullnonlineartransformationofthesuperdeterminantfollowsfrom(5.2.46): ( EŠ 1)= EŠ 1ei =( ei EŠ 1eŠ i )(1 ei ).(5.2 .59) Bythesam emethodusedt oderivethisr esultfrom E A= ei EAeŠ i ,from EA= eHEAeŠ H(5.2.60a) wehave EŠ 1= EŠ 1eH(5.2.60b) andhence EŠ 1= eŠ H EŠ 1eH(1 eŠ H).(5.2 .60c) Substituting(5.2.60)into(5.2.57)we“nd 4 n=[ E (1 eŠ H)]n +1,(5. 2.61) or,usingtheoriginalconstraint(5.2.51), 4 n=4 n(1 eŠ H)n +1.(5. 2.62) Finally,substituting(5.2.62)and(5.2.51)into(5.2.49)we“nd 4 n=4( n +1)(1 eŠ H)( n +1)22 n En +1,(5. 2.63) or =[(1 eŠ H)( n +1)28 n En +1 4 ]Š 1.(5. 2.64)

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5.2.Prepotentials259Thuswehavesolvedforintermsof H .Usingtheser esults,wecanrewritethe n =0 supergravityaction(5.2.48)intermsoftheunconstrainedsuper“eld HM: SSG= 1 n 2 d4xd4 [ En(1 eŠ H)n +1 2 ].(5.2 .65) (Thefactor1 n givestheappropriatenormalizationforthephysicalcomponentactions andforthesupersymmetric-gaugepropagators: En=(1+)n 1+ n ;see,e.g., (7 .2 26) .) Thisactionisinvariantunderthegroupoftransformationsrestrictedby (5.2.53). c.Densitycompensators Formany purposes,e.g.,quantization,itisawkwardtoworkwiththeconstrainedgaugegroup.Asdescribedinsec3.10wecanenlargetheinvariancegroupofa theorybyintroducingcompensating“elds.Inthiscase,theconstraint(5.2.53)was in tr o ducedbytherelation(5.2.51),whichisnotcovariantunderthefullgaugegroup. We ch oo seourcompensatorstorestorethecovarianceof(5.2.51)underthefullgroup. Forthe n = Š1 3 caseasuitablecompensating“eldisa chiraldensity thattransformsas =[ i ]+1 3 ( DŠ m m) D€ =0.(5. 2.66) (Thefactor1 3 gives thesameweightasadensitymatte rmulti plet:seebelow.)From (5.2.46,50,66),itfollowsthatthecovariantversionof(5.2.51)is: Š4 3 = 2E2 3 .(5. 2.67) Eq.(5.2.66)canberewrittenas ( 3)=( 3) ei ch,ch= mi m+iD.(5. 2.68) Thechiralparameter(1 i ch)= Š m m+ Dcanbeusedtoscale arbitrarily:In particular,ifwe choosethegauge =1,from(5. 2.67,68)werecovertheconstraints (5.2.51,54),r espectively. Inthe n = Š1 3 case,theconstrainedobjectisthelinearsuper“eldexpression(cf.

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2605.CLASSICALN=1SUPERGRAVITY(5.2.53,54)) D€€+(n +1 3 n +1 ) D2DL.Hen ce,asuitablecompensatorisacomplexlinear super“eld, D2=0.Wedete rmineitstransformationpropertiesbyrequiringthatits variationis linearandthatcanbescaledto1.(I tsho uldberecalledfromthediscussioninsec.4thatacomplexlinearsuper“eldcanalwaysbeexpressedintermsofan unconstrainedspinorsuper“eld:= D€ €(4.5.4).)Sincetheproductofachiralanda linearsuper“eldislinear,wecanalwayshaveaterm( DŠ m m)in .Toscale to1,wen eedaterm( D€€), butsincetheproductoftwolinearsuper“eldsis not linear,suchatermmustcomefromthecombination( D€€) Š € D€= D€(€). Thisleadsuniquelyto =[ i ,]+( D€€)+(n +1 3 n +1 )( DŠ m m)(5.2 .69) and 4 n=3 n +1En +1.(5. 2.70) Asintheminimal n = Š1 3 case,choosingthegauge=1leadsbacktotheconstraints (5.2.53,51).Weobservethatfor n =0,(5. 2.70)implies=(1 eŠ H).(Wec ande “ne a linearcompensator Š3 n +3 3 n +1 intermsofbothand ;thise nlar gesthegauge groupbyachiralscaletransformation,andresultsin n -independenttransformationlaws forboth and). Wecanr epeatthecomputationsofeqs.(5.2.57-64)includingthecompensators;we “nd n = Š1 3 ,=[n Š 1n +1]Š (3 n +1 8 n )[(1 eŠ H)n +1 E2 n]Š (n +1 8 n ), n = Š1 3 ,= Š 11 2 (1 eŠ H)1 6 EŠ1 6 .(5. 2.71) The n =0superg ravityaction(5.2.48,65)takestheform n = Š1 3 SSG( H ,)=1 n 2 d4xd4 En(1 eŠ H)n +1 2 []3 n +1 2 n = Š1 3 SSG( H )= Š3 2 d4xd4 EŠ1 3 (1 eŠ H)1 3 .(5. 2.72)

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5.2.Prepotentials261Theseareinvariantunderthefullgaugegroup;after -integration,theybecomethe usualPoincar es up er gravitycomponentactionforthegraviton,gravitino,andauxiliary “elds.The n =0casew illbediscu ssedlater. d.Gaugechoices Thecomponent“eldcontentoftheactions(5.2.72)ismanifestinaWess-Zumino gaugewherethetransformationshavebeenusedtoremovealgebraicallythegauge componentsofthesuper“elds.Sincethegroupisnowunconstrained,wecanchoose thegauge H=0and H mwithonly h , A asdescribedfollowing(5.2.10).Therewe foundthattheremaini ng gaugefreedomisparametrizedbythe Im , Re + iIm and componentsof L;theseco rrespondtoLorentz,coordinate,localsupersymmetry, scale+ i chiral,and S -supersymmetrytransformationsrespectively. For n = Š1 3 ,wede“nethe(lineari zed)componentsof by u = | u= D | S =S+ i P= D2 | .(5. 2.73) UnderthegaugetransformationsthatremainintheWess-Zuminogauge(weneedthe compensatingtransformations(5.2.18)andinaddition L1 €=1 2 C €in(5.2.10)), thesecomponent strans formas u = Š1 3 ( + i a a), u=1 3 ( + i €L1 €), S = i1 3 aL2 a.(5. 2.74) Inwritingthesetransformationlawswehavelinearizedthefullin“nitesimaltransformationof(5.2.66)andkeptonlythosetermsindependentof H aand .( Thisisanalogous to th ea pproximationgivenin(5.2.7)ascomparedtothefullin“nitesimaltransformationgivenin(5.2.17).)Thescaleandaxialtransformationsparametrizedby canbe usedtoscale u to1,andthe S -supersymmetrytransformationcanbeusedtogauge uaway.Thusweseethat actsasacompensatorfortheconstrainedpartofthe group,i.e., u and uarecompensatorsforthecomponentsuperscaletransformations (scale,chiral U (1),and S -supersymmetry).Setting u Š 1= u=0restoresthe

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2625.CLASSICALN=1SUPERGRAVITYconstraintandleavesonlythecomponentsuper-Poincar etransfo rmations.Theremaining“elds S S alongwith h ,and A from H marethecomponent“eldsofminimal supergravity. Withintheframeworkofthe n = Š1 3 theory,thech oiceofcompensatorisnot unique.Anysuper“eldthatcontainsonlysuperspin0(components u and u)canbe usedasacompensator.Inglobalsupersymmetrywefoundanumberofsuchmultiplets: thevariantrepresentationso fs ec.4.5.d.Thuswecanreplace 3Š 1in(5. 2.66-72)by thechiral“eldstrengths= D2V of(4.5.56)or= D2Dof(4.5.66).Thenew components u uofthenewmultipletsarecompletelyequivalenttothecorresponding componentsof .Onthe otherhandthe S componentischanged.Wethushavetwo casesina dditi onto S = D2 | =S+ i P (2) S = D2 | = D2 D2V | =1 2 { D2, D2} V | +1 2 [ D2, D2] V | =1 2 { D2, D2} V | + i €[ D, D€] V | =S+ i aP a,( 5.2.75a) or (3) S = D2 D2D| = Š i €D2 D€| = aS a+ i aP a.(5. 2.75b) Incase(2)theauxiliary“eldsaretherealscalarSandthedivergenceoftheaxialvector P a(insteadofthepseudoscalarP).Alsoi ncase( 2)wecanmakethereplacement V = iV.Thee ectofthisatthecomponentlevelisthat S takestheform S = aS a+ i P. Theauxiliary“eldsarethedivergenceofavector S aandthe pseudo sc alarP.Incase(3),bothauxiliary“eldsaredivergences. For n = Š1 3 ,beforeweintro ducedcompensators,therestrictedgaugegroup (5.2.54)couldbeusedtoeliminateallbutthe h , A and S componentsof H m,where S mD2H m| (cf.thediscussionafter(5.2.56)).Inthepresenceofthecompensators, thefullgaugegroupwasusedtogaugeaway S from H m; S appearedinthecompensatorinstead.However,onlyincase(3)aboveis S (inthecomp ensator)adivergence; ther eforethisi stheonlyca seinwh ichthetheory with thecompensatoriscompletely equivalentto thetheory without thecompensator. For n = Š1 3 ,0,wecan alsogototheWess-Zuminogaugeand“ndthecomponents

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5.2.Prepotentials263ofthenonminimaltheor y.Wede “nethe(linearized)componentsof: u = | u= D | S = D2 | = D |Š (4 n +1 3 n ) D | V a= D D€ | €= D2 D€ | .(5. 2.76) Thetransformationsanalogousto(5.2.74)are u = Š Š (n +1 3 n +1 )( Š i m m), u= S = i (n +1 3 n +1 ) mL2 m, = Š i € €+ i (n +1 3 n +1 ) €L1 €, V a= Š i1 2 a Š i € €€, €= €.(5. 2.77) Asbeforewe canscale u to1andgauge uto zero.Theremainingcomponents V a, aretheadditionalauxiliary“eldsofnonminimalsupergravity. On ceagainwenotethatthecase n =0isdiere nt;thetran sformation u isindepe ndentof Im (theaxialrotations)sothatthisgaugeinvariance survives intheW essZuminogaugefor.Since=(1 eŠ H),wehave S = €= €=0and V a= bT[ a b ], where T[ a b ]isarealantisymmetrictensorgauge“eld. e.Su mmary Wesu mmarizeheresomeofthequantitie sthatwehavecon structedsofar: E€= D€, E= eŠ HDeH, H = HMiDM, E€= E€, E= E,= eŠ H eH, E€= Š i { E, E€} ,[ EM, EN} = CMN R ER.( 5.2.78a)

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2645.CLASSICALN=1SUPERGRAVITYE€= N€€ E€, E= N E, E€= N€ €( E€+ i1 2 C€ ,( € )€ E+ i1 2 C (€ € ) E€), N€ €= N N€€, det ( N€€)=1.(5.2 .78b) Thefactor N isequalto inasuitableLorentzframe.Thesuperscalecompensator takesth eform n = Š1 3 := Š 11 2 (1 eŠ H)1 6 EŠ1 6 n = Š1 3 ,0:=[n Š 1n +1]Š (3 n +1 8 n )[(1 eŠ H)n +1 E2 n]Š (n +1 8 n ).(5. 2.78c) Thetildequantitiesarechiralrepresentationhermitianconjugatesde“nedbyanalogy withthewayisde “nedfrom.TheLorentzconnectionsuper“eldsA andA€€takesimpleformswhenexpressedasfunction softhea nholonomycoecientsde“nedby [ EA, EB} = CAB CEC.Theyaregivenby = Š1 4 [ C ,( € )€Š ( C ), d d], €€= Š1 4 C (€ € ), a = Š i [ E€ + E€ Š C ,€ Š C ,€€€ + € +€ ],(5.2 .78d) and€€€,€ a€€areobtainedbychiral-representationcomplexconjugation(i.e., thetildeoperation). f.Torsionsandcurvatures Fromthecov ariantderi vatives A= EA+A( M )wede “netorsionsandcurvatures [ A, B} = TAB CC+ RAB( M ).(5.2 .79)

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5.2.Prepotentials265Usingtheexplicitform,we“ndthatthetorsionsandcurvaturesidenticallysatisfythe followingconstraints: T €= T ,€ cŠ i €€= T c=0, R ,€= R €€= T ,( b c )Š1 2 b cT d d=0 .( 5.2.80a) Wealso “ndfurtherconstraintswhos eformdep endsonthevalueof n .Theseare n = Š1 3 : T b c=0, n = Š1 3 ,0: T b c=1 2 €€ T d d, R = Šn 3 n +1 ( €+n Š 1 2(3 n +1) T€) T€,(5. 2.80b) where R = i1 4 T€ ,€ and T= T b b.Weref erto theseas conformalb reaking constraints.Ina treatmentthatdoesnotusecompensators,theseconstraintshavetobe imposeddirectly,tobreakconformalsupergravitydowntoPoincar esupergravity.Inthe compensatorapproachtheyarisenaturallywhenagaugechoiceismadetobreakthe conformalinvariance;seesec.5.3.b.6,7. Anotherwaytoex presstheconstraintsonthetorsionsandcurvaturesistowrite thegradedcommutatorofthecovariantderivatives.Fortheminimaltheory( n = Š1 3 ) we “nd {, } = Š 2 RM, {, €} = i €, [ b]= Š iC[ R €Š G€] + iC[ W€€€ M€€Š ( G€) M ] Š i ( € R ) M, [ a, b]={[ C€€W + C( €G€) Š C€€( R ) ] + iCG€€

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2665.CLASSICALN=1SUPERGRAVITY+[ C€€( W +( 2 R +2 R R ) C ) Š C( €G€)] M }+ h c .,(5.2 .81) where W, G a,and R arei ndependentLorentzirreducible“eldstrengths de“ned by theseequations. Fornonmi nimaltheories( n = Š1 3 ,0)thegr adedcommutatorstaketheforms {, } =1 2 T( )Š 2 RM, {, €} = i €, [ b]=1 2 T€Š iC[ R +1 4 T] €+ i [ CG€Š1 2 C(( +1 2 T) T€)+1 2 ( €T) ] Š i [ C( G€) M +(( €Š T€) R ) M] + iC[ W€€€ M€€+ i1 3 W€ M€€],(5.2 .82) where W, G a,and Tarethei ndependenttensors.Intheseequations, R and Ware de“nedintermsof T.Thequantity R wasde “nedin(5. 2.80b)and Wisgivenby W= i [1 2 €( €+1 2 T€)+ R ] T(5.2.83) TheexpressionforthecommutatoroftwovectorialcovariantderivativesinthenonminimaltheorycanbecalculatedfromtheBianchiidentity; [ a, {, €} ]= { €,[ a, ] } + {,[ a, €] } .(5. 2.84)

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5.3.Covariantapproachtosupergravity2675.3.Covariantapproachtosupergravity a.Choice ofconstraints Intheprevioussectionweshowedhowtostartwithunconstrainedprepotentials andconstructoutofthemacovariantsuper“eldformulationofsupergravity.Herewe dothereverse:Startingwithamanifestlycovariantbuthighlyreduciblerepresentation ofsupersymmetry,weimposeconstraintsonthegeometryandsolvethemintermsof theprepotentials.Prepotentialsareessentialforsuper“eldquantization,whereasmanifestlycovariantf ormulationsmakecouplingtomatterstraightforwardandallowusto developanecientandpowerfulbackground“eldmethodforthequantumtheory. a.1.Compensators Asdiscussedinsec.3.10,itisoftenusefultointroduce(additional)localsymmetriesrealizedthroughcompensators.Asdiscussedinsec.5.1,ingravitationaltheories therearetwotyp esofcompensators:(1)densitycompensators,whichtransformnoncova riantlyunderthefulllocalsymmetrygroup,andthusappearincovariantquantities onlyincombinationwithother“elds;(2)tensorcompensators,whichtransformcovariantly,andthusallowtherealizationofsymmetriesthatmaynotbeinvariancesofthe entiretheory.Densitycompensatorsallo wthe linearrealizationofsymmetriesthat wouldotherwisebereali zednonlinearly(as,forexample,innonlinear models,or Lorentzi nvarianceingravitywith spinors).Whensuchsymmetriesareglobalsymmetriesofsometheory(atleaston-shell),the (super)s pacetimederivativeofthedensity compensatorcanappearasagaugeconnection.(Ifsome“eldtransformsas = then itiseasytoconstr uctaconnectionas since = .) Ontheotherhand, ifthesymmetriesonewantstorealizelinearlyandlocallyare not evenglobalsymmetriesoftheentiretheory,itisnecessarytointroducetensorcompensat orstocancelarbitrarygaugetransformationsofthistypeintermsthatarenot invariant.Thisphenomenonwasillustratedinsec.3.10.binthediscussionofthe CP (1) model,andin(5.1.35),wherethecompensator pe rmitsageneralizationoftheEinstein-Hilbertactiontoanact ionwithanadditionallocalscaleinvariance.(Thisdoes not implyanynewphysics.Thegaugeinvariancewithrespecttoscaletransformations mustbe“xedju stasanygaugeinvarianceandthemostconvenientchoiceis =1.)

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2685.CLASSICALN=1SUPERGRAVITYIngravitationaltheories,somesymmetriesneed both typesofc ompensators.This isbecausethesymmetriesarere alizedtwice.Forexample,inordinarygravity,thedensitycompensatoreŠ 1transformsas eŠ 1= Š m mŠ 4 ,where m mrepresentsthelocal scaletransformationpartofthegeneralcoordinatetransformationparametrizedby m, and isanindependentlocaltangentspaceparameter.ThequantityeŠ 1isthusa gauge “eldforsca letransformationsin m, butalsoa compensator forthetransf ormations parametrizedby .Ita llowsthefull minvariancetoberealizedlinearly;italsoallows localscaletransformation stoberea lizedlinearlyvia .Howev er,mostgravitational theori esarenotlocally(tangentspace)scaleinvariant:Thus,ifwestillwanttorepresent localscaletransformations,wemustintroduceatensorcompensator = ,tocancelthe transformationofnoninvarianttermsinanaction.Tosummarize:(1)eŠ 1isa density compensatorforlocalscaletransfo rmations,allowingthemtoberealized linearly inlocallyscale-invarianttheories;while(2) isa tensor compensatorforlocalscale transformations,allowingthem toberealizedlin early(wheneŠ 1isalsopr esent)in noninvariant theori es.NotethateŠ 1tr an sformsunderboth m mand ,whereas transformsonlyunder (inthelinearizedtransformation ).Thereisalsothecombination eŠ 14,t ransformingonlyunder m mas eŠ 14= Š m m,whichis usefulinconstructinginvariantactions.ItshouldbenotedthateŠ 14is,inasense,a“ eldstrengthfor the gaugetransformations.Anothersuch“eldstrengthistheintegrandofthe expressionin (5.1.35). We “ndasim ilarsituationinsupergravity.Thereweintroducenotonlylocal (real)scaleinvariance,butalsolocal(chiral) U (1)invariance(asageneralizationofthe globalR-invarianceofpuresupergravity)tosimplifytheanalysisofconstraintsand Bianchiidentitiesasmucha spo ssible,andweincludeitsgeneratorinthecovariant derivatives.(Inextendedsupergravity,thecorrespondingextrainvarianceis U ( N ):i.e., thelargestinternalsymmetryoftheon-shell theory.S eesecs.3.2and3.12.)Inthepresentapplication,theresultofsuchanapproachisthatthetorsionsandcurvaturescontainfewertensorsthantheywouldwithout theenlargedtangentspace.(Themissing tensorsrea ppearas“eldstrengthsofa tensor compensator.)Theselattertensorsare generallythetensorsoflowestdimension,sotheireliminationfromthetorsionsandcurvaturesa llowsgreatsimpli“cationintheanalysisoftheBianchiidentities,asdiscussed below(s ec.5.4),andsimpli“esouranalysisofconstraints.

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5.3.Covariantapproachtosupergravity269Sinceneither U (1)(R-invariance)norscaleinva rianceisasymmetryofgeneral theoriesofsupergravity+matter,weintroduceacomplexscalar tensor compensatorto compensateforboth(exceptfor n =0,whereR-inva rianceismaintained,sothecompensat orisreal).Inanalogytogravity,thereisalsoacomplexscalar density compensatorforthesetransformations:Itisthedensityoftheprevioussection,now unconstrained ,whichw illappearwhensolvingtheconstraints.Thequantityisthedirect analogofeŠ 1ofgravity,andthetensorcompensatoristheanalogofgravityscomponent “eldcompensator .Wew illfurthermore“ndaspecialsi gni“cancefortheanalogof gravityscombinationeŠ 14:Itisthesupe rspacedensitycompensator or,whichsatis“esasimple(noncovariant)constraint.T hetensorcompensatorsatis“esthedirect covariantizationofthisconstraint.Theanalogofgravitys misM,andth atof is thescaleparameter L and U (1)parameter K5.Afeatureofs upergravityn otappearing ingravityisthatofaglobalsymmetry,namely U (1),withbothdensityandtensorcompensat ors,whosedensitycompensator()isusedtoconstructa U (1)-gaugeconnection thattriviallygaugesthesymmetryinsuperspace(asinnonlinear models).(However, asingravity,itisnotusefultointroducegaugeconnectionsforscaletransformations, sinceglobalscaletransformations,unlikeR-symmetry,arenotanunbrokeninvarianceof theclassicaltheory(evenwithoutmatter)). Aftercompletingouranalysisofthe U (1)-covariantderivat ivesandt ensorcompensators,w ew illobtainthe( n =0) U (1)-noncovariant derivativesoftheprevioussection, whicharemoreconvenientforsomeapplications.Thisisachievedby“rstgaugingthe tensorcomp ensatorto1,whichexpressesintermsof H and or,andthenby droppingthe U (1)connection,shoulditnotdisappearautomatically. Webeginwit hthecovariantd erivat ives(cf.(5.2.20)) A= EA+A( M ) Š i AY [ A, B} = TAB CC+ RAB( M ) Š iFABY ,(5. 3.1) where Y = Yisthe U (1)gen erator,whosetangentspaceactioncanbesummarizedby [ Y A]=1 2 w ( A ) A,orexp licitly: [ Y ]= Š1 2 ,[ Y €]=1 2 €,[ Y a]=0.(5 .3.2)

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2705.CLASSICALN=1SUPERGRAVITYThecovariantderivativestransformas A= eiKAeŠ iK, K = KMiDM+( K iM + K€€i M€€)+ K5Y .(5. 3.3) a.2.Conf ormalsupergravityconstraints Thecovariantderivativesde“nearealizat ionoflocalsupersymmetry.However,it ishighlyreducibleandcontainsmuchmorethanthesupergravitymultiplet.Therefore, by analogywithYang-Mills,weimposecovariantconstraintsonthesederivativesto e liminateunwantedrepresentations.Thesupergravityconstraintscanbeexpressedin thesimpleform €= Š i {, €} ,(5. 3.4a) T = T b b= T (€ € )=0,(5. 3.4b) {, } =0 when =0;(5. 3.4c) (andtheirhermitianconjugates)or,intermsofthe“eldstrengths, T€ c= i €€, T€ = R€ = F€=0,(5. 3.5a) T = T b b= T (€ € )=0,(5. 3.5b) T c= T€=0.(5. 3.5c) Wehave dividedtheconstraintsintothreecategories:(a)conventionalconstraints thatdeterminethevectorLorentzcomponentofthecovariantderivative, a,inte rmsof thespi norcomponents €;(b)conv entionalconstraintsthatdeterminethespinor connectionsand(andtheirhermitianco njugates)intermsofthespinorvielbein E;and(c)representatio n-preservingconstraintsthatareneededforconsistencywith thede“nitionofchiralsuper“eldsincurvedsuperspace.Asforsuper-Yang-Millsand ordinarygravity,conventionalconstraintscanbeinterpretedaseithersettingcertain “eldstrengthstozero,oraseliminatingthemfromthetheoryby“eldrede“nitions. The“rstsetofconventionalconstraintsisofthesameformasforsuper-Yang-Millstheory,whilethesecondisanalogoustotheconstraintsofordinarygravity.The

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5.3.Covariantapproachtosupergravity271representation-preservingconstraintsarealsoofthesameformasforsuper-Yang-Mills theory.Althoughwehaveonlyrequiredtheexistenceofchiralscalars(i.e.,scalarmultiplets),thissetofconstraintsisalsosucienttoallowtheexistenceofchiralundotted sp inors(forexample,the“eldstrengthsofsuper-Yang-Mills),witharbitrary U (1) char ge.(ThesecondtypeofconstraintalreadydeterminesthespinorialLorentzand U (1)conn ectio ns.) Theconstraintsactuallyhavealargerinvariancegroupthanthatimpliedby (5 .3 .3 ): in a dditiontobeinginvariantunderthetransformationsgeneratedby(5.3.3), th eyareinvariantunderlocal superscale transformations.Inthecompensatorapproach, weuseconstra intsthatdetermineonlythe conformal partoftheP oincar esupergravity mult iplet.Therestofthemultiplet(thesuperscalepart)iscontainedinthecompensatori tself,andthereforetheparticularformofthePoincar esupergravit ymulti plet dependsonthechoiceofcompensatormultiplet. Todiscoverthee xplicitformoftheadditionalinvariance,we“rstnotethatthe i n“nitesimalvariationofthespinorialvielbeinunderscaletransformationsmustbeof theform LE=1 2 LE(5.3.6) where L isarealunconstrainedsupe r“eldwhichparametrizes thescaletransformation (see(5.3.4c)).Next,to“ndthesuperscalevariationof A,weuse(5.3 .6),vary( M ), ,and E aarbitrarily,and demandthat(5.3.5)issatis“ed.Thisdeterminestheremainingvariations.Theres ultscanbesummarizedas L=1 2 L +2( L ) M +3( L ) Y ,(5. 3.7a) L€= L €Š 2 i ( €L ) Š 2 i ( L ) €Š 2 i ( €L ) M Š i 2( €L ) M€€+3 i ([ €] L ) Y ,(5. 3.7b) andconsequently LEŠ 1= Š 2 LEŠ 1.(5. 3.8)

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2725.CLASSICALN=1SUPERGRAVITYThegaugesymmetriesofthetheory,(5.3.3,7),andtheconstraints(5.3.4)aresucienttoreducethecovariantderivativessothattheydescribean irreducible mult iplet: conformalsupergravity.Toseethis,westudythescalingpropertiesoftheremaining “eldstrengths.Thesearenotallindependent;UsingtheBianchiidentities,asweshow insec.5.4,allnontrivial“eldstrengthscanbeexpressedintermsofthreetensors R G a, and W.Forconv enience,wealsointroducethe(dependent) U (1)“eldstrength W. Theseobjectscanbede“nedby R =1 6 R€€€€=1 4 iT€€ , G a= iT a W=1 12 iR€ ( € )= Š1 12 T( € € ), W=1 2 iF€ €.(5. 3.9) From(5.3 .2,7)wehavethe U (1)andsuperscaletransformationsof (and hence €by hermitianconjugation),and a.Wecanthende terminethetransformationsofthese “eldstrengthsbyeva luatingcommutators.Theresultis: [ Y R ]= R LR = LR Š 2 2L ; [ Y G a]=0, LG a= LG aŠ 2[ €, ] L ; [ Y W]=1 2 W, LW=3 2 LW; [ Y W]=1 2 W, LW=3 2 LW+6 i ( 2+ R ) L .(5. 3.10) Thusthesuperscaleand U (1)transformationscanbeusedtogaugeawaypartsofthese tensors,leavin gonlyt he“eld Wofconformalsupergravity.Atthelinearizedlevel, thiscontainsth e puresuperspin3 2 projectionof H mdiscussedins ec.5.2.a.1.

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5.3.Covariantapproachtosupergravity273a.3.Cont ortion Thereisnothinguniqueaboutthesetofconventionalconstraintsweuse.Anyset thatallowsustoexpr essthevectorderivativeandspinorconnectionsintermsofthe spinorvi elbeins E, E€isequallysuitable.Forexample,wecoulduse T a b c=0instead of R€ c d=0tode termine a b c.Thiswou ldgivea a b cwhosecorresponding aisan equallygoodcovariantderivative.Thedierence Š isatensor(the contor tion tensor).Addingcontortionstoconnectionsdoesnotchangethephysicsandsimply amountstoarede“nitionofminimalcoupling.Indeed,formostfamiliarmodelstheconnectionsdonotenteratall:Forthe scalarmultiplettheLagrangian andthechirality constraint € =0arei ndependentoftheconnection.The“eldstrengthsofsuperYa ng -M illstheoryare FAB= [ AB )+[ AB )Š TAB CC= E[ AB )+[ AB )Š CAB CC(5.3.11) andarealsoindepe ndentofthesupergravityconnections.Finally,thesupergravity Lagrangian(for n =0)is EŠ 1,alsoi ndependentoftheconnections. Furthe rmore,anyothersetofcons traintsthatd etermines E aiscorr ect:Wecan always rede“ne E a Mbywrit ing E a M= E a M+ g a E M+ g a€E€ M,(5. 3.12) where g a isacovariantobjectconstructedoutofthe“eldstrengthsof A.There fore, insuperspace,inadditiontothecontortiontensorfortheLorentzconnection,wehavea contortionthatchanges E a M.H ow ever,thisdoesnotaectthephysicsas,onceagain,it amountssimplytoa rede“nitionofminimalcoupling. Thereisanotherambiguityinthechoiceofconstraints,which,however,leadsto nomodi“cationofthetheoryatall:SincetheBianchiidentitiesrelatevarious“eld strengths,therearemanywaystoexpressanyparticularconstraint.Forexample,since allcurva turesand U (1)“eldstrengt hscanbeexpressedintermsoftorsions(seesec. 5.4),anyconstraintonacurvatureor U (1)“eldstrengthcanbe expressedintermsof torsions.W hichformischosenispurelyamatterofconvenience.

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2745.CLASSICALN=1SUPERGRAVITYa.4.Poincar esupergravit yconst raints Thesuperspin3 2 superconformalmultipletwith“eldstrength Wisnots ucient todescribeo-s hellPo incar esupergravity.Wemu stincludelowersuperspinsuper“elds toobtainaconsistentactio n.Wecandothisintwoways:Byintroducingextraconformalrepresentationsascompensators,orbydirectlyrestrictingthegaugegroupsothat somelowersuperspinconformalrepresentationscontainedin HMcannotbegauged away.Suchrestriction sonthe gaugegroupareintroducedbyimposingconstraintsthat ar en ot in va riantunderthefullgroup.Theseconstraintsappearnaturallywhenweuse thefullsuperconformaltransformationstogaugethecompensatorsawayandrequire thattheremainingtransformationspreservetheresultingsuperconformalgauge. Therearethreetypesoftensorcompensa torsthatcanbecoupledtoconformal supergravityandcanbeusedtoreduceittoPoincar esupergravity.Th epossiblecompensat orsarerestrictedbytherequirementth attheymustha vedime nsionlessscalar “eldstrengthstocompensatefor L of(5.3.6,7,8,10).(Thus,thecompensator“eld strength X hastheusuallinearizedcompensatortransformation X = L X | isthena scalarwithaction(5.1.35).Theremainingtypeofconformalmattermultiplet,thevectormultiplet,c annotbeusedasacompensatorbeca useitsonly scalar“eldstrength Whasthewrongdimensionanditsprepotentialisinertundersuperscaletransformations.)Theyareparametr i zedbythecomplexnumber n :(1)thescala rmulti plet ( n = Š1 3 ),(2)thenonminimalscalarmultiplet(any n except0or Š1 3 ),and(3)the tensormultiplet G ( n =0).Thesem ultipletscanbede“nedbyconstraintsandcanbe expressedexplic itlyintermsofunconstrainedsuper“elds(prepotentials): €=0,=( 2+ R );(5.3.13a) ( 2+ R )=0,= € €;(5. 3.13b) ( 2+ R ) G =0, G = G =1 2 ( 2+ R )+ h c .;(5.3 .13c) where R isa“eldstrength(see(5.3.9)andsec.5.4)and 2+ R givesachiralsuper“eld whenactingonasuper“eldwithoutdottedspinorindices(seebelow).The U (1)and superscaletransformationsforwhichtheycompensateare

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5.3.Covariantapproachtosupergravity275[ Y ,]=1 3 L= L ; (5.3.14a) [ Y ,]= Š2 n 3 n +1 L=2 3 n +1 L ;(5.3 .14b) [ Y G ]=0, LG =2 LG .(5. 3.14c) Insec.5.3.b.7wewillbreakthesuperconformalsymmetryby“xingthecompensators. Intheresultingsuper-Poincar etheory( 5.3.13)becomeadditional,conformalbreaking constraintsonthecovariantderivatives(i.e.,onthetorsionsandcurvatures). Thescaleweightofisarbitrary(sincewecouldreplacebymandstillsatisfy (5.3.13a)).However,theratioofthe U (1)chargetot hescalewei ghtforachiralsuper“eldis“xed.Thiscanbeseenbyasimpleargument.Consideranarbitrarychiral super“eld € =0.Wewriteitss caletransformationin termsofthedilatational generator d d d d (see(3.3.34)): L = L [ d d d d ].Ifweperformascalevariationofthede“ning conditionforachiral“eld,anduse(5.3.7a),we“nd: 0=( L €) + €( L ) = Š 3( €L )[ Y ]+ €( L [ d d d d ]) =( €L )[ Š 3 Y + d d d d ], (5.3.15a) andhence 0=[ Š 3 Y + d d d d ].(5.3 .15b) Thusthe U (1)chargeandthedilatationalcharge always satisfytherule d d d d Š 3 Y =0for chiralsuper“elds .Thisiss eenfor W, W,and R in(5.3.10)andforin(5.3.14a). (Actuallyfor R thisisonlyclearifthetransformationlawiswrittenintheform LR =3 LR Š 2( 2+ R ) L .)Therelationofthechiralchargetothedilatationchargefor chiralsupe r“eldsin N =1supersy mmetryisaspecialcaseofthegeneralrelationnoted insec.3.5. Inpreciselythesamemanner,startingfromthede“ningcondition(5.3.13b,c)fora linearsuper“eld,wecan showthat thecondition d d d d Š 3 Y =2mustbe satis“edfor all linearsuper“elds.Thisisseenforand G in(5.3.14b,c).Inthecaseofin(5.3.14b)we havechosenac onvenientparametrizationforitsscaleweight.Thetensormultipletis

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2765.CLASSICALN=1SUPERGRAVITYneutralbecauseits“eldstrengthisreal( Y =0),andthusthe n =0theory (withthe G compensator)retainsitslocal U (1)invarianceaftersuperscaleinvarianceisbroken. Therelationofthedilatationalchargeandthechiral U (1)chargeforchiraland linearsuper“eldsimp liesthatc ombi ned L and K5transformationsonarbitrarychiral andlinearsuper“elds, andrespectively,taketheforms = L [ d d d d ]+ iK5[ Y ], = d ( L + i1 3 K5) ,( 5.3.16a) = L [ d d d d ,]+ iK5[ Y ,], =[ dL + i1 3 ( dŠ 2) K5].(5.3.16b) Thequantities d and darethescaleweightsofthesuper“elds. b.Solutiontoconstraints b.1.Conven tionalconstraints The“rstconstraintintheform(5.3.4a)isalreadyexplicitlysolved(aswasthe ca se fo rs up er -Y ang-Mills). Webeginour analysisofthesecondconstraintbyextractingfrom(5.3.1)the explicitformofthetorsion.Insections5.1,2wede“nedthecoecientsofanholonomy CAB Cby [ EA, EB} = CAB CEC(5.3.17) Theycanbeexpressedexplicitlyintermsofthe EA Mandtheir deriva tives.Wethen have TAB C= CAB C+[ AB ) CŠ i1 2 w ( C )[ AB ) C;(5. 3.18) i.e., T€= C€, T c= C c,

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5.3.Covariantapproachtosupergravity277T€ c= C€ c, T b€= C b€, T a b = C a b€, T = C +( ) +1 2 i ( ), T€€= C€€+€€Š1 2 i €€, T a = C a + a +1 2 i a, T b c= C b c+ €€+€€ T a b c= C a b c+( a €€+ h c Š b a );(5.3 .19) aswellasthecomplexconjugates. Byusingt heseequationsthe“rstconstraintof(5.3.4b)canbesolveddirectly(notingthatA istra celessinitslasttwoindices): T =0 =1 2 ( CŠ C ( )) Š1 2 iC ( ).(5. 3.20) However,solvingthelasttwoequationsof(5.3.4b)forand€€,resp ectively, isless straightforward,since C b citselfdepe ndsonthemthrough a.(Ontheo ther hand, (5.3.4a)introdu cesnodependenceof aon .)Tosolvetheseconstraintsweintroduce,asins ec.5.2.a.3, EA=( E, E€, E a) ( E, E€, Š i { E, E€} ).(5.3 .21) Wede “ne CAB Cby[ EA, EB} = CAB C EC.Weem phasizethat,since E aisstilldependenton€€and, CAB Cisalso.Incontrast, CAB Ciscompletelydeterminedinterms of Eand E€.Webeginby expressing E aintermsof E aan d (th ea sy etundetermined) C€ M: E a= Š i [ { E, E€} +(€€Š1 2 i €€) E€+(€ +1 2 i € ) E]

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2785.CLASSICALN=1SUPERGRAVITY= E€Š i (€€Š1 2 i €€) E€Š i (€ +1 2 i € ) E,(5. 3.22) whereweuse(5.3.4a).Thencomputingthecommutator[ E, E b]= C b DEDwe “nd C b c= C b cŠ (€€Š1 2 i €€)+ i (€ +1 2 i €) C c.(5. 3.23) Aswewillseeshortly,thenextconstraintweimpose(eq.(5.3.4c))willset C c=0,and ther eforethenonlineartermdropsout.Consequently,thelasttwoconstraintsof (5.3.4b)(inc ombi nedform) 0= T € €= C € €+2€€=( C € €Š €€+1 2 i €€)+2€€(5.3.24) give €€= Š1 2 C (€ € ),= i C b b.(5. 3.25) Thiscompletesthesolutionoftheconventionalconstraints(5.3.4a,b).Wehavenow determined E a,A,andAintermsof Eand E€.Furthe rmore,fromtheformof (5.3.22)weimmediatelyobtain E = sdetEA M= sdet EA M.(5. 3.26) (Thelasttermsin(5.3.22)givenocont ributiontothesuperdeterminant.) b.2.Representationpreservingconstraints Havingdeterminedallquantitiesintermsof E,wehaveare alizationoflocal su pe rsymmetrywith512ordinarycomponent“elds.IntheYang-Millscase,further reductionwasachievedbyimposing repres entation-preserving constraints:Toensurethe existenceof(anti)chiralscalarsuper“elds(de“nedby =0),werequired {, } =0.(5. 3.27) Insupergravity(assuming[ Y ]=0forsimp licity),we“nd(5.3.27)implies T DD = T + T€€ + T c c =0.(5. 3.28) Therefore,toallowtheexistenceofchirals calarsinsupergravitywemustenforcethe

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5.3.Covariantapproachtosupergravity279constraints(5.3.5c): T€= T c=0.(5. 3.29) From(5.3 .19),thisimplies: { E, E} = C E.(5. 3.30) (Equivalentlysince = E =0itfo llowsthat { E, E} =0whic hi mmediately leadsto(5.3.30).)Thus E= E MDMisabasisfortangentvectorsthatlieinacomplextwo-dimensionalsubspaceofthefullsuperspace:Alloperators Egenerate complex translationswithanalgebrathatcloses.We canalsoparametrizethesetranslations byabasis ofderivativeswithrespecttocoordinates( 1, 2): E= A ,where A is anarbitrarymatrixor zweibein. Wecanalwaysexpre ssthecoordinates as complex supercoordinatetransformsoftheusual -coordinates: = eŠ De,where =MiDM = isarbit rary.Ourfullsolutionoftheconstraints(5.3.4c)isthus E= A eŠ De eŠ A De; =MiDM, A = N ;(5. 3.31) wherew ehavesplit A intoacomplexscalefactor andaLoren tzrotation N ( detN =1 ).ThissolutioniscloselyanalogoustotheYang-Millssolution = eŠ Deto {, } =0,ex ceptfortheintroductionof A .I nf act,theofYang-Millscanbe interpretedasacomplextra nslationinthegroupmanifold.Wenowhaveadescription ofsupergravityintermsof,,and N .Howev er, N canbegaugedawaybya Lorentztransformation(withparameter K ). b.3.The gaugeg roup Atthispo int,wecanmakecontactwiththeprevioussection:Thesolutionofthe constraintsimposedsofarhasintroducedanewgaugegroupasaninvarianceof E= eŠ A De.Thevielbein Eremainsunchangedunderthetransformations ( e)= ei e,

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2805.CLASSICALN=1SUPERGRAVITY( A D)= ei ( A D) eŠ i .(5. 3.32) with = MiDM,[ D]= Š i ( D ) D;(5. 3.33) providedthat D €= D €Š i €=0, arbitrary ;(5. 3.34) or €= D2 L€, €= Š iD L€.(5. 3.35) Thefactthat iscompletelyarbitraryimpliesthatthepartofthe -gaugegroup parametrizedby canalwaysbecompensatedawaybyarede“nitionof N .Thus asintheYang-Millstheorysolvingaconstraint( F=0)givesrise toanewgauge group.Thetransformationon A canberewrittenas = ei (1 eŠ i )Š1 2 eŠ i ( N D)= ei (1 eŠ i )1 2 N DeŠ i ;(5. 3.36) wherethefactor(1 eŠ i ),= iD ,isthesuper-Jaco bian ofthetransformation(c.f. (5.2.59)). West illhavethe realK = KMiDMcoordinatetransforma tionsofthetheory(see (5.3.3)),aswellasthetangentspaceLorentzand U (1)rotations: E = eiKEeŠ iKis realizedby ( e)= eeŠ iK,( A )= A ;(5. 3.37) while E = eŠ1 2 iK5K Eisrealizedby ( e)= e,( A )=( eeŠ1 2 iK5K eŠ ) A .(5. 3.38) The K transformationscanberewrittenas = Š iK + O (, K ).(5.3 .39)

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5.3.Covariantapproachtosupergravity281Since K = K ,thisimp liesthat Š canbe gaugedaway.Intheresultinggauge, =1 2 H H = H = HMiDM.(5. 3.40) Sim ilarly,theLorentztransformationcanbeusedtogauge N to .Inthere sult ing gaugewehave E= eŠ1 2 H De1 2 H, E€= e1 2 H D€eŠ1 2 H.(5. 3.41) However,itismoreconvenienttoeliminatet herealpartofbygoingtoachiralreprese nt ation(asforsuper-Yang-Mills),asdiscusedinsec.5.3.b.5below. b.4.Eval uati onof and R Wecannow “ndsimpleformsfor(5.3.25)and R (5.3.9).Theresultsarecontainedin(5.3.52,53,56).Thedetailsofthederivationarenotessentialforfurtherreading, butpresentsomeusefulgeneraltechniques.Tosolvefor,weusetheidentity EŠ 1 A= Š EŠ 1( Š )BTAB B,(5. 3.42) whichholdsindependentlyofanyconstraints,foranysuperspace,foranytangentspace. Incaseswhere( Š )BTAB Bvanishes(ash ere),itallowscovariantintegrationbyparts, since dzEŠ 1AX = Š dzEŠ 1 AX =0.(5. 3.43) Toderivethis identitywewillsaveourselvesalotoftroublebynotingthatattheendof acalculati onthesignsresultingfromgradedstatisticscaneasilybedeterminedifthe i ndicesofeachcontractedpairareadjacent,withthecontravariantindex“rst.Thenet signchangeisthenjustthatresultingfromthegradedreorderingoftheindicesofthe initialexpression.Usingthisfacttoignorethesignsfromgradingatintermediatesteps ofthecalculation,wehave(inthebasis EA= EA MM) ( Š )BTAB B= EM B[ A, B} zM= EM B[ AEB ) M

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2825.CLASSICALN=1SUPERGRAVITY= EM BAEB MŠ EM BBEA M.(5. 3.44) We evaluatethesecondtermbyuseoftheidentity(againignoringgradingsigns)for arbitrarysup erfunctions X and Y XY A= X [ Y A]+ X AY = X AY + X AY ,(5. 3.45) wherewehaveused(5.1.26b)toevaluatethecommutator.(5.3.44)nowbecomes EM BAEB MŠ EM BEA M B+ EM B BEA M= AlnE Š 1 A+0,(5. 3.46) whichleadsto(5.3.42).Theevaluationofthe“rsttermusedtheusualexpressionfor thederivativeofthelogarithmofadeterminant(see(5.1.28);fortangentspacegroups whichincludescaletransformations,( Š )BAB B =0,sothatt hescalegener atoractsnontriviallyon E ).Thelasttermv anishesb ecause EM B B= EM BEB N( N+N( M) Š i NY).Ther efore M N( N+N( M) Š i NY) = M N N=0. Actually,itissimplerforourpurposestousetheformof(5.3.42)intermsof EAinsteadof A.Using E = E ,wehave: EŠ 1 E A= Š EŠ 1( Š )B CAB B.(5. 3.47) Fromtheexp ression(5.3.25)for, E= E,and C€ = C€€=0weobtain Š i =( Š )B C B B+ C = Š EŠ 1E E + C .(5. 3.48) Usingtheexpression(5.3.31)for EintheLorentzgauge N = (thegen eralLorentz gaugewillbeeasilyrestoredattheend),we“nd C = ( E )ln ,sothisexpression b ecomes Š i = Š EŠ 1E E +3 Eln = Š 1 eD eŠ + ElnE 2,(5. 3.49) wherew ehaveused E= eŠ De(5.3.50a) whichimplies

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5.3.Covariantapproachtosupergravity283 E = eD eŠ .(5. 3.50b) Weuset heidentity ,va lidforanyfunction f andlinearoperator X feX=(1 eXeŠ X) feX=(1 eX)( eŠ XfeX) =(1 eX)( eXfeŠ X) =(1 eX)( eXf ), (5.3.51a) toderivetherelation 1=(1 eŠ X) eX=(1 eX)[ eX(1 eŠ X)].(5.3.51b) Thesetworesultsmakeitpossibletorewrite(5.3.49)as Š i = Š (1 eŠ ) eŠ D(1 e)+ ElnE 2= Š (1 eŠ ) eŠ De(1 eŠ )Š 1+ ElnE 2= T T,(5. 3.52) wherewehaveintroduceda(noncovariant)scalar densityT : T ln [ E 2(1 eŠ )].(5.3.53) Animmediateconsequenceof(5.3.52)is F=0(see (5.3.1)). Wenowsolvefor R ,where {, } = Š 2 RM,(5. 3.54) asfollowsfromtheBianchiidentities(sec.5.4).Usingthesameformfor Easinthe previouscalculation,andusingtheresultfor,we “ndfrom(5 .3.20) = Š1 2 ( ( E )ln 2+ i ))= Š1 2 ( E )ln ( eŠ T 2) = Š1 2 ( E )ln [(1 eŠ )Š 1EŠ 1].(5.3 .55)

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2845.CLASSICALN=1SUPERGRAVITYWethen “nd(€€doesnotcontribute) R = e T( E€)2( eŠ T2)= e T( E€)2(1 e )Š 1EŠ 1= EŠ 1 ( E€)2e T(1 e )Š 1.(5. 3.56) where( E€)2=1 2 ( E€)( E€). *** Itisusefultoderivetheexplicitformoftheoperatorthatgivesachiralscalar fromagener alscalar f .Forth ecase[ Y f ]=0,asimpl ecalculati onusing(5.3.55)for theconnectiongiv estheresultthat( 2+ R ) f = fEŠ 1( E€)2e T(1 e )Š 1iscovariantly chiral.Thisres ultcanthenmosteasilybeextendedtoarbitrary U (1)charge [ Y f ]=1 2 wf byusingt heexpression (5.3.53)fortowrite ( 2+ R ) f =[ e1 2 w T( 2+ R ) eŠ1 2 w Tf ](1 e )Š 1,(5. 3.57) where 2istheformof 2onaneutralscalar(asimpliedby(5.3.57)for1 2 w =0).We thusobtain ( 2+ R ) f = feŠ1 2 w TEŠ 1( E€)2e(1+1 2 w ) T(1 e )Š 1.(5. 3.58) Thisquantityiscovariantlychiralwith U (1)charge1+1 2 w b.5.Chiralrepresentation Duetotheformof Ein(5.3.31),itispossibletode“nelocalrepresentationsthat arechiralwithrespecttothesupergravity“elds.(Theseareanalogoustochiralreprese nt ationsinsuperYang-Mills(4.2.78)aswellasinglobalsupersymmetry(3.4.8).)On allquantities F weperforma(n onunitary)similaritytransformation F(+)= eŠ Fe .(5. 3.59) (Antichiralrepresentatio nscanalsobede“ned,with Š .)Inthisrepresentation, asforsuper-Yang-Mills,allquantitiesareinvariantunder KMtransformations,andthe covariantderivativestransformexplicitlyundertransformations.Furthermore,we

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5.3.Covariantapproachtosupergravity285choosetheL orentzgauge N = ,which forces K€€toequal €€of(5.2.27)tomaintainthegauge.Inthechiralrepresentation,thevielbeinbecomes: E(+)€= D€, E(+) = eŠ H DeH=eŠ HDeH; eH= ee .(5. 3.60) Thisispreciselywhatwehadconstructedintheprevioussection((5.2.27)and(5.2.28)). Thetransformationof H canbeobtainedfromthatof: ( eH)= ei eHeŠ i .(5. 3.61) (Notethat,asinsuper-Yang-Mills,(5.3.60)canbeusedtode“ne H inany K -gauge;it is K invariant.Alternatelythe and€transformationscanbeusedtogaugeaway Hand H€.) Itispossiblet ogotoarepresentationthatisalsochiralwithrespectto U (1). From(5 .3.53)wehave = iET ,€= Š i E€T;( 5.3.62a) where T= eŠ H TeHisthechiral-representati onhermitianconjugateof T (cf.(5.2.28)). Using(5.3.2),wecanwrite EŠ i Y = eŠ TYeŠ1 2 TEeTY, E€Š i €Y = eTYeŠ1 2 T E€eŠ TY;(5. 3.62b) Inadditiontothetransformation T = Š iK5,t heseexpressionsareinvariantunder T = i 5,where5ischiral.Wecanusethisgaugefreedomtoreplace T by 5= T +3 ln ,(5. 3.62c) thusintro ducingforsubsequentusethechiraldensity D€ =0.Wenowgo t oachiralrepresentationnotonlywithrespecttoMDMand N , butalsowithrespectto5, bymakingth ea ppropri atenonunitary U (1)transformation,andobtain: E€Š i €Y = EŠ1 2 D€,

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2865.CLASSICALN=1SUPERGRAVITYEŠ i Y = eŠ V5YEŠ1 2 (1 eŠ H)Š1 2 EeV5Y= E EŠ1 2 E+( E EŠ1 2 EV5) Y eV5 e5e5= E3 EŠ 1(1 eŠ H) 33=(€€EŠ 1E )3,€€EŠ 1 EŠ1 3 (1 eŠ H)1 3 ;(5. 3.63) whereweh aveused E =2 2 E (5.2.49)toreplacewith E .Thise ectivelyreplaces with E assuperscaledensitycompensator:Inthe U (1)-chiralrepresentation,the U (1)densi tycomp ensator5nolongerappears.Forthisreasonthischiralrepresentationisus efulfor n =0superg ravity,whereatruelocal U (1)invariancer emains,butnot veryus efulforother n .Howev er,itdoesbearacloserelationshiptothe n = Š1 3 results oftheprevioussection: n = Š1 3 canbeobtainedbyconstrainingAtovanishidentically.Inthisrepresentation,theresultissimplythat V5vanishes,a ndhence EŠ 1=€€EŠ 1, inagreementwith(5.2.72).Insection5.5,thi sresultw illbeusedtowritea“rst-order formalismfor n =0combin edwith n = Š1 3 b.6.Densitycompensators Aftergaugingaway Hand H€,wehaven owdete rminedallthegeometrical super“eldsin Aintermsof H mand ,whichconta in64and32component“elds, respectively.Theaxialvectorprepotential H mcontainsthecomponentgauge“elds. Thesuper“eld isthesupercon formal(densitytype)compensator:Byscaling arbitrarily (witho uttransforming H m),wegeneratecomplexscaletransformations(realscale U (1))ofthevielbein.Thus,thecomplexscaletransformationpropertiesofanyquantityexpressesits dep endence.Thesetransformationsmustberestrictedandtherepresentationreducedfurther,sinceEinsteintheoryisincludedinPoincar esupergravity andis not scaleinvariant.Wenowconsiderthe(scalar)tensor-typecompensators ,, G (5 .3 13) ,w hichalsotransformunderthesecombinedtransformations.Fixing thegaugesofthesetransformationsby“xingthecompensatorisaconvenientwayof determining,sinceitseparatesthelowersuperspinmultipletsfromtheconformal supergravitymu ltipletinacovariantway. Itisconvenienttosolvetheconstraints(5.3.13)onthetensorcompensatorsin termsofcorresponding density compensatorsthatsatisfythecorresponding ”at-space

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5.3.Covariantapproachtosupergravity287constraints; thisallowsusto“ndanexplicitsolutionfor. b.6.i.Minimal( n = Š1 3 )supergravity We“rstco nsiderthe(covariantly)chiralscalarcompensator.Theconstraintis, us ing(5.3.62a), €=( E€Š1 3 i €)= e1 3 T E€eŠ1 3 T= 0( 5.3.64a) andissolvedby = e1 3 T =[4 2 E (1 e )]1 3 E€ =0.(5. 3.64b) Ittransformsunderscaletransformationsasin(5.3.14).Here isa”atspacechiral super“eld,inthechiralrepresentation,asfollowsfromthede“nitionof E.Ifwechoose thegauge=1,then T = Š 3 ln andweobtain = Š 1 1 2 EŠ1 6 (1 eŠ )1 6 (1 e )Š1 3 .(5. 3.65) (Wehaveagainused E =2 2 E .)Inthisgauge €=0imp lies €=0;bycomplex conjugationand(5.3.4a)A=0,andth usthe“elds trengthoftheaxial U (1)transformations(see(5.3.9))vanishes: W=( 2+ R )=0.Therela tion(5.3.58)becomes 3( 2+ R ) f = fEŠ 1( E€)2(1 e )Š 1.( 5.3.66a) Inthechiralrepresentation,thissimpli“esto 3( 2+ R ) f = D2( EŠ 1f ).(5.3 .66b) Thetheoryisnowdescribedby H mand andtheonlysuperspacegaugefreedom leftisthatofsuper-Poincar etransfo rmations.Asin(5.2.75),thedensitycompensator canbereplacedbyoneofitsvariants. b.6.ii.Nonminimal( n = Š1 3 )supergravity Fo rt he no nm inimalscalarmultiplet,weagain“ndasolutionintermsofadensity compensator.Theconstraintis,using(5.3.58), ( 2+ R )= EŠ 1e2 n 3 n +1 T( E€)2eŠ2 n 3 n +1 T(1 e )Š 1=0 (5.3.67a)

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2885.CLASSICALN=1SUPERGRAVITYandissolvedby = eŠ2 n 3 n +1 TE (1 e )=[4 2 E (1 e )]n +1 3 n +1 Š 2,( E€)2=0.(5.3 .67b) Hereisa”at-space-linearsuper“eld(asopposedtothecovariantlylinearsuper“eld ):Inthechiralrepresentation D2=0.A gain,inthegauge=1,weobtainthe solutionfor: =[n Š 1 n +1]Š 3 n +1 8 n [ E2 n(1 eŠ )n +1(1 e )n Š 1]Š n +1 8 n .(5. 3.68) Inthisgaugewehave T i asanewtensor(appearinginarbitrarygaugesas ),intermsofwhich R and Waredetermined.Thesolution(5.3.68)doesnot a pplytothefollowingcases:(1) n = Š1 3 ,forwhichthemi nimalscalarmultipletisused instead;(2) n =0 ,f orwhichthesolutionof(5.3.67a)ismoresubtleandwillbediscussednext;and(3) n = ,which doesnotleadtoasensibletheory.Theparameter n canalsobegeneralizedtocomplexvalues,buttheconstraintsthenviolateparity(o shell),andwedonotdiscussthemhere. b.6.iii.Axial( n =0 )supergravity Theconstraint(5.3.13c)for n =0ismostea silysolvedbyexpressingthecompensator G intermsofacovariantlychiralspinor =( 2+ R ).Usingtherelation = E ( EŠ 1E )(asfo llowsfromintegrationbypartsond4xd4 EŠ 1f =d4xd4 EŠ 1Ef forany f ),wehaveinthechiralrepresentation(using(5.3.63)): G =1 2 ( + € €)=1 2 E ( EŠ1 2 E + h c .) E G ,(5. 3.69) sothat G isafunctionofonly H and .Inthechiral representation,butinthe Lorentzgauge€€=0, is”at-spac echiral: D€=0.(This gaugeexistsbecause R€€=0,asfo llowsfromtheBianchiidentities,seesec.5.4,whichimpliesthat€€anditsconjugatearepuregauge.)Insuchagauge N = X = , butdependsonly on H .Ontheot herhand,inth eLoren tzgauge N = ,where€€dependsonlyon H upt oafactoro f(see (5.3.25)), is( XŠ 1) timesa”at-spac echirals pinor.

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5.3.Covariantapproachtosupergravity289IntheWeylgauge G =1weobtain EŠ 1= G ( H )=1 2 ( EŠ1 2 E + h c .).(5. 3.70) Inthisgauge,(5.3.13c)impliestheconstraint R =0.Furth ermore,inthechiralrepresentation(5.3.70)maybecombinedwith(5.3.63)toreplacewith E asthecompensatorf orthe n =0covariantd erivat ives: = EŠ1 2 = G1 2 ,= E EŠ1 2 = GŠ 1 EŠ1 2 .(5. 3.71) b.7.De gauging Thetheoryandthecovariantderivati veswehav econstru ctedsofarcontain exp licit U (1)gen eratorsandconnections.For n =0,the U (1)symmetryisagenuine local symmetryofthetheory attheclassicallevel(thereareanomaliesatthequantum level,seesec.7.10)andtherefore n =0superg ravityonlycouplestoR-invariantmatter systems(3.6.14,4.1.15).For n =0,thesupersca lecomp ensatorcanbeusedto remove the U (1)chargeofanymultiplet:Bymultiplyingthesuper“eldbyanappropriatepower ofthecompensator(see(5.3.12a,b))wecanalwaysconstructa U (1)neutral object.If wedothis toallquantities(vielbein,connections,matter),the U (1)generatorsdonot actandcanbedro ppedfromthetheory.Theresultingformalismisapplicabletomattermultiplets without de“nite U (1)chargea ndhencetosystemswithoutglobalR-invariance(seesec.5.5).Theprocedurewearefo llowingissimilartowhatonedoesinordinaryspontaneouslybroken gaugetheories.Onegoestoa U-gauge eitherbyusingthe Goldstone“eldtode“negaugeinvariantquantitiesaswejustdid,orbygaugingthe Goldstone“eldaway,aswedonow:Insteadofrescaling“eldsbythecompensator,we cangaugeitaway,and“xthe U (1)(andsuperscale)gaugeasdiscussedinsec.b.6. above.For n = Š1 3 ,thissetsA=0andthe Y generatordropsfromthecovariant derivati ves.For n = Š1 3 ,0,the U (1)conn ectionbecomesacovarianttensorwith respecttotheremaining(super-Poincar e)group.Thereforewecaneliminate Y by a ddingacontortionterm AAŠ i AY andthus,by(5.3.18), TAB C TAB CŠ i1 2 w ( C )[ AB ) C( butwithnochangeinthecurvatures).Theonlymodi“edtorsi onsare(where iT,see(5.3 .52)):

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2905.CLASSICALN=1SUPERGRAVITYT T Š1 2 ( T ), T€€ T€€+1 2 €€T, T a T a Š1 2 i ( T€Š €T+ T T€).(5.3 .72) Furthe rmore, R and Whavethefollowing explicitexpressionsintermsof Tandthe Y i ndependentor degauged : R = Š Š 1 2 Šn 3 n +1 ( €+n Š 1 2(3 n +1) T€) T€,(5. 3.73) W=( 2+ R ) i [1 2 €( €+1 2 T€)+ R ] T.(5. 3.74) Weem phasizethatfor n = Š1 3 wecansimplydropa llreferenceto U (1)wit hout anyothermodi “cations. Althoughwehaveem phasizedthecompensatorapproachtothebreakingofthe superconformalinvariance,weshouldpointoutthat“xingtheconformalgaugebysettingthecompensatorto1iscompletelyequivalenttoimposingadditional,conformalbreakingconstraintsonthecovariantderivatives.After U (1)degauging,theconstraint equations(5.3.13)or(5.3.64a,67a)become,w he nt he compensatorsare“xed,conditions onthecovariantderivatives.Theseconditionsaretheconstraintsontorsionsandcurvaturesgivenin(5.2.80b). Atthispo intwehaveadescri ptionofPoincar esupergravit yinte rmsof H andone ofthedensitycompensators.Theycompensatefor component conformaltransformations.Forexample,the -independentof canbeidenti“edwiththecomponent of (5.1.33).Similarly,thelinear component,aspinor,compensatesfor S -supersymmetry. Afterdegauging,thesuperconformalinvarianceofthesupergravityconstraintsis destroyedforarbitrarysuper“elds L and K5.Nev erthelessaremnantofsuperconformal invarianceremains.Thisisbecausetheuncon strainedsuper“eldsthatdescribePoincar e supergavityare H mandsomescalarsuper“eldcompensator.Thussuper“eldsupergravity,unlikeordinarygravity, always containsacomponent compensatorof(5.1.33). (Thisist hereasonwhyatthecomponentlevelsu perconfo rmalsymmetryissouseful.)

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5.3.Covariantapproachtosupergravity291Obviouslyarede“nitionofthedensitycompensatingmultiplet(onceitstypehasbeen speci“ed)cannotaectthePoincar esupergravityc onstraints.Thisisrealizedbyan invariancegroupoftheconstraintsinadditiontothatparametrizedby KMand K Thetransformationsofthisinvariancegroupareexactlythesameasthoseoftheconformalgroup,butwiththeimpor tantrestrictionthat L and K5arenolongerarbitrary. Thesimplestwaytoobtaintheformofthesere strictedconformalt ransformationsisto usethetensortypecompensators,,and G Beforegaugingthecompensatorsto1wecansimplymakeanarbitraryrede“nition ofthetensortypecompensators.Wehave(i): = Š ,( ii) = Š ,a nd (iii) G = Š G ,whereisaco variantsuper “eld.Ineachcasetherede“nitionmustbe suchthattheproductofthecompensatortimessatis“esthesamedierentialequation astheoriginalcompensator(5.3.13)(i.e.,(i) €=0,( ii)( 2+ R )()=0,and(iii) ( 2+ R )( G )=0,= ).Thetransformationsaect only thecompensators,not thecovariantderivativesnormattersuper“elds,whereas L and K5transformations a ectall“elds.Thecombinedtransformationofthetensorcompensatorsunder L K5, andisthus n = Š1 3 : =( L + i1 3 K5) Š ,( 5.3.75a) n = Š1 3 ,0: =[(2 3 n +1 ) L Š i (2 n 3 n +1 ) K5] Š ,(5.3.75b) n =0: G =2 LG Š G .(5. 3.75c) Wenowde gaugebysettingthecompensatortoone.Inordertomaintainthis gaugeconditionwemustsetthetotalvariationofthecompensatortozeroandwe“nd that L and K5satisfytheconstraints L =1 2 (+ ), K5=3 2 i ( Š );(5.3.76a) L =3 n +1 4 (+ ), K5=3 n +1 4 n i ( Š );(5.3.76b) L =1 2 ,= ;(5.3 .76c) respectively.

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2925.CLASSICALN=1SUPERGRAVITY5.4.SolutiontoBianchiidentities Inanygaugetheorythe“eldstrengthssatisfyBianchiidentitiesthatareaconsequenceoftheJacobiidentitiesforthecovariantderivatives,ormoregenerally,forsuperforms,aconsequenceofthePoincar etheorem.Asexp lainedinsec.4.2,theBianchi identities containnousefulinformation unless someofthe“eldstrengthshavebeenconstrained.Inthatcasetheymakeitpossibletoexpressallthe“eldstrengthsintermsof anirreducibleset(thatmaystillsatisfy dierential constraintsthatarealsocalled Bianchiidentities).Insec.4.2wegaveadetailedexampleofthisprocedureforsuper Ya ng -M illstheories;hereweconsidersupergravity. Webeginina generalcontext,withcovariantderivativesforarbitrary N andarbitraryinternalsy mmetrygeneratorsi(cf.(5.2.20,5.3.1)): A= EA MDM+A M +A€€ M€€+A ii(5.4.1) where EA M,A,andAarethevielbein,Lorentzconnection,andgaugepotential, respectively.Wede“ne“eldstrengths:torsions TAB C,curvatures RAB and RAB€€,and gauge“eldstrengths FAB iintermsofthegradedcommutator [ A, B} = TAB CC+ RAB M + RAB€€ M€€+ FAB ii.(5. 4.2) Thegeometryofsuperspaceimplicitin(5.4.1)givesnontrivialrelationsamongthe “eldstrengths T R F .Wehavechosent heacti onoftheLorentzgrouptobereducible intangents pace:Itdoesnotmixthe( V, V€, V a)parts ofasupervector VA,andit rotatesthevectorandthespinorpartsbythesametransformation;i.e., VA A BVB,where A B=( €€, €€+ €€ )(cf.(5. 2.19,5.3.3)).Consequently,thereareno connectionssuchasA corA €,andA b c=A €€+ h c .,et c. Wecanviewthi srestric tionasaconstraint:Ithasthe consequencethattheBianchi identitiesnowgivealgebraicrelationsamo ngthe“eldstrengths.BecausewehavechosentheactionoftheLorentzgrouptobere ducib le,wehaveimposed theconstraints RAB c = RAB = RAB€ = RAB d=0, RAB = RAB c d, RAB c d= RAB €€+ RAB€€ ,(5. 4.3) andtheir hermitianconjugates.Theseconstraintsaresucienttoexpressallofthe

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5.4.SolutiontoBianchiidentities293curvatures R andgauge“eldstrengths F intermsofthetorsions T ifwea ssumethat EA Mtr an sformsundertheactionofi;otherwise Firemainsasanindependentobject. Inparticular,inthepresenceofcentralchargesthecorresponding“eldstrengthscan remainasindependentquantities.Wewri tethetransformationintermsofmatrices (i)A B: [i, A]=(i)A BB(5.4.4) wheretheonlynonvanishing(i)A Bare (i)a b ,(i)a b€€,(i) a b,(i)a b=((i)a b)(5.4.5) TheBianchiidentitiesfollowfromtheJacobiidentities0= [[ [ A, B} C )} = BABC EE+ BABCandare: BABC EŠ ABC E+ R[ AB C ) E+ F[ AB C ) E=0,(5. 4.6a) BABC( M ) Š[ ARBC )( M )+ T[ AB | DRD | C )( M )=0,(5 .4.6b) BABC iŠ[ AFBC ) i+ T[ AB | DFD | C ) i=0(5.4 .6c) where ABC E[ ATBC ) EŠ T[ AB | DTD | C ) E, FABC E FAB i(i)C E.(5. 4.7) TheBianchiidentitiesaresatis“edidenticallysimplybecausethe“eldstrengthsareconstructedoutofthepotentials EA M,A,andA.In(5. 4.6a)bydecomposing BABC Einto i rreduciblepiecesundertheLorentzandinternalsymmetrygroups,weexpress F and R intermsof T ;thissol utionautomatic allysatis“es BABC( M )= BABC i=0,sothat (5.4.6b-c)containnousefulinformation. Toor ganizetheanalysisoftheBianchiidentities,weclassifytheidentitiesby (ma ss)dimension.Thelowestdimensionidentitieshavedimension1 2 : B d= B € d=0andhe rmitianconjugates.Thehighestdimensionidentitieshave dimension3: B a b c i= B a b c( M )=0.Ordi nar ily,westartwiththelowestdimensionidentitiesandworko urwayup;however,thedimension1 2 B sarerelationsamongthetorsionsonly(theyareindependentofthecurvaturesand“eldstrengths).Therefore,to determine R and F westartwitht hedimension1 B s.Forexample, B € €=0imp lies

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2945.CLASSICALN=1SUPERGRAVITYR ,€€c e+ F c e€€= € €(5.4.8) (Thegradedsymmetrizationdropsoutbecauseoftheconstraints(5.4.3)and(5.4.5)). Toextract F c eand R ,€€from(5.4 .8),wedecomposeitintoLorentzirreducible pieces.Thisgives: F c e=1 2 ,€ c€ e(5.4.9a) R ,€€=1 2 N ,(€ c€ ) c(5.4.9b) Proceedinginasimilarmanner,wedetermine all theremainingcurvaturesandgauge “eldstrengthsinte rmsofthetorsions: R =1 4 ,( € )€,(5. 4.9c) F c d=1 2 , d€ c€.(5. 4.9d) Furthe rmorewe“nd RAB =1 2 N ABd ( d ),(5. 4.9e) FABc d=1 2 ABc d ,(5. 4.9f) for( A B )=(€ ,€ ),(€ b ),and( a b ),and “nally R B =1 4 [(1 N +1 )(B ,( a | | c )( c )+1 3 + B a ( C ) ) +(1 N Š 1 )(B ,[ a | | c ]( c )+Š B a ( C ))],(5. 4.9g) F Bc d=1 2 [1 3 B ,( a | | c ) d +B ,[ a | | c ] d ] Š1 2 [1 3 R( a | B | c ) dŠ R[ a | B | c ] d],(5.4 .9h) + B =B ,( a | | c ) c ,Š B =B ,[ a | | c ] c for( B )=(€ )and( b ).Thissolutionautomaticallysatis“es(5.4.6b-c),ascanbeveri“ed

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5.4.SolutiontoBianchiidentities295bydir ectcomputation.FromnowonweneedonlyconsidertheBianchiidentitiesin (5.4.6a)thatareindependentof R and F Wenows p ecializeto N =1superg ravity:OurtangentspacetransformationscontainonlytheLorentzgroupand U (1);i.e.,i= Y .Weimpose: T = T€= T c= T€€= T€ cŠ i €€=0 ,( 5.4.10a) T ,( € )€= T (€ € )= T b b= F€=0.(5. 4.10b) We proceedasabove,startingwiththelowestdimensionidentities.Forexample, B ,€ d= T ,€ d+ T€ d+ €T d+ T ETE ,€ d+ T ,€ ETE d+ T€ ETE d(5.4.11) but T E=0and T ,€ E= i €€whichimplies 0= B ,€ d= Š i €€T € €Š i €€T € €.(5. 4.12) Nextwed ecompose T b cintoirreduciblerepresenta tionsoftheLo rentzgroup, T b c= C€€[ f1 + C ( f2 )+ Cf3 ] +[ f4 ( )(€€ )+ C ( f5 )(€€ )+ Cf6 (€€ )](5. 4.13) andnote T b b=0imp lies f3=0, T ,( € )€=0imp lies f2= f1=0,and “nally T (€ € )=0imp lies f6=0.Nows ubsti tuting(5.4.13)into(5.4.12)leadsto 0= Š i 2 f4 ( )(€€ )Š i 2 C ( f5 )(€€ ),(5. 4.14) whichyields f4 ( )(€€ )= f5 (€€ )=0.Inotherwords T b cvanishesid entically. ThisexampleshowshowwedecomposethetorsionsintoirreduciblerepresentationsoftheLorentzg roup,andthensolvetheconstraintsbylookingatwhattheyimply aboutthevariousirreduc ibleparts.Fortwocomponentspinorsthisdecompositionsimplyconsistsofsymmetrizingandantisymmetr izinginallpossibleways.Otherexamples are

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2965.CLASSICALN=1SUPERGRAVITYT€ € = C€€CX1+ CX2 (€€ )+ C€€X3 ( )+ X4 (€€ )( ), T€ € ,€= X1 (€€€ )+ X2 (€C€ )€+ X3 €C€€.(5. 4.15) TheseexpressionsarenowsubstitutedintotheBianchiidentitieswhichseparateinto severalequations.Thesearethensolved,withtheresultthatsomeoftheirreducible partsarezero,whileothersareexp ressibleintermsofaminimalset. Thecompleteanalysisisstraightforwardbuttedious.We“ndthatalltheBianchi identitiesandconstraintsaresatis“edby {, €} = i €, { €, €} = Š 2 R M€€, [ €, i €]= C€€[ Š R Š G€ €+( €G€) M€€Š iWY Š1 3 iWM + W M ]+( R ) M€€, [ i €, i €]= C€€fŠ h c .(5. 4.16) wheretheoperator fisde “nedby f= Š i1 2 G( € )€Š1 2 ( ( R Š i1 3 W( ) )+ W Š1 2 ( ( G )€) €Š i1 2 ( ( W )) Y Š WMŠ i1 8 [( ( €G )€) M + ] +( 2 R +2 R R + i1 6 W) M+1 2 ( ( €G )€) M€€,(5. 4.17) and W1 4! ( W ). Theindependenttensors R G a,and W,andthedep endentone Wsatisfytherelations G a= G a, €R = €W= €W=0,

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5.4.SolutiontoBianchiidentities297 €G€= R + iW, W+1 3 i ( W )=1 2 i ( €G )€, W+ € W€=0.(5. 4.18) Therefore,allthetorsions,curvatures,and“eldstrengthsof(5.4.2)areexpressiblein termsofthethreecova riantsuper“elds W, G€,and R ,andthei rderivat ives. Byconsid eringthecoecientof Monbothsideso ftheeq uationforthecommutatoroftwovectoriald erivatives,weconcludethatthesuperspaceanalogofthedecompositionin (5.1.21)takestheform R a b = C€€[ W Š1 2 ( ) ( 2 R +2 R R + i1 6 W)+1 2 ( ( X ) )] + C1 4 ( (€( G )€ )),(5.4 .19) where X isde “nedby X= Š i1 8 ( €G )€.(5. 4.20) Bycomparingthisto(5.1.21),weseethereisarepresentation Xpresentinthesupercovariantc urvature R a b thatwasabsen tinthecompon entcurvature r a b .This o ccursbecausetheconstraintsthatwehavechosenimplythereisnontrivial x -spacetorsion T a b c a b c dG dpresent. Wenowc hoosethescale+U (1)gaugewherethecompensatorequals1.For n =0 theonlyresultingmodi“cationof(5.4.16)isthatweset R =0 (thus,for n =0,the spinor derivatives(butnotthevectorderivatives!)obeytheglobalsupersymmetryalgebra).F orother n itisnecessarytodropthe Y partofthecov ariantderiva tives.However,for n = Š1 3 ,Avanishesid entically inthisgauge,sotheonlymodi“cationof (5.4.16)istoset W=0(seedisc ussionbefore(5.3 .72)).Inthiscase,(5.4.16)reducesto (5.2.81).For n =0, Š1 3 themodi“cationsareslightlymorecomplicated:(1)Thespinor U (1)conn ectionisnowcovariant;toavo idconf usion,wede“ne TŠ i ( degauged ).(2)Notensorsar esetto zero,butnow R isdetermi nedbythe compensatorconstraint(see(5.3.73)),and Wbyitsexp licitform(see(5.3.74)).(3)

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2985.CLASSICALN=1SUPERGRAVITYSeparatingoutthe Y partscausesshiftsinafewofthetorsions(see(5.3.72))bythe covariantquantity T.Theresult ingformof(5.4.16)is {, €} = i €Š1 2 ( T €+ T€) { €, €} =1 2 T(€ € )Š 2 R M€€[ €, i €]= C€€[ R + G€ €] Š1 2 ( T€Š €T+ T T€) €+ C€€[ Š ( €G€) M€€Š W M + i1 3 WM ]+(( + T) R ) M€€,(5. 4.21) with(5.4.18)modi“edby + TY and € €Š T€Y .Thisisj ustthe U (1) degaugingdescribe dins ec.5.3.b.7. Thisresultcorrespondstooneofthemanyformsofnonminimal n = Š1 3 N =1 supergravity.Asexplainedinsec.5.3,covariantderivativescanberede“nedbyshifting withcontortions.Asanexample,wenotethatthetwoanticommutatorsin(5.4.21)can besimp li“edbytheshift €€Š i1 2 ( T €+ T€),(5.4 .22) whichgives(5.2.82).Fortheremainderofthebook,unlessotherwisestated,ourcovariantderivativesfor N =1superg ravitywillbeinthegaugewiththetensorcompensator settoone,andfor n =0,withthe Y partsdropped.

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5.5.Actions2995.5.Actions Inthissectionweconstructanddiscuss superspaceaction sforma ttersystems coupledtosupergravity,andforsupergravityitself. a.Reviewofvectorandchiralrepresentations Inthevectorrepresentation(where,e.g., €=( )),acovariantsuper“eld X€ ...tr an sformsunderthegaugetransformationsoflocalsupersymmetry(hermitian supercoordinateandtangentspacetransformations)as X= eiKXeŠ iK(5.5.1a) with K = K = KMiDM+ K iM + K€€i M€€(cf.(5.3.3)).Inchiral( € D€)or antichiral( D)representatio ns,thetransfo rmationlawsare X(+) = ei X(+)eŠ i ,(5. 5.1b) X( Š ) = ei X( Š )eŠ i ,(5. 5.1c) respectively,with, givenby,e .g.,(5.3.33-35),and X(+)= eŠ HX( Š )eH= eŠ Xe .(5. 5.2) (Recallthatthehermitianconjugateofano bj ectinthechiralrepresentationisinthe antichiralrepresentationandtransformswith (5. 5.1c).Therefore,justasin(5.2.28) andinYang-Millstheory(4.2.21),wemustconverttheconjugatetoanobjecttransformingwith( 5.5.1b),andde“nethechiralrepresentationconjugate X(+)= eŠ H( X(+))eH.)Thetransformationpropertiesof EŠ 1inthevector,chiral,and antichiralrepresentationsare EŠ 1 = EŠ 1eiK(5.5.3a) E(+) Š 1 = E(+) Š 1ei (5.5.3b) E( Š ) Š 1 = E( Š ) Š 1ei (5.5.3c) respectively.

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3005.CLASSICALN=1SUPERGRAVITYb.Thegeneralmeasure Usingtheresultsoft heprevioussections,itisstraightforwardtoconstruct locallysupersymmetricactions:Wecovariantizeallderivatives(withpossiblysome ambiguityinwhetherweuseminimalcouplin goraddcontor tionterms),includingthe derivativesusedtode“neconstrainedmatter“elds(suchaschiral“elds),andwecovariantizethemeasure.Forintegralsoverallsuperspace,byanalogywithordinaryspace (see5.1.23-5),weuse EŠ 1asadensitytode“neacovariantmeasure,andwriteactions oftheform: S = d4xd4 EŠ 1ILgen,(5. 5.4) where ILgenisageneralrealscalarsuper“eldconstructedoutofcovariantmatter“elds, derivatives,etc.Sincebyconstruction ILgenmusttran sformasin(5.5.1),andsince EŠ 1transformsasin(5.5.3),theexpressionin(5.5.4)isinvariant.(Recallthatcoordinate invarianceisde“nedonlyuptosurfaceterms(5.1.27).)Thistypeofexpressionisthe integratedversionofwhatisreferredtoasaD-typedensityformula. c.Tensorcompensators Justasingravity(sec.5.1.d),wecangeneralizethecoordinateinvariantmeasure (5.5.4)toascale(and U (1))invariantmeasurebyintroducingtensorcompensators.For an ILgenthathasscaleweight d (therealityoftheactionimpliesthatitmustbe U (1) invariant),wehave(see(5.3.8,14)) n = Š1 3 : S = d4xd4 EŠ 1( )1 Šd 2 ILgen,(5. 5.5a) n = Š1 3 ,0: S = d4xd4 EŠ 1( )3 n +1 2 (1 Šd 2 )ILgen,(5. 5.5b) n =0: S = d4xd4 EŠ 1G1 Šd 2 ILgen.(5. 5.5c) Theseactionsreduceto(5.5.4)inthegaugewherethecompensatoris1.(Theanalogousexpressioninordinarygravityisd4xd4 eŠ 14 Š dL where isthetensor-type componentscalecompensatorintroducedin(5.1.33)and L isascaleweight d Lagrangian.)

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5.5.Actions301d.Thechira lmea sure Inthechiralrepresentation,acovariantlychiralscalarsuper“eld (+), €(+)=0isch iralinthe”atsuperspacesense D€(+)=0.There fore,its transformation (5.5.2b)canbewrittenas (+) = ei (+)eŠ i = ei ch(+)eŠ i ch(5.5.6a) where ch=(for€=0)= mi m+iD.(5. 5.6b) Sincefor n = Š1 3 thetransformationof 3is(5.2.68) 3 = 3ei ch, ch=(for€=0)=( mi m+iD ),(5.5 .7) thequantity 3isasuitablechiraldensitytocovariantizethe”atspacechiralmeasure (sees ec.5.2.c). Sn = Š1 3 = d4xd23ILchiral.(5. 5.8) ThisistheintegratedversionofanF-typedensitymultiplet.Forothervaluesof n nodimensionlesschiraldensityexists.Wedescribehowthissituationishandledbelow. e.Representationindependentformofthechiralmeasure For n = Š1 3 ,wecanwritethe chiralme asureintermsoftherealmeasure.From (5.3.66b)wehave,in chiralrepresentation, D2EŠ 1IL = 3( 2+ R ) IL .(5. 5.9) Thus d4xd4 EŠ 1ILgen= d4xd23( 2+ R ) ILgen.(5. 5.10) Fr om(5.3.66a)wecan“ndthevectorrepresentationof(5.5.10): d4xd4 EŠ 1ILgen= d4xd2 eŠ 3( 2+ R ) ILgen.(5. 5.11) Ifwechoose ILgen= RŠ 1ILchiral,since ILchiral= R =0,

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3025.CLASSICALN=1SUPERGRAVITY d4xd23ILchiral= d4xd4 EŠ 1RŠ 1ILchiral.(5. 5.12) Theformofthechiralmeasure d4xd4 EŠ 1RŠ 1isvalidin all representationssinceitdoes notdependontheexistenceofachiraldensity.Itismanifestlycovariantand,inprinciple,couldbeusedforall n =0( R =0for n =0).ThusFtypede nsitymultiplets alsoexistfor nonminimal supergravity.However,unless n = Š1 3 ,(5. 5.12)leadstocompo ne nt actionscontaininginversepowersoftheauxiliary“elds(withtheexceptionofRinvariantsystems:seebelow). The U (1)-covariantformofthechiralmeasure( n = Š1 3 )issomewhatm ores ubtle: U (1)invariancealonegivestheanalogof(5.5.12)as S = d4xd4 EŠ 1RŠ 13(1 Š1 2 w )ILchiral(5.5.13) when[ Y ILchiral]=1 2 wILchiral.Inparticular,su perconfo rmalactionsalwayshave w =2. However,scaleinvarianceisnotsostraightforward:Using(5.3.10),weseethat R hasan inhomogeneousterminitstransformationlawproportionalto 2L butbecauseofthe chira lityof R ,,and ILchiralthistermvanishesuponintegrationbyparts.Thusthechiralityofthecompensatorisessentialforconstructinggeneralchiralactions.Sinceinthe ( U (1)-)chiralrepresentation= ,u sing(5.3.64a),wereobtain(5.5.8).Theexpression (5.5.13)canbeusedfor n = Š1 3 ,0if w =2,sinceth entheactionisindependentand consequentlysuperconformal. f.Scalarmultiplet Todisc ussspeci“ccouplingstomatter,we“rstconsiderthe U (1)-covariantform. Accordingtoourgeneralprescription,thedir ectcovariantizationoftheaction(4.1.1)for thefre escalarmul tipletis S = d4xd4 EŠ 1 ,( 5.5.14a) withcovariantlychiral € =0,(5. 5.14b) Thisformoftheactionisvalidforany n .Ifwea ssignscaleweight d =1to (see

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5.5.Actions303(5.5.5)),theactionissuperconformalsinceitisindependentofthetensortypecompensators.Inparticular,itisinvariantundertherestrictedsuperconformaltransformationsthats urvivein Poincar esupergravity( 5.3.76).Atthecomponentlevelthisaction leadstoconformallycoupledscalar“eldswithactionsasin(5.1.35)butwiththeoppositesign.(Compensatorsgenerallyhaveacti onswithanoverallminussignrelativeto physicalsystems).Sincetheactionissuperconformalevenwithoutthecompensators,it isclearthatthescalarsofthemultipletareconformallycoupledtogravitywithoutthe needforacomponentcalculation. Afterdegauging,theaction(5.5.14a)andde“ningconditionremainunchangedfor n = Š1 3 .Forthen onminimaltheories, weuset hesameactionbutif wewantthecomponentscal arstobeconformallycoupledtogravit ywemustc hangethede“ningcondition((5.5.18)with w =2 3 ;seebelow).A lternativelyifthede“ni ngconditionisnotmodi“edandthe operatorin(5.5.14b)isforadegaugednonminimaltheory,thenthe actionof(5.5.14a)doesnothaveconformallycoupledscalars. f.1.Superconformalinteractions Forsuper conformallyinvariantactions,thedensitycompensator canbegauged away(i.e .,removedbya“eldre de“nitionwhichisasupersca letransformation).Inthis case,theaction(5 .5.8)writtenfor n = Š1 3 makessenseforany n .Forex ample, since EŠ 1= EŠ1 3 (1 eŠ H)1 3 (5.2.72),theconformallyinvariantactionforacovariantlychiral scalarsuper“eldis S = d8zEŠ 1 +( 1 3! d6z 33+ h c .).(5. 5.15) Atthecomponent level,thetermsproportionalto describequartics elf-interactions andYukawacouplingsforthecomponent“el dsofthematterchiralmultipletjustasin theglobalcase. Therescaling removes fromtheactio nentirely,an dtheresult isvalidforany n ( isachiraldensityofweight w =2 3 ).Thiscanbeg eneralized s lightly:Toremove fromthechiralintegrands,fullsuperconformalinvarianceisnot required;R-invariance(3.6.14)issucient.Then appearsonlyinthe fullsuperspace integrand,andonlyi nthecomb ination ;inthatcase,the n = Š1 3 compensator can

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3045.CLASSICALN=1SUPERGRAVITYberepl acedby n =0orno nminimalcompensators.Forexample,inthecaseofasingle chiralmu ltiplet,wecangeneralizetoarescaledchiraldensity witharbitraryweight w ; see(5.5.20)andtheparagraphafter(5.5.32)below. f.2.Conformallynoninvariantactions Nonconformalcouplingsofthescalarmultipletarealsopossible.Wecanalways addthesupersymmetricterm Snonconf= d4xd4 EŠ 1( 2+ 2)(5. 5.16) Thisactuallyvanishesfor n =0.(Itisalso po ssibletowriteaCPnon-conservingterm bytaking i timethedierenceinsteadofthesumin(5.5.16).)Atthecomponentlevel, (5.5.16)generatesd is-improvementterms r (A2Š B2)+ ... withoppositecontributions forthescal arandpseudoscalar“elds.Therefore,(5.5.16)cannotbeusedtoeliminate theimprovementtermsofboth“elds.For n = Š1 3 ,wecanrewrit e(5. 5.16)asthechiral integral Snonconf= d4xd23R 2+ h c .(5. 5.17) Fornonmi nimal( n = Š1 3 ,0)supergravit y(5. 5.16)alsointroducesdisimprovem enttermsbutthereexistsanotherwayofintroducingsuchnonconformal termsforthescalarmultiplet.Beforedegauging,ifthescaleweightof isnot1 ,then theonlywaytowriteasuperconformalkineticactionforthechiralmultipletistointroduceoneofthedensitycompensatorsof(5.5.5).Thustheactionwithoutthecompensatorsisnotsuperconformal.Afterdegaugingthe U (1)invariance(seesec.5.3.b.8),we canusethenewtensor Ttode “neamodi“edchiralcondition.Wecanreplace (5.5.14b)by ( €+1 2 w T€) =0.(5. 5.18) Thekineticactionisstillgivenby(5.5.14a),andif w =2 3 ,itisnotsuperc onformalwithoutoneofthecompensatorsof(5.5.5).Eventhoughthe U (1)groupis nolonger gauged,itstillexistsasaglobalR-invarianceoftheaction(5.5.14),andtheconstraint (5 .5 18) is covariantevenunderlocaltransformations

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5.5.Actions305[ Y ]=1 2 w .(5. 5.19) Themodi“edchiralityconditionof(5.5.18)intermsofunconstrainedsuper“elds leadstoamorecomplicatedactionfort hescal armultiplet.Wecanexpress interms ofa”at chiraldensity eŠ1 2 w T E€ =0(inthechiral representation D€ =0).In termsofunconstrainedsuper“elds,theaction(5.5.14a)becomes(inthegaugewithcompensat orssetequaltoone) S = d4xd4 En [(1 eŠ )(1 e )]n+1 2 ,(5. 5.20) where nisde “nedintermsof w by w =2n Š n3 n +1 .Only n= Š1 3 issuperconformaland hastheconventionalconformalimprovementterms;then w =2 3 .Thusinthe nonminimaltheories,conformalcouplingforthescalarsisachievedbyreplacing(5.5.14b)by (5.5.18)with w =2 3 f.3.Chiralself-interactions Thecovariantizationofanyglobalchir alpolynomialself-interactiontermsP( )is straightforward.Fromourgeneralprescription(5.5.12)wehave(for n =0) Sint= d4xd4 EŠ 1RŠ 1P( )+ h c ..(5.5 .21) Theexpression(5.5.21)remainslocallysupersymmetricfor“eldssatisfyingthemodi“ed chira litycondition(5.5.18)orinthe U (1)-covariantforma lismwitht heusual € =0 forarbitrarychiralweight.For n = Š1 3 Sintispolynomialinthecomponent“eldsafter theeliminationofthesupergravityauxiliary“eldswheneverP( )ispol ynomial.For n = Š1 3 ,itisingen eralnonpolynomial,exceptforP( )= 2 w (forasinglechiralmultiplet;formoremultiplets,theconditionisgivenbelow).Thus,asmentionedearlier, althoughF-typedensitiesexistfornonminimaltheories,ingeneralthesewillleadto no np olynomialityaftertheeliminationofauxiliary“elds. Inthechiralrepresentation,for n = Š1 3 ,(5. 5.21)canberewrittenintheform (5.5.8),andfor n = Š1 3 ,intheform

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3065.CLASSICALN=1SUPERGRAVITYS = d4xd2 eŠ TP( )+ h c .(5. 5.22) whenthechiralchargeofP( )is1 2 w =1.Thisfo llowsfrom(5.5.13),sincefor n = Š1 3 S mustbei ndependent.(Forthespecialinteractiongivenabove,thiscanbewritten asd4xd2 2 w ,withnodep endenceonthesupergravity“elds.) g.Vectormultiplet Thevectormultipletcanbecoupledtosupergravitybysimplyde“ningderivativesthatareco variantwithresp ecttothelocalinvariancesofbothsupergravityand su pe r-Yang-Mills: A= EA+(A M +A€€ M€€) Š i AATA; A = eiKAeŠ iK, K = KMiDM+( K iM + h c .)+ KATA;(5. 5.23) whereAAistheYang-Millspotentialand KAitsgaugeparameter.Fieldstrengthsfor bo thsupergravityandYang-Millsarede“nedbythegradedcommutatorsofthecovariantderivativesasusual,andthesamesupergravityandYang-Millsconstraintsare imposed(see(4.2.66)and(5.3.4,5,13)).Thesolutiontotheconstraintscanbegivenby expressing Aintermsofthepures upergravityc ovariantderivatives Aandthe usual Ya ng -M illssuperpotential=ATA: = eŠ e, €= Š i {, €} .(5. 5.24) Alternatively,thesolutioncanbewrittenwith ofthesameformas butnowwith =MiDM+(inanalogyto U U + V intheglobalcase).TheYang-Mills“eld strengthis W= i [ €, { €, } ](5. 5.25) andtheactionis S = gŠ 2tr d4xd4 EŠ 1RŠ 1W2.(5. 5.26)

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5.5.Actions307For n = Š1 3 intheYang-Millschiralrepresentation,usingtheBianchiidentities (5.4.16,18)wecanrewrite(5.5.25)as W= i ( 2+ R ) eŠ VeV,(5. 5.27) andtheactionis S = gŠ 2tr d4xd23W2.(5. 5.28) Asfortheconformalcouplingofthescalarmultipletin(5.5.8),theactionsin(5.5.26,28) areinvariantwithrespecttotheconformaltransformationsparametrizedbyarbitrary L and K5super“elds.Thisensuresthatthe dependenceof W issuchthatit cancelsin theactionof(5.5.28).Moregenerally,alsoasaconsequenceofconformalinvariance, (5.5.26)isindependentof or T For n =0thefor m(5. 5.26)cannotbeusedsince R =0and( 5.5.28)cannotbe usedsince onlyoccursinthe n =1 3 theory.T hecorrectactionis S = gŠ 2tr d4xd4 EŠ 1( WŠ1 6 [ €,€])+ h c .(5. 5.29) where Wisgivenby(5.5.25)and,€areobtainedfrom(5.5.24)using Š i A= AŠA.The gaugeinvarianceoftheaction(5.5.29)followsfromtheBianchi identity W+ € W€=0.Wenotetha tthisform( va lidforall n andinallrepresentations)issimilartothethree-dimensionalgaugeinvariantmassterm(2.4.38). Alternatively,itispossibletouse(5.5.28)forall n ,ifweareon lyinterestedinthe explicitdependenceonthesupergravityprepotential H m.The H mdependenceanddensitytypecompensatorindependenceofany truly conformalactionisindependentof n h.Generalmattermodels Wenowconsi derageneralclassofmattermulti plets(chir alandgauge)coupled to n = Š1 3 supergravity.A globally supersymmetricgaugeinvariantaction,restricted onlybytherequirementth atnobosonictermswithmorethantwoderivativesor fermionictermswithmorethanonederivat iveappearinthecomponentLagrangianis

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3085.CLASSICALN=1SUPERGRAVITYS = d4xd4 [ IK (i,j)+ trV ] + d4xd2 [P(i)+1 4 QAB(i) WAWB]+ h c .(5. 5.30) where j= k( eV)j k, WA= i D2( eŠ VDeV)A(5.5.31) andP(i)and QAB(i)= AB+ O ()arechir al.Theterm trV isthe(global)FayetIliopoulosterm(4.3.3).Asexplainedinsec.(4.1.b), IK canbeinterpretedastheK¨ ahler potentialofa ninterna lspacema nifold. Thecorresponding locally supersymmetricaction, including ( n = Š1 3 )supergravity, is S = Š3 2 d8zEŠ 1eŠ 23 [ IK (i,j)+ trV ]+ d6z 3[P(i)+1 4 QAB(i) WAWB]+ h c .(5. 5.32) Inthelimit 0, E and 1,thisreducestotheglobalaction(5.5.30).ThecovariantFayet-Iliopoulostermis(Yang-Mills)gaugeinvariantonlyifthechiralactionisgloballyR-invariant.Underagaugetransformation ( trV )= itr ( Š ), EŠ 1exp ( Š1 3 2 trV )isinvaria ntifwesimu ltaneouslyperformthe(restricted)complex superscaletransformationdiscusseda ttheendofs ec.5.3withchiralparameter L + i1 3 K5= Š i26 tr .Theinvarianceofthe chiral integralin(5.5.32)followsfromRinvarianceof(thechiralpieceof)theglobalaction. Ingeneralthecouplingsofthescalarmultipletinsuperspaceinvolveconformal couplingofthespinzerocomponent“eldstogravity.Thereisonespecialchoiceofthe K¨ ahlerpotential,however,whereallsuchconformalcouplingcanbeeliminatedforthe componentscalar“elds.Thisspecialchoiceisgivenby IK (i,j)=ii. ForR-invaria nttheori essuperscaletransformationscanbeusedtorescalethematter“eldsan dremove fromthe chiral integral;asmentionedabove,theresultingaction dependsonlyonthecombination ,andweca nrewriteitforany n ,e.g .,usingduality

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5.5.Actions309transformationsofthecompensatoraswillbe describedinsec.5.5.ibelow.Inparticular,ifweperformadualitytransformationtothe n =0theory,theac tion(5.5.32) b ecomes S = d8zEŠ 1[ Š1 2 V5+ IK (i,j)+ trV ] + d4xd2 [P(i)+1 4 QAB(i) WAWB]+ h c .(5. 5.33) whereiand Waresuita blyde“neddensities(the -independentquantitieswede“ned tomakethedualitytransformationpossible). Althoughwehaveconcentratedhereonthe n = Š1 3 theory coupledtovectorand chiralsc alarmultiplets,moregeneralsystemsalsocanbeconsidered.Aswestated above,couplingofotherversionsofsuperg ravitycanbeobtainedbyperformingduality transformations.Asdescribedinchapter4,therearealargenumberofscalarmultipletsandmanyothermattermultiplets.Thesemaybecoupledtosupergravitybyuse theprescriptionof(5.5.4,12). i.Supergravityactions i.1.Poincar e For n =0,thePoincar esupergravitya ctionisobtainedfrom(5.5.4)(or(5.5.5), for d =0)bychoosing Lgen=( n 2)Š 1.For n = Š1 3 ,thiscanbere writtenas S = Š 3 Š 2 d4xd23R .(5. 5.34) For n =0,theobvi ouschoice S = d8zEŠ 1,(oritss cale invariantformwiththe tensorcompensator(see(5.5.5c))and Lgen= Š 2)vanishes:With thecompensator G =1,thechiral curvature R =0 (seesec.5.3.b.7.iii),and(e.g.,inthechiralrepresentation)(5.3.56)implies D2EŠ 1=0.Iftheacti onvanishesinonegauge,itmustdosoin allgauges,includingoneswherethecompensatorhasnotbeengaugedaway.However, the n =0theoryha sadime nsionless U (1)prepotential V5thatallowsustowritean action:Since D2EŠ 1=0,int he gauge G =1 th ef o llo wingactionisinvariantunder

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3105.CLASSICALN=1SUPERGRAVITYU (1)gaugetransformations E =0, V5= i ( 5Š 5): Sn =0= Š1 2 d4xd4 EŠ 1V5.(5. 5.35) Inthechiralrepresentation,thiscanberewritten,using(5.3.63),as Sn =0=3 2 d4xd4 EŠ 1ln [ EŠ 1 E1 3 (1 eŠ H)Š1 3 ].(5.5 .36) Sinceinthegauge G =1wehave EŠ 1= G (5.3.70),thisisthecovariantizationofthe ”atspaceaction(4.4.46)fortheimprovedte nsormult iplet.Wesawthat(4.4.46)could bewri ttenina“rst-orderformthatmademani festthedualitybetweenthescalarand tensormultiplet.Thisconstructioncarriesovertothelocalcase,andwe“ndthat n =0 supergravity(withatensorcompensator)isdualto n = Š1 3 supergravity(withachiral scalarcompensator). Wewritea“rst-o rderactionas S = Š3 2 d4xd4€€EŠ 1( e€€XŠ€€G€€X ),€€G =1 2 (€€+ €€ € €), €€ €=0;(5. 5.37) where€€X isanindepe ndent,unconstrained,realsuper“eld,and all objects(€€EŠ 1,€€)are thoseof n = Š1 3 .Thisi sjust n = Š1 3 supergravitycoupledtothe“rst-orderformof theimprovedtensormultiplet(4 .4.45).Ifwevarywithrespectto€€X ,ands ubsti tutethe resultbackinto(5.5.37),we“ndthe n =0acti on;ontheotherha nd,ifwevarywith respectto ,we “ndthe n = Š1 3 action.Indetail,wehave,fromthevariationwith respectto€€X€€X = ln€€G (5.5.38) andhence(5.5.37)becomes Sn =0=3 2 d4xd4€€EŠ 1(€€Gln€€G Š€€G )(5. 5.39) B ecause€€G islinear,thesecondterm canbedropped.Since€€EŠ 1€€G = G = EŠ 1G ,(see

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5.5.Actions311(5.3.63)),using(5.3.63)weobtaintheaction(5.5.35)withthecompensator G inageneralgauge: Sn =0=3 2 d4xd4 EŠ 1G ( lnG Š1 3 V5).(5.5 .40) Thisactionisscaleand U (1)invar iant. Alternatively,vari ationwithrespectto gives( €€ 2+€€R )€€€€X =0andh ence€€X = ln + ln €€ €= 0,sothatagainusingthelinearityof€€G toeliminatetheterms€€Gln + h c .,weobtain S = Š3 2 d4xd4€€EŠ 1 ,(5.5 .41) i.e.,the n = Š1 3 action(5.5.5a). Thedualitytransformationfromthe n = Š1 3 supergravitytheorytothe n =0theory,asdescribedabove,canbereversedthro ughastraightforwardc ovaria ntizat ionof thereversedualtransform(4.4.38)(compareto(4.4.42)).Bothformsoftheduality transformcanbeperformedeveninsystemswh erethesupergravity mult ipletiscoupled tomattermultiplets(justasins ec.4.4.c.2);however,thoughany n =0sy stemcanbe conv ertedtoan n = Š1 3 system,thereversetransformationispossibleonlyifthe n = Š1 3 systemisR-invariant,andhencetheactioncanbewrittensothatitdepends onthe n = Š1 3 compensator(tensorordensitytype)inthecombination or Analogousdualitytransformationsthatarethecovariantizationofthosedescribed attheendofsec.4.5.b.canbeusedtorelate n = Š1 3 andnonminimalsupergravitysystems. Theformofthesuperspaceactionfor n =0revealsach aracteristiccommonto mostextendedsupersymmetrictheories.Naively,wemightexpectactionstotakeageometricalformdzEŠ 1IL ( fieldstrengths ).However, wecaneas ilyseethatfor N 3, evenif dz isachiralmeasure,therearenoquantitiesofproperdimensionstoformsuch anactionforglobalorlocalsupersymmetry.Ourexperiencewiththe n =0theory showsthatitispossible,aftersolvingconstraints,to“ndquantitieslike V5thatwemay

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3125.CLASSICALN=1SUPERGRAVITYcallsemiprepotentialsorprecurvatures,withoutwhichtheactioncannotbewritten. Thus,theunconstrainedsuper“eldapproachbecomesincreasinglyimportant,sincesuch precurvaturesareactuallyfoundasintermediatestepsinsolvingconstraints. i.2.Cosmologicalterm TothePoi ncar esupergravitya ctionwecanaddasupersymmetriccosmological term(foradiscussionofglobaldeSittersupersymmetry,seesec.5.7).For n = Š1 3 ,we have Scosmo= Š 2 d4xd23+ h c .(5. 5.42) For n = Š1 3 ,0wecouldwriteatypeo fcosmol ogicaltermusingtheform(5.5.12), but thattermcontainsinversepowersofthescalarauxiliary“eld;for n =0, R =0and henceitisimpossibletowriteacosmologicalterm(thesedicultiesarisebecausethe cosmologicaltermis not R-invarian t).Oneotherinterestingfeatureofthecosmological termfornonminimalsupergravityisthatthesumof(5.5.12)(with ILchiral=1)and (5.5.5b)(with ILgen=1)leadst oaspontaneous breakingofsupersymmetry.Theresulting“eldequationsaresuchthatitisnotpossibletoconstructananti-deSitterbackgroundwhichissupersymmetric(seesec.5.7). i.3.Conformalsupergravity Nextweconsidertheactionforconformalsupergravity.Itisjustthecovariantizationofthelin earizedexpression(5.2.6): Sconf= d4xd23( W)2.(5. 5.43) Theconformal“eldstrength Wdependson onlythroughaproportionalityfactor Š3 2 ,soall dependencecancels.Thef ormof(5.5.43)isvalidonlyfortheminimaltheory.Itcanbeextendedtothenonminimaltheorybytheuseof(5.5.12).For n =0ev en thisinsucient,againbecause R =0.Thisdoe snotimplyt hatconformalsupergravity doesnotexist;itis n -independent.Insteadtheactionforconformalsupergravitytakes aformsim ilartothethree-dimensionalsupergravitytopologicalmasstermwith Wtakingtheplaceofth ethr ee-dimensional G,and G€and R theplaceof thet hree-

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5.5.Actions313dimensional R (see(2.6.47)). j.Fieldequations Toobtaincovari ant“eldequationsfromtheactionbyfunctionaldierentiation withrespecttothesupergravitysuper“elds,whicharenotcovariantthemselves,we de “ne am od i“edfunctionalvariation,aswedidforsuper-Yang-Mills(see(4.2.48)): H eŠ H eHor eŠ e, ( e ) eŠ ;( 5.5.44a) ( 3).(5.5 .44b) Theequationsofmotionforsupergravitywithactiongivenby(5.5.4)andthecosmologicalterm(5.5.42)canthenbeshowntobe S H a = Š Š 2G a=0, S = Š Š 2( R Š )=0.(5 .5.45) Thecovariantized“eldequationfor H€isthesam easthato btainedbythebackground “eldmethod(thevariationisthesameasthebackground-quantumsplittinglinearizedinthequantum“eld). Thederivationofthis“eldequationwillbedescribedin moredetailwhenwedescribethissplittinginsec.7.2.The equationisea s ilyobtained using(5. 2.71,5.3.56,5.5.34,42). To obtaincovariant“eldequationsforacovariantlychiralsuper“eld,itisnecessarytode“neasuitablefunctionalderivative.Thiscanbedoneinanyofthreeways: (1)by“rstusingthe” at-space de“nitionfordierentiationby ,and usingtherelation (5.5.2);(2)bycovariantizingthe”at-spaceforminawaythatsatis“esthecorrect covariantchiralitycondition;or(3)byexpressingthechiralsuper“eldasthe“eld strengthofageneralsuper“eld.Theresultis: ( z ) ( z) =( 2+ R ) 8( z Š z),(5.5 .46) where,for n = Š1 3 is U (1)covariant.Theresulting“eldequationsforascalarmultipletarethusthesameasintheglobalcaseexceptthat D2isrepla cedwith 2+ R The“eldequationsforsupergravitycoupledtoascalarmultipletare(for n = Š1 3 ,and usingtheacti on(5.5.14))

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3145.CLASSICALN=1SUPERGRAVITYŠ 2G€= 1 3 [ i€ Š ( € )( )+ G€], Š 2R =1 3 ( 2+ R ) =0.(5. 5.47) Whenaself-interactiontermisadded,the G equationisunchanged,but R b ecomes nonzero(exceptforthesuperconformalcoupling 3).Inthelastequat ionwehav eused theequationofmotionofthescalarmultiplet.Alternatively,termsin“eldequations prop ortionaltoother“eldequationscanberemovedingeneralevenoshellby“eld rede“nitionsintheaction.(Toremovetermsproportionaltothe“eldequationsof 2fromthe“elde quationsof 1,a“eldre de“nitionof theform 1= 1 2= 2 + f modi“esthe“eldequationsto S 1 = S 1 + S 2 f 2 .)Inthiscase,the appropriate“eld rede“nitionis = Š 1= Š + ... ,whichre movesall dependencefromthe scalar-multipletaction(seesec.7.10.c). Fo rt he co upl ed su pe rgravity-Yang-Millssystem(sec.5.5.h),the“eldequationsfor Ya ng -M illsarestill {, W} =0,wh ilethesupergravityequationsare Š 2G€= gŠ 2tr W€W, Š 2R =0.(5. 5.48) Wehave droppedtermsinthe G eq ua tionproportionaltotheYang-Mills“eldequation. Theseterms,whichinthiscasearenotYang-Millsgaugecovariant,canagainbeelimina tedbya“eldrede“nition(againseesec.7.10.c). Althoughwehaveonlyconsidered n = Š1 3 forsimp licity,covariantvariationwith respecttothecompensatorsfortheotherversionsofsupergravitycanalsobede“ned analogously.Inboth n =0andn onminimaltheoriestheimportantpointtonotein de“ningthecovariantvariationsisthattheunconstrainedcompensatorsforboththeoriesarespinors(= D€€forthenonminimaltheoryand for n =0).Thus functionaldierentiationinthesecasesle adtospinorialequationsofmotion.

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5.6.Fromsuperspacetocomponents3155.6.Fromsuperspacetocomponents a.Generalconsiderations Sofarinourdiscussionofsupergravitywehaveconcentratedexclusivelyon superspaceandsuper“elds.Ontheotherhand,somewhereinthisformalismasupergravitytheoryinordinaryspacetimeisbei ngdescribed.Thequestionariseshowto extractfromasuperspaceformulationinformationaboutcomponent“elds.Weknow howtodothisintheglobalsupersymmetryc ase,andherewewilldescribethecorrespondingprocedureinlocalsupersymmetry,andderivethe tensorcalculus ofcomponent supergravity.Wecannotuse D and D tode “nethecomponentsofsuper“eldsbyprojectionasinglobalsuperspace,sincethiswouldnotbecovariantwithrespecttolocal supersymmetry. Todisc usscomponentsupergravity,wemust“rstchooseaWess-Zuminogaugein whichthe K -transformationshavebeenusedtosettozeroallsupergravitycomponents thatcanbegaugedawayalgebraically.AWess-Zuminogaugeisnecessarysothat resultsfornoncovariantquantities(i.e.gauge“elds)canbederivedalongwiththosefor covariantquantities.Wecanthenderivetra nsformationlawsfortheremainingsupergravitycomponentsaswellascomponentsofothersuper“eldsandexhibitsupercovariantizationandthecommutatoralgebraofloc alsupersymmetryatthecomponentlevel. Wederivemult iplicationrulesforlocal(covariantlychiral)scalarmultiplets,andwrite thecomponentformoftheintegrationmeasures(densityformulae),fromwhichcomponentactionscanbeobtained.Alltheresultsre”ecttheunderlyingsuperspacegeometry andcanbeobtainedforany N ,imposingasf ewconstraintsaspossible(preferably none).Thisimpliesthatsuperspacegeometryismoregeneralthanacomponenttensor calculuswhichfo llowsfromachoiceofconstraintsonsuperspacetorsionsand/orcurvatures.The “nalformofthetensorcalculusisdeterminedbywhichsolutionofthe Bianchiidentitiesisutilized. Webeginwitha generalsuperspacefor N -extendedsupergravity.(Wewillspecializeto N =1when everneeded.)Insuchasuperspacewehaveavielbein EA Mwhich describessupergravity.Wealsointroduceanumberofconnectionsuper“eldsA for tangentspacesymmetriessuchasLorentzrotations,scaletransformations, SU ( N )-rotations,centralcharges,et c.Thesesuper“eldsarecombinedwithoperators DMand M

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3165.CLASSICALN=1SUPERGRAVITYtoformasuper covariantderivative A= EA+A M EA= EA MDM,(5. 6.1) where M arethegeneratorsofthetangentspacesymmetries.The -subscripti salabel thatrunsoverallthegeneratorsofthetangentspacesymmetries.Forinstance,in N =1, n =0superg ravity M =( M, M€€).Therealizationofthesegeneratorsis speci“edbygivingtheiractiononanarbitrarytangentvector XA.Thus,fors omesetof matrices( M )A Bwehave [ M XA]=( M )A BXB.(5. 6.2) Wewrite DM= DM N zN +M M for “xed matrices DM NandM where DM NŠ M NandM vanishat =0(sees ec.3.4.c).Weassumethevielbeinisinvertible;speci“cally,weassumethatwecanalways“ndacoordinatesystem(orgauge)inwhichwecan write A= A+A.(5. 6.3) Thegaugetransformationsof Aaregivenasusualby A= eiKAeŠ iK.(5. 6.4) Theparameter K isasuper“eldwhichisalsoexpandedover iDMand iM K = KMiDM+ K iM ,(5. 6.5a) andissubjecttoarealitycondition K = ( K ).Wecanequallywell expandtheparameter K overthecovariantderivatives Aand M : K = KAi A+( K Š KAA ) iM = KAi A+ K iM .(5. 6.5b) A gaugetransformationofan arbitrary covariantsuper“eldquantityisalwaysgenerated byacti ngwith iK as in (5 .6 .4 ). Fo ri n“nitesimaltransformationsofsupercovariant quantitiesthisimpliesthatwesimplyactonthequantitywiththeoperator iK .

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5.6.Fromsuperspacetocomponents317b. We ss-Zuminogaugeforsupergravity Wede “necomponentsof covariantq uantit ies (matter“elds,torsionsandcurvatures)usingthelocalgeneralizationofthe covariantprojectionmethodintroducedina globalcontext:thesecomponentsarethe i ndependentprojectionsofthesuper“elds andtheircovariantderivatives.Wede“necomponentsof gauge“eldsEA M,A by choosingasp ecialgaugeandthenprojectingasoncovariantquantities.ThisWessZuminogaugechoicereducesthesuperspacegaugetransformationstocomponentgauge transformations:Itusesallbutthe i ndependentpartof K toalgebraicallygauge aw ay th en oncovariantpiecesofthehighercomponentsofthegauge“elds(thelowest componentsremainasthespacetimecomponentgauge“elds).Weusethenotation X | tomeanthe i ndependentpartofanysuper“eldquantity X ;if X isanoperator XMiDM+ X iM ,then X | istheoperator XM| i M+ X | iM ;weuse DM| = Manddo not set , € to zero.Inparticular,wede“nethecomponentsofthecovariantderivativesby C| C| C| €C| ,etc. Wede “netheusualco mponentgauge“eldsby a| e a m m+ a + a€ € + a M D D a+ a + a€ € ,(5. 6.6) where e a misthecomponentinversevierbein, a , a€ arethecomponentgravitino “elds,and a arethecomponentgauge“eldsofthecomponenttangentspacesymmetries.(For M =( M M€€)these gauge“eldsaretheLorentzspinconnections a and a€€.) Fr om th ei n“nitesimaltransformationlaw a=[ iK a]= Š i aK + ... weseethatt hesecomponentstransformasspacetim egradientsof thegaugeparameters, whichjusti“esthede“nition.Wehavealsointroducedtheordinaryspacetimecovariant deriva tive D D a= e a+ a M (cf.5.1.15).Covariantlytransformingcomponentsofthe su pe rgravitymultiplet(e.g.auxiliary“elds)appearascomponentsofthetorsionsand curvatures. Wenowderivet he“rstfewcomponentsofthecovariantderivatives.Webeginby exploitingtheexistenceofaWess-Zuminogauge.Fromthein“nitesimaltransformation law =[ iK ],using(5.6.3)we“nd

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3185.CLASSICALN=1SUPERGRAVITY | = Š i K + ... =[ iK ] | K(1) (5.6.7) andhence,byusingthe componentof K i.e., K(1) ,wecanchoo seagauge | =0 or | = .(5. 6.8) Wehaveth usdetermined A| .W ec anproceedto“ndthehigher-ordertermsina straightforwardmanner.Thusto“nd | westartwith ( )=[ iK ].(5.6 .9) Then ( ) | = Š i K + ... .(5. 6.10) Since { } =0weca n gaugeaway[ ] butnot { } .Howev er,the latter iscovariant:Itcanbeexpressedintermsof torsionsandc urvatures.Henceinthis gaugewe“ndthe componentof : | =1 2 { }| =1 2 T CC| +1 2 R | M .(5. 6.11) Inthesameway,we“ndthe € componentof : € | =1 2 { € }| =1 2 T€ CC| +1 2 R€ | M .(5. 6.12) Similarly,we“ndthenextcomponentof b;we“rstobserve thatb ecause | = ,wehave b| = D D b+ b | + b€ € | .(5. 6.13) Thenwecompute b| =[ b] | + b | =[ b] | + D D b | + b | + b€ € | .(5. 6.14) Using(5.6.8,11,12)weobtain

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5.6.Fromsuperspacetocomponents319 b| = T b CC| + R b | M + b [ M | ] +1 2 b [T CC| + R | M ]+1 2 b€ [T€ CC| + R€ | M ](5.6.15) Thuswehavefound B| .Wecan “ndhighercomponents,butwhatwehaveissucientfortheapplicat ionswegivebelow.Wehaveobtainedtheseformulaewithout imposing any constraints. Theprocedurewehavedescribedusesallthehighercomponents(projectionswith more s)of K to eliminatethenoncovariantpiecesof and € andde “nestheWessZuminogauge.Theremaininggaugetransformations,determinedbythe i ndependent term K | ,arejusttheu sualcomponenttransformations.Coordinatetransformationsare determinedby iKGC| = Š m( x ) m(5.6.16) (orequivalentlycovarianttranslations iKCT| = Š a( x ) D D a= Š m mŠ a( a M )). Tangents pace gaugetransformationsaredeterminedby iKTS| = Š ( x ) M ,(5. 6.17) andsupersymmetrytransfo rmationsaredeterminedby iKQ| = Š ( x ) Š € ( x ) € = Š ( x ) |Š € ( x ) € | .(5. 6.18) However,tostayintheW ess-Zuminogauge,the K transformationsmustbe restricted:thehighercomponentsareexpressedintermsof K | .Forex ample, | = implies: =0=[ iK ] | ,( 5.6.19a) sothat 0= Š [ KBB, ] |Š [ K M ] | = Š KB[ B, }| +[ KB}B|Š K [ M ] | +[ K ] | M ,(5. 6.19b) andhence

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3205.CLASSICALN=1SUPERGRAVITY KB| = KCTC B| + K | ( M ) B K | = KCRC | .(5. 6.20) Similarly,wecan“ndthehighercomponentsof K fromthehigherc omponentsof andtherequirementt hattheWess-Zuminogaugeismaintained.Itturnsoutthatinthe We ss-Zuminogauge(5.6.16)holdstoallordersin i.e., KGChasnohighercomponents, whereasboth KTSand KQhavehighercomponentsdependingonthecomponent“elds, thegaugeparameters ,andin general,the gradients oftheparameters.Thus,inthe localcas e,wecannotwrite Š iKQ= Q + € Q€ forsomeoperator Q .The higher ordertermsin iKTSarealwaysproportionaltothematrices( M ) ,( M )€ € ;hen cefor internalsym metries ascomparedtotangentspacesymmetries, iKTShasnohighercomponentsa nd(5.6 .17)isexact. c.Commutatoralgebra Asanotherapplicationoft heuseoftheWess-Zuminogaugesupersymmetrygenerator,wederivethecommutatoralgebraoflocalcomponentsupersymmetry.Inthis gaugeweusethe dierentialoperator iKQasthelocalsupersymmetrygeneratorforthe componentformulationofsupergravity.Sincethesupersymmetrygeneratoris“eld dependent,wecanindicatethisbywriting iKQ( ; )where denotesallofthe x -space “eldscon tainedin iKQ.Thismea nsthatcaremustbe takeninde“ningthecommutator oftwosuchtransformations.Letusi magineperformingsequentiallyon twosupersymmetrytransformationswithparameters 2and 1.The “rsttransforma tionisobtained from iKQ( 2; ) ,wherewe havedroppedthecommutatornotation,keepinginmind that iKQisanoperator.Thesecondtransformationisimplementedby iKQ( 1; + 2 ) iKQ( 2; ) .Therefo re,thecorrectwaytocomputethecommutator algebraisfro mthe de“nition [ iKQ1, iKQ2] iKQ( 1; + 2 ) iKQ( 2; ) Š (1 2) iK12.(5. 6.21) However,bylookingattheformofthesupersymmetrygeneratorin(5.6.18)wenote that iKQ| hasno“elddependentterms.Thisimplies[ iKQ1, iKQ2] | correspo ndsto the usualcommutator[ iKQ( 1, ), iKQ( 2, )],andthisisallweneedto“ndthecomponent commutatoralgebra.Takingtheexpressionfor iKQfrom(5.6 .18)andusingtheWess-

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5.6.Fromsuperspacetocomponents321Zuminogauge-preservingcondition[ iK ] | =0(see (5.6.19)),weobtain [ iKQ1, iKQ2]= 1 2 { } + 1€ 2€ { € € } +( 1 2€ + 1€ 2 ) { € }| (5.6.22) Comparingtherighthandsideoftheaboveequationto iKGC| iKTS| and iKQ| we “nd iK12 iKGC( m)+ iKTS( )+ iKQ( ), m= Š [( 1 2€ + 1€ 2 ) T € c+ 1 2 T c+ 1€ 2€ T€ € c] e c m, = Š [( 1 2€ + 1€ 2 )( R € + T € c c ) + 1 2 ( R + T c c )+ 1€ 2€ ( R€ € + T€ € c c )], = Š [( 1 2€ + 1€ 2 )( T € + T € c c ) + 1 2 ( T + T c c )+ 1€ 2€ ( T€ € + T€ € c c )].(5. 6.23) Theseresultsshowhowthecommutatoralgebraoflocalsupersymmetryiscompletelydeterminedbysuperspacegeometry.Inparticularthe“elddependenceofthe localalgebraisaconsequenceofonlyconsideringcomponent“eldswhicharepresentin theWZgauge.Thefullresultfor(5.6.21),toallordersin isgivenby [ iKQ1, iKQ2]= iKGC( m)+ iKTS( ; )+ iKQ( ; ), + 2 Š 1 .(5. 6.24) d.Localsupersymmetryandcomponentgauge“elds Wenowderivet hesupersymmetryvar iationofthecomponentgauge“elds.We obtainthesebyevaluatingasuper“eldequationat =0 a| =[ iKQ, a] | .(5. 6.25) From(5 .6.13)wehave

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3225.CLASSICALN=1SUPERGRAVITY[ iK a] | = iK a|Š aiK | = Š ( + € € ) a|Š D D aiK |Š a iK |Š a€ € iK | .(5. 6.26) UsingtheWess-Zuminogaugecondition(5.6.19a,17),werewritethisas [ iK a] | = Š ( + € € ) a|Š D D aiK |Š a iK |Š a€ iK € | = Š ( a| + € € a| )+ D D a( + € € ) | + a ( + € € ) | + a€ ( + € € ) € | .(5. 6.27) Expa ndingover m, | and M ,a ndusing(5.6.10,11,15)we“nd: Qe a m= Š [ T a d+ € T€ a d+( € a + a€ ) T€ d+ a T d+ € a€ T€ € d] e d m, Q a = D D a Š ( T a + T a e e ) Š € ( T€ a + T€ a e e ) Š ( € a + a€ )( T € + T € e e ) Š a ( T + T e e ) Š € a€ ( T€ € + T€ € e e ), Q a = Š ( R a + T a e e ) Š € ( R€ a + T€ a e e ) Š ( € a + a€ )( R € + T € e e ) Š a ( R + T e e ) Š € a€ ( R€ € + T€ € e e ).(5.6 .28) Theseresultscanbespecializedto N =1, n = Š1 3 superspace,andusingthesolutionto constraintsandBianchiidentitieswecandeducethetransformationlawfor e a mand a :

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5.6.Fromsuperspacetocomponents323Qe a m= Š i ( a€+ € a ) e€ m,( 5.6.29a) Q a = D D a+ i €S Š i A€Š i ( € a + a€) € .(5. 6.29b) Thetransformationlawofthegravitinocanbesimpli“edsomewhatbyconsideringthe supersymmetryvariationof m .(Thelasttermin( 5.6.29b)isabsenta sacons equence of(5.6.29a).)Theauxiliary“elds S and A aarede “nedas R | and G a| respectively;consequently,theirtransformationscanbefounddirectlybecause R and G aarecovariant (seebelow).Thesecovariantde“nitionsoftheminimalauxiliary“eldsarethegeneralizationsofthelinearizedexpressionsof(5.2.8)and(5.2.73). Weshouldpo intoutthattheresultsfor Q a arevalidforgaugedinternalsymmetries(suchas U (1), SU (2),etc.)also.Inthiscase( M ) =0,( M )€ € =0andthe quantities RAB arethe “eldstrengthsfortheinternalsymmetrygaugesuper“eld. Thereforetheformulaein(5.6.28)containspartofthetensorcalculusforamattervectormultiplet.T hecovariantcomponentsofsuchamultipletaretreatedjustlikethose ofanycovariantmultiplet,e.g.,achiralscalarmultiplet. e.Superspace“eldstrengths Tosimp lifycalculationswithcomponentgauge“eldsitisconvenienttode“ne supercovariant“e ldstrengths(quantitieswhichtransformwithoutderivativesofthe localsup ersymmetryparameter).Webeginbycomputing [ a| b| ]=[ D D a, D D b]+( D D[ a b ] ) +( D D[ a b ]€ ) € +( [ a | | b ] ) +( [ a | €€ | b ]€ ) € .(5. 6.30) Theordinaryspacetimetorsionsandcurvaturesarede“nedby(see(5.1.17)) [ D D a, D D b]= t a b cD D c+ r a b M ,( 5.6.31a) where t a b c= c a b c+ [ a ( M ) b ] c,(5. 6.31b)

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3245.CLASSICALN=1SUPERGRAVITY[ e a, e b]= c a b ce c,(5. 6.31c) r a b = e[ a b ] Š c a b c c + a 1 b 2f 1 2 ,(5. 6.31d) and[ M 1, M 2]= f 1 2 3M 3.Wede “neacurvaturefor a by t a b D D[ a b ] Š t a b d d = e[ a b ] Š c a b d d Š [ b a ] .(5. 6.32) Wenowhaveallthe x -space“eldstrengths.Fromthefactthat e a m, a ,and a are gauge“elds, t a b c, t a b ,and r a b aretheappropriate“eldstr engths.The“eldstrength associatedwith M(and M€€), r a b = Š r a b ,€€istheRiemanncurvaturetensor.With thesede“nitionswecan expressthesuperspacetorsionandcurvatures at =0as T a b C= t a b C+ [ a T b ] C+ [ a€ T€ b ] C+ [ a b ] €T € C+ a b T C+ a€ b €T€ € C,(5. 6.33) R a b = r a b + [ a R b ] + [ a€ R€ b ] + [ a b ] €R € + a b R + a€ b €R€ € .(5. 6.34) Wehave used [ a, b] | =[ a| b| ]+ [ a b ]| + [ a€ € b ]| (5.6.35) and(5.6.15).Weseethesetensorsdierfromtheir x -spaceanalogs(5.1.17,18)byadditionalgravitinoterms.Thesuperspace“e ldstrengthsarecovariant:Thereforethe =0proj ectionsof(5.6.33,34)arethesupercovariant x -space“eldstrengths. Fo rt hegauge“eldsofinternalsymmetries,covariant“eldstrengthsarealsonecessary.These“eldstrengthsarede“nedbyexactlythesameformulaeasthecurvatures above.

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5.6.Fromsuperspacetocomponents325f.Supercovariantsupergravity“eldstrengths Wenowusethee xplicitsolutionofthe n = Š1 3 Bianchiidentitiestoobtainfrom (5.6.33,34)thecomponent“eldstrengths.ThesolutionoftheBianchiidentitiescontains allofthenecessaryinformationaboutthetors ionsandc urvatures.Forthetorsionsand curvatureswithatleastonelowerspinorialindex,wesubstitutefrom(5.2.81)intothe leftha ndsideof(5 .6.33). Considering“rst T a b we “nd T a b = t a b + i ( a G€Š b G€) Š i ( a€ Š b€ ) R .(5. 6.36) Thisequationiscorrectto -independentorderandthusasupercovariantgravitino“eld strength, f a b ,isde “nedby f a b = T a b | .(5. 6.37) For T a b cweuse(5. 6.33)inaslightlydierentway.Alongwiththetorsionswith atleastonelowerspi norialindex,wealsosubstitutefor T a b contheright side.This yields t a b c+ i [ a b ]€= i ( C€€G€Š C€€ G€).(5.6 .38) Nowwecantakethisresult,useittosolveforthecomponentspin-connection,andthus obtainasecondorderformalism.Beforedo ingthisiti sconvenientt oobservethat i ( CC€€G€Š C€€CG€)= Š a b c dG d,(5. 6.39) sothat a b ccanbeexpressedas a b c= (e) a b c+ i1 2 ( [ b c ]€+ [ a c ]€Š [ a b ]€) Š1 2 a b c dA d.(5. 6.40) where (e) a b cisde“nedin(5.1.19). FinallythesupercovariantizedRiemanncurvaturetensoristreatedanalogouslyto T a b .Webeginbys ubsti tutingfrom(5.2.81)forthecurvatureswithatleastonespinorialindex. R a b = r a b + R a ( b )+ i { [ a€W

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3265.CLASSICALN=1SUPERGRAVITYŠ1 2 a ( C( G )€+( € R ) C ( C ) )] Š ( a b ) } .(5. 6.41) However,wemustcarryoutonefurtherste pbeforewehaveanexpressionwhichcanbe evaluatedintermsofcomponent“elds.Wemusteliminate G€, € R ,and Wfrom thisexpr ession.Thiscanbedonebyconsideringthecoecientsof onbothsidesof (5.2.81).Onthele fthandsidewe“nd f a b ,wh ileontherig htha ndside W, €G€, and R appear.Wecanthereforesolveforthesequantitiesintermsof f a b whichis expressedintermsofcomponent“eldsin(5.6.36). W=1 12 f( € € ),( 5.6.42a) G b= Š1 2 [ f€ € ,€Š1 3 C f€ € ,€],(5.6 .42b) R = Š1 3 f€ € .(5. 6.42c) Theseexpressionscannowbesubstitutedinto(5.6.41)whichresultsinawellde“ned(at thecomponentlevel)supercovariantizedRiemanncurvaturetensor. Asaby-pro ductofthisprocesswehavealsoderivedthecomponentsupersymmetrytransformationlawoftheauxiliary“elds A aand S where S R | and A a G a| Thesearesupercovariantsandhencetheir supersymmetryvariationsaregivenby, QA a= iKQG a| = Š1 2 [ f€ € ,€+1 3 f€ €€]+ h c ., (5.6.43a) QS = iKQR | = Š1 3 f€ € .(5. 6.43b) g.Tensor calculus Thecomponentrulesforthemanipulationoflocallysupersymmetricquantities arecalledthetensorca lcul usforsupergravitytheories.Theserulesgiveacomponentby componentdescriptionofsupersymmetrictheories.Super“eldsontheotherhandprovideaconcise descriptionofthesetheoriesinmu chthesam ewaythat v ectorno tation pr ov idesamoreconcisedescriptionofMaxwellsequations.Super“eldscanalwaysbe reducedtotheircomponent“eldcontentinthecaseofglobalsupersymmetryandinthis sectionwediscusstheanalogousprocedureinthelocallysupersymmetriccase.

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5.6.Fromsuperspacetocomponents327Asanexample,letusconsiderfor N =1superg ravityalocalscalarmultiplet describedbyacovariantlychiralsuper“eld € =0.Thecomp onent“eldsofthis mult ipletarede“nedbyprojection A | | F 2 | .(5. 6.44) Thein“nitesimalsupersymmetrytransformationsof all quantitiesareobtainedbycommuta tionwith iKQ( ).Thus,usi ng(5.6 .18) QA = iKQ( ) | = Š | = Š Q= Š ( + € €) | = Š[(1 2 {, } + C2)+ €{, €}] | .(5. 6.45) At thispoint,nospeci“cchoiceofauxiliary“eldsforsupergravityhasbeenmade.The onlyconstraintsonthesupers pacetorsionsnecessaryarethosewhichfollowasconsistencyreq uirementsfortheexistenceofchiralsuper“elds,i.e.,therepresentation-preservingconstraints.UsingthesolutiontotheBianchiidentitiesforthecaseof N =1, n = Š1 3 supergravityweobtain Q= F Š i €( € ) | .(5. 6.46) Using(5.6.6),wehave Q= F Š i €( D D€A + € ).(5.6 .47) Thelastexpressionillustratestheconceptofa supercovariantderivative atthecomponentlevel.Thecombination[ D D aA + a ],whichgeneralizestheordinarycovariant deriva tive D D aA ,transfo rmswithoutatermproportionalto D D a.Thus,this combinationof“eldsiscovariantwithrespecttoal ocalcomp onentsupersymmetrytransformation.Finally,forthetransformationlawfortheauxiliary“eldwe“nd

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3285.CLASSICALN=1SUPERGRAVITYQF = Š ( + € €) 2 | = Š €[ i D D€Š A€Š i €F Š €€( D D€A + € )]+ S = Š €[ i D D€Š A€] + S .(5. 6.48) (Analogoustransformationsforachiralscalarmultipletcanbefoundfor n = Š1 3 by usingtheappropriatesolutiontotheBianchiidentities.)Onthesecondlineabovewe haveintroducedthenotation D D aforthesupercovariantderivativeofthespinormatter“eld. Wecanalso “ndthecomponentsoftheproductoftwodierentmultipletsin termsofthecomponentsoftheoriginalmultiplets.Thus,forexample,aproductoftwo chiralsc alarmultipletsdescribe dbychirals uper“elds 1, 2isthescalarmultiplet describedbythechiralsuper“eld 3= 12: A3= 12| = A1A2, ( 3)= ( 12) | =([ 1] 2+ 2[ 2]) | =( 1)A2+ A1( 2), F3= 2( 12) | =([ 21] 2+[ 1][ 2]+ 1[ 22]) | = F1A2+( 1)( 2)+ A1F2.(5. 6.49) Thecomponentsof3transformaccordingto(5.6.45,47,48).Thismultiplicationlawis just likeintheglobalcase(3.6.11). Anotherpossi bleproductoftwoscalarmultipletsisfoundbytakingtheproductof achirals uper“eld 1andanantichi ralsuper“eld 2;thisgivesthec omplexgeneral scalarsuper“eld= 1 2: | = A1 A2, | = 1 A2, 2 | = F1 A2,

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5.6.Fromsuperspacetocomponents329[ €] | =2 1 2€, ( 2+ R ) | = F21 + i 2€( D D€A1), ( 2+ R ) | = Š 1 ( i D D€+ A€) 2€+ 2€( Š i D D€+2 A€) 1 +2 F2F1Š ( D D€ A2)( D D€A1).(5.6 .50) whereweh aveused( 2+ R ) =0whichcan beobtained from(5.4 .16).Notethe appearanceofthesupercovariantderivative D D a. Wecanalsog ivethecompon entsofachiralsuper“eldmadeoutofanantichiral one 1=( 2+ R ) .Thisi ssometim escalledthekineticmultiplet.Itscomponents are: A1=( 2+ R ) | = F + S A ( 1)= Š ( i D D€+ A€) €Š1 3 f€ € A F1=(( D D a+ i 3 A a) D D aŠ1 3 R€ €€€Š 4 S S ) A Š 4 SF Š1 3 f€ € ,€ €.(5. 6.51) wherew ehavemadeuseoftheresult 2R +2 R R = Š1 6 R€ €€€.(5. 6.52) The x -spacesupercovariantcurvature R€ €€€in(5.6.51)isgivenby(5.6.41).The computationoftheseresultsisstraightforwardbuttedious.Alloftheaboveresultshave madeextensiveuseofthecommutatoralgebrain(5.2.82). Asafurther exampleofcomponenttensorcalculus,weconsiderthevectormultiplet.Thelocalcomponentsarede“nedinthesamewayasintheglobalcase(4.3.5), butwith Areplacedby A,t he su pe rgravityandYang-Millscovariantderivative.The “eldstrengthsandBianchiidentitie sforthev ectormult iplettaketheforms FYM = FYM €=0,

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3305.CLASSICALN=1SUPERGRAVITYFYM b= Š iC W€ YM, FYM a b= C€€f+ C f€€, €W YM= W YM+ € W€ YM=0.(5. 6.53) Thequantity fisasuper covariant“eldstrength(seebelow).Thelocalcomponentsof themultipletar ethusde “nedby v a= a| = W YM| ,D= Š i1 2 W YM| .(5. 6.54) Thesupersymmetryvariationsofthecovariantcomponents, andD,areobta inedas withthecomponentsofthechiralmultiplet(see(5.6.46)). Q = Š f+ i D, QD=1 2 ( D D€ €Š €D D€),(5.6 .55) where D D€ €isthesup ercovariantderivativeof D D€ €= D D€ €Š €€( f€€Š iC€€D).(5.6 .56) Thisfollowsfrom(5.6.13)andtheBianchiidentitiesofthevectormultiplet(4.2.90) whicharevalidinacurvedsuperspace.Thequantity f(anditsconjugate f€€)canbe calculatedinthesamewayas(5.6.41)from(5.6.34,53) f=1 2 [ fYM € €+ i ( ( € ) €)+ i ( ( € ,€ ))],(5. 6.57) where fYM a bistheordinary x -s pa ce Ya ng -M ills“eldstrength.Forthetransformation lawof v a,weuse(5.6 .28).(Eventhoughthederivationofthatresultwasforthegauge “eldsfortangentspacesymmetries,italso appliestothegauge“eldsforinternalsymmetries.) Qv a= i ( €+ €) Š i ( a€+ € a ) v€.(5. 6.58) Justasforthegravitinotansformationlawin(5.6.29b),thelasttwotermsaboveare

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5.6.Fromsuperspacetocomponents331absentifweconsiderthetransformationlawof v m. Wehave presentedtheaboveresultsfor N =1, n = Š1 3 supergravity;theycanbe generalizedtoall n byusingt heappropriateset ofBianchiidentities(5.4.16-17). Inourdiscussionofglobalsupermultipletswefoundalargenumberofgaugemultipletsw herethecomponent gauge“eldwasnotaspin-one“eld(forexamplethetensor mult iplet).Sincewegaveacompletelygeometri caltreatmentofthesemultipletsusing p -formswithinglobalsupersy mmetry,theirextensiontothelocallysupersymmetriccase (i.e.,transformationlaws,supercovariant“eldstrengths,etc.)isobtainedbythe straightforwardgeneralizationofthemethodswhichweusedtotreatthespin-onecase. Theonlycomplicationthatcanoccuristhattheexistenceoftheunconstrainedprepotentialmustbecon sistentwiththesetofconstraintsthatdescribethesupergravity background.Anexampleofan N =1mult ipletforwhichthesuper“eldextensionto localsupersymmetryisnotknownisthemattergravitinomultiplet.Thisisnotsurprisingsinceasecondsupersymmetry(i.e., N =2supersy mmetry)isrequiredfortheconsistencyoftheequationsofmotionforthemattergravitino. h.Componentactions Finallywegiveformulaetoobtaincomponentactionsfromthe N =1 n = Š1 3 superspaceactions S1= d4xd23ILchiral,( 5.6.59a) S2= d4xd4 EŠ 1ILgeneral.(5. 6.59b) We“rst have S1= d4x eŠ 1[ 2+ i €€+3 S +1 2 (€ |€ |€ )€] ILchiral| (5.6.60) Toderive(5.6 .60)thereareseveralsteps.Fir st,using(5.5.9)andchoosing IL = RŠ 1,we seethat 3= D2EŠ 1RŠ 1.T hisisadensityunder x -spacecoordinatetransformations. Butin x -space,adensityiseŠ 1mult ipliedpossiblybya dimensionless x -spacescalar.

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3325.CLASSICALN=1SUPERGRAVITYNosuchdime nsionlessscalarscanbeconstructedintheminimaltheory.Therefore 3at lowestorderin mustbepr oportionaltoeŠ 1.(Itshouldbenote dthatt hereareno exp licitfactorsof anywhereexcepttha tmulti plyingthesupergravityaction.)Thissituationisnottrueforthenonminimaltheor ies,inwhichitispo ssibletoconstructa di mensionlessscalarfromsomeoftheadditionalauxiliary“elds.Thisispreciselywhat happensfortheF-typedensityforthenonminimaltheory,andisresponsibleforthe nonpolynom ialitydiscussedinsubsec.5.5.f.3. Onceweknowth atthelowestcomponentof 3iseŠ 1,wederive(5. 6.60)bymultiplyingeŠ 1bythehighest component F ofachirals uper“eldandperformingasupersymmetrytransformation.Thisgen eratesatermproportionalto F timesthegravitino, whichwecancancelbyaddingtoeŠ 1F atermpropo rtionalto €€.Thisnewterm generatessupersymmetryvariationsproportionalto D DA timesthegravitino.Thesecan becan celedbyaddingatermproportionalto 2A tothestartingpoint.Finally,we determinethecontributionofthe SA termbycancelingvariationsproportionalto S Bydime nsionalanalysis,therecanbenoothercontributions,andwehaveobtainedthe densityformula of(5.6.60). To “ndthecorrespondingexpressionfor S2,weuse S2= d4xd23( 2+ R ) ILgeneral(5.6.61) andtheformula(5.6.60)forthechiralcase.Thecovariantderivativesactonthesuper“eldsintheL agrangianandprojectoutthecomponents. Asasimpleexample,wecomputethemasstermforachiralsuper“eld : S =1 2 m d4xd232=1 2 m d4x eŠ 1[2 FA + + i 2 €€A +(3 S +1 2 (€ |€ |€ )€) A2].(5.6 .62) Asas econdexamplewecomputethe N =1, n = Š1 3 componentsupergravityaction andcosmologicalterm.

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5.6.Fromsuperspacetocomponents333From(5.5 .4,34)wehave SSG=1 2 d4x eŠ 1[1 2 r€ € ,€€Š a b c d a ,€D D c d Š 3 | S |2].(5.6 .63) Thisisthecomponentformofthesupergravityactionwiththeimprovedspinconnection.Theaxialvectorauxiliary“eldispresentimplicitlyinthe“rsttermsincethespin connection,asde“nedin(5.6.40),dependson A a.Ifwesepar ateoutfrom ( e ) a b cthe contributionof A aitappears only quadraticallyintheaction.Inparticular,thereisa cancellationamongtermsoftheform A a b d whichcomefromthe“rsttwotermsin theaction. Forthecosmol ogicaltermfrom(5.5.42)andusing(5.6.60),wehave Scosmo= Š 2 d4xd23+ h c = Š 2 d4x eŠ 1[3 S +1 2 (€ |€ |€ )€+ h c .](5.6 .64) Thecosmologicaltermcontainsatthecomp onentlevelanapparentmasstermforthe gravitino.However,inthedeSitterbackgroundgeometrythegravitinoisactuallymassless,sinceitisstillagauge“eld. Inclosingwemaketwoobservations:Although(5.6.61)wascomputedafterthe constraintswereimposedonthecovariantderivatives,inprincipleonecancomputesuch anactionformulaw itho utimposing any constraintsa ta ll.Thisfollowsbecausethe transformationlawsforthecomponentsofthetotallyunconstrainedsuperspaceare di rectlyobtainablefrom(5.6.28)and,formattermultiplets,fromequationsanalogousto (5.6.45-48).Alargenumberofauxiliary“eldsde“nedasthe =0valueoft hevarious su pe rspacetorsionswillentersuchaconstruction.Amongtheseoccursanauxiliary“eld whichisaLagrangemultiplierthatmultiplieseŠ 1.(Thevariati onofthisLagrangemultiplierwillconstrainthegeometryof x -space.)Clearlythisisunacceptable,andwehave seenhow,for N =1 supergravity,thiscanbeavoided.However,understandingtherole of such“eldsmaybenecessarytounderstand N > 4o-shellt heories. Thesecondpointisthatwelackatpresentadirectmethodforcomputingdensity fo rmulaeanalogousto(5.6.60).Wecanalwayscomputesuchaformulabyhand:We startwitheŠ 12 N(whereisan arbitrarysuper “eld)andperformsupersymmetry

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3345.CLASSICALN=1SUPERGRAVITYvariationsto obtainanentiredensitym ultiplet.Whatislackingisawaytoobtainthis resultwithoutlaboriouscalculation.

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5.7.DeSittersupersymmetry3355.7.DeSitters upe rsymmetry Insec.3.2.f,wediscussedthesuper-deSitteralgebra(3.2.14).Herewedescribe howsupersymmetricdeSittercovariantderivativescanbeobtainedfromsupergravity covariantderivatives.We“rstdiscussthenonsupersymmetricanalog.NonsupersymmetricdeSittercovariantderivativescanbeobtainedfromgravitationalcovariantderivativesbyeliminatingall“eldcomponentsexceptthe(density)compensating“eld(i.e., thedeterminantofthemetricorvierbein).T hisfo llowsfromthefactthatindeSitter spacetheWeyltensorvanishes,whichsaysthatthereisnoconformal(spin2)partto themetric:Itisconformally”at.Ontheotherhand,thescalarcurvaturetensoris anon zeroconstant r =2 2(thisisthegravity“eldequation). Wecanwrite e a m= Š 1 a mwhere isthecompensatorof(5.1.33).Aftersetting theothercomponentstozero,theactionfordeSittergravity(Poincar epluscosmo logical term)isjusttheactionforama sslessscalar“eldwithaquarticself-interactionterm. (Therestofgravity,theconformalpart,issimplythelocallyconformalcouplingofgravitytothisscalar.)T heequationofmotioncorrespondingtothecovariantequation r =2 2, =2 23,(5. 7.1) hasthesolution,withappropriateboundaryconditions, Š 1=1 Š 2x2.(5. 7.2) ThedeSittercovariantderivativesarenowobtainedfromthegravitycovariantderivativesofsec.5.1bysubstituting e a m= Š 1 a m,with Š 1givenby(5.7.2). Inthesupersymmetriccase,westartwiththesupergravityactionandacosmologicalterm(5.5.16).Weset H tozero,andsolveforthechiraldensitycompensator :In super-deSitterspace Wvanishes(asdoes G a),while R = TheactionforthecompensatoristhemasslessWess-Zuminoaction(againaconformalaction,whosesuperconformalcouplingto H givesthe deSittersupergravity action).The“eldequationsinthechiralrepresentation D2 = 2(5.7.3) havethesolution

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3365.CLASSICALN=1SUPERGRAVITYŠ 1=1 Š x2+ 2.(5. 7.4) The(realpartofthe) =0compon entof isthusthegravitycompensatorof(5.7.1,2). Thesuper-deSittercovariantderivativesareobtainedbysubstitutingthissolutionfor (with H =0)intotheex pressionsf orthesupergravitycovariantderivativesgiveninsec. 5.2. Theprecedingdiscussioninvolvedthe n = Š1 3 compensator .For other n ,we “ndstrangepathologies:deSitte rspace cannotbedescribedfor n = Š1 3 inaglobally (deSitter)supersymmetricway.For n = Š1 3 ,emptyde Sitterspaceisdescribedby R = G a= W=0, butfornonminimal n wewouldrequire G a= W=0with T .Thisfo llowsfromthefactthatthecommutatorsofcovariantderivativesmust takethefollowingformtode scribedeSittersuperspace {, } = Š 2 M, {, €} = i €,[ b]= Š i C €, [ a, b]=2 ( C M€€+ C€€M).(5.7 .5) Thisrequires spontaneousbreakdown of N =1supers ymmetry,since Tisatensor: T| =0wouldimply T=0ifgl obal(deSitterorother)supersymmetryweremaintained.( Tmustbeno nzerofor R tobeno nzerointhenonminimaltheory.See (5.2.80b)).For n =0, G a= W=0alre adyimpliesMinkowskispace.

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Contentsof 6.QUANTUMGLOBALSUPERFIELDS 6.1.Introductiontosupergraphs337 6.2.Gauge“xingandghosts340 a.OrdinaryYang-Millstheory340 b. Sup er sy mmetricYang-Millstheory343 c.Othergaugemultiplets346 6.3.Supergraphrules348 a.DerivationofFeynmanrules348 b.Asamplecal culation353 c.Theeectiveaction357 d.Divergences358 e.D-algebra360 6.4.Examples364 6.5.Thebackground“eldmethod373 a.OrdinaryYang-Mills373 b. Sup er sy mmetricYang-Mills377 c.CovariantFeynmanrules382 d.Exam ples 389 6.6.Regularization393 a.General393 b.Dimensionalreduction394 c.Othermethods398 6.7.AnomaliesinYang-Millscurrents401

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6.QUANTUMGLOBALSUPERFIELDS 6.1.Introductiontosupergraphs Aswehaves eeninpreviouschapters,atthecomponentlevelsupersymmetric modelsaredescribedbyordinary“eldtheoryLagrangians,andtheirquantizationand renormalizationusesconventionalmethods.Evidentlythequantumtheoryshouldbe renormalizedinamannerthatpreservessupe rsymmetry.Unlessamanifestlysupersymmetricregularizationmethodisused,thisrequiresapplyingtheWard-Takahashiidentitiesofsupersymmetryateachorderofperturbationtheory. Supersymmetricmodelsareingenerallessdivergentthannaivecomponentpower countingindicates,andthiscanbetracedtotheequalityofnumbersofbosonicand fermionicde greesoffreedom,togetherwithrelationsbetweencouplingconstantsthat areimposedbysupersymmetry.We“ndthatthevacuumenergy(or,when(super)gravit yi sp resent,thecosmologicalterm)receivesnoradiativecorrections,andthat,in renormalizablemodels,acommo nwave-f unctionrenormalizationc onstantissucientto renormalizetermsinvolvingonlyscalarmultiplet“elds(the norenormalization theorem).Arelatedresultisatheoremthatifthe classicalpotentialhasasupersymmetric minimum(nospontaneoussupersymmetrybreaking),sodoestheeectivepotentialto allordersofperturbationtheory(noCole man-Weinbergmechanism:seesec.8.3.b). Improvedconvergenceduetosupersymmetryisalsoevidentinsupergravity.For all N ,the S-matrixof(extended)supergravityis“niteatthe“rsttwoloops;weargue insec.7.7thatitisalso“niteatlessthan N Š 1loops.Ins uitablesupersymmetric gaugesthis“nite nessalsoholdsfortheo-shellGreenfunctions. Insupersymmetrictheoriestheone-loo psuper conformalanomalies(traceofthe energy-momentumtensor, -traceofthecomponentsupersymmetrycurrent,andthe divergenceoftheaxialcurrent)formasupers ymmetricmultiplet,thesupertrace,so thattheircoecientsareequal.Thereexistotheranomaliesaswell.Weshowinsec. 7.10thatinnonminimal N =1superg ravity( n = Š1 3 ),anomaliesmaybepresentinthe Wardidenti tiesoflo calsupersymmetry.Thus,ingeneral,onlyminimal N =1supergravityisconsistentatthequantumlevel( butextendedtheoriesthathavenonminimal N =1superg ravityasasubmultipletareconsistentbecauseofanomalycancellation

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3386.QUANTUMGLOBALSUPERFIELDSmechanisms). Super“eldsgreatlysi mplifyclassicalcalculations:Supersymmetricactionscanbe easilyconstructed,andthetensorcalculusofsupersymmetrybecomestrivial.However, thegreatestadvantagesofsup er“eldsappearatthequantumlevel.Therearealgebraic simpli“cationsinsupersymmetricFeynmang raph(supergraph)calculationsfora numberofre asons:(1)compactnessofnotation,(2)decreaseinthenumberofindices (e.g.,thevector“eld A aishi ddeninsidethescalarsuper“eld V ),and(3)automaticcancellationofcomponentgraphsrelatedbysup ersymmetry(whichwou ldrequireseparate calculationincomponentformulations).Furthermore,theuseofsuper“eldsleadsto power-counti ngruleswhichexplainmanycomponentresultsandcanbeusedtoderive additional“nitenesspredictions,especiallywhencombinedwithsupersymmetricbackgr o und-“eldmethods. Renormalizat ionismuchsimplerinthesuper“eldformalism.Supersymmetryis manifestand,aswediscusslater, any regularizationmethodthatpreservestranslational invarianceinsuperspacewillmaintainit.Forgaugetheorieswecanusesupersymmetric gauge-“xingterms.BycontrastcomponentWe ss-Zuminogaugecalcul ationsexplicitly breaksupersymmetryandhavethedisadvantagethattheWard-Takahashiidentitiesfor globalsupersymmetrycannotbedirectlyappliedduetotheirnonlinearity. Inthischapterandthenextonewediscussthequantizationof N =1super“eld theori es.Weconsiderclassicalsuper“eldactions S ()and usefunctionalmethodsto constructthegeneratingfunctional Z ( J )andthee ectiveaction().Ifisagauge “eldwequantizecovariantl y,intro ducinggau ge-“xingterms,gaugeaveraging,and super“eldFaddeev-Popovghosts.WethenderiveFeynmanrulesforsupergraphsusing superspacepropagators( x x, ).Themethodsarecompletelyanalogoustothose forcomponent“elds,butsomenewfeaturesarepresent:Wemustdealwithconstrained (chiral)super“elds,a ndweencounternotonly i a= p aoperators,butalsospinor deriva tives Dactingontheargumentsofpropagatorsorexternallines.Weshowhow theseoperatorsaremanipulatedandhow,foranygraph,the -integralsateachvertex canbedone,leavingu swithoneoverall -integralforthewholegraph(theeective actionislocalin ),andordinaryloop-momentumintegrals.Atallstepsofthecalculationsmanifestsupersymmetryismaintained.

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6.1.Introductiontosupergraphs339We di sc us sn ex tt he ba ck ground“eldmethodforsupersymmetricYang-Millstheories.Thisissimilartothatforcomponenttheories,withonesigni“cantdierence:The quantum-backgroundsp littingisnonlinear,re”ectingthenonlinearitiesofthegauge transformationsofthesuper“eld V .Themethod simpli“esma nycalculat ionsandcan beusedtost udyhigher-loop“nitenessquestions. Forsupergr a phsthesimplestregularizationprocedureistousedimensionalregularizationofmomentumintegrals after thecontributionfromagraphhasbeenreduced toasingle integral.Theresultingeectiveaction,whichisa(localin )f unction alof theexternals uper“elds,ismanifestlysupersymme tric.How ever,thisregularization methodcorrespondsto(component)regular izationbydimensionalreduction,whichis knowntobeinconsi stent.Thesuper“eldresults,alth oughsupersymmetric,mayre”ect thisinconsistencybyexhibitingambiguitiesassociatedwiththeorderinwhichsomeof the -integrationshavebeencarriedout.Wealsodiscussalternativeregularizationprocedures.Besidesgivingpowercountingruleswedonotdiscussthedetailsoftherenormalizationofsuper“eldtheories.WeworkwithWick-rotatedtimecoordinates: d4x id4x ,so eŠ iS eS.(Themetric a bhassignature( Š + ++) (++ ++),so + ,etc.Notethatino urconventions, i isoppositeinsignfromusualconventions:Thuspositive-energystatesaredescribedby ei tandpropagatorsare ( p2+ m2+ i )Š 1.Wefurtherwarn thereaderthatthegaugecouplingconstant g is 2 timesthe usual g (seepage55).)

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3406.QUANTUMGLOBALSUPERFIELDS6.2.Gauge“xi ngandghosts Thequantizationofsupersymmetricgaugetheoriesissimilartothatofordinary gaugetheories.Therearetworelatedaspectsofthesituation:(a)Theactionisinvarian t undergaugetransformationsandthereforethefunctionalintegrationshouldbe restrictedtothesubsetofgaugeinequivalent“elds.(b)Thekineticoperatorisnot invertibleoverthespaceofall“eldcon“gurationssothatthepropagator,neededfor doingperturbationtheory,cannotbede“nedunlessthesetof“eldsisrestricted.In componentgaugetheories,imposinganalgeb raicrestrictionexplicitlyinthefunctional integralleadstoanaxialgaugewhichbreaksm anifestLorentzinvariance.Alternatively, wecanquantizecovar iantlyusingtheFaddeev-Popov procedure:Weintroducegauge “xingfunction(s),weightedgaugesandFadd eev-Popovghosts.Insupersymmetricgauge theoriestheanalogoftheaxialgaugeistheWess-Zuminogauge.Inthisgauge,quantizationbreaksmanifestsupersymmetry.Inco ntrast,covariantsuper“eldquantization maintainsmanifestsupersymmetry. a.OrdinaryYang-Millstheory Fo ro rientationwebrie”yrecallthequantizationmethodforordinaryYang-Mills theory.TheYang-Millsgaugeactionis SYM=1 g2 tr d4x [Š 1 8 f a bf a b], f a b= [ aA b ]Š i [ A a, A b],(6.2 .1) withgaugeinvarianceunderthetransformation A a A a = ei [ A a+ i a] eŠ i ,(6. 2.2a) or ,i n“nitesimally, A a= a = a + i [ A a].(6.2 .2b) Here isanelementofthegaugealgebra.Both A and arematricesintheadjoint representation.Weobservethatthekinetic(quadratic)partoftheLagrangiancanbe written(afterrescaling A gA )intheform1 2 A TA where(T) a b= a bŠ1 2 a b Š 1isatransverseprojectionoperator(see(3.11.2)). Westartwiththe normalizedfuncti onalintegral

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6.2.Gauge“xingandghosts341Z = N IDA aeSinv,(6. 2.3) wherewehaveincludedin Sinvpo ssibletermswithsourcescoupledtogaugeinvariant operators.Wede“nethegaugeinvariantintegraloverthegroupmanifold F( A a)= ID [ F ( A a ) Š f ( x )],(6.2.4) where f ( x )isanarbit rary“eld-independentfunction,and F isagauge-variantfunction suchthat F = f forsomev alueof .Itisimportant toverifythatthisisthecase.We introduceafactorof1inthefunctionalintegral,intheformF Š 1F: Z = N IDA aF Š 1( A a) ID [ F ( A a ) Š f ]eSinv= N IDA aF Š 1( A a) ID [ F ( A a) Š f ]eSinv,(6. 2.5) wherethelastformfollowsfromachangeofvariablesthatisagaugetransformation, andthegaugeinvarianceofFand S .The in te gr al nowgivesaconstantin“nitefactorthatwea bsorbintothenormalization N,leadi ngto theform Z = N IDA aF Š 1( A a) [ F ( A a) Š f ]eSinv.(6. 2.6) Byconstr uction Z isindepe ndentof F and f ,and hencewecan averageover f withan arbitrary(normalized)weightingfactor.Inparticular,ifweintroduceafactor 1= NIDfexp(Š1 g2 trd4xf2),the ( F ( A ) Š f )factorc anbeusedtocarryout theintegrationandleadstotheform Z = N IDA aF Š 1eSinv+ SGF, SGF= Š1 g2 tr d4x [ F ( A a)]2.(6. 2.7) wherew ehaveabsorbed Ninto N. Wecanpar ametrizethegaugegroupbyagaugeparameter ( x )sucht hat F ( Aa )= f ( x )for =0.Then

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3426.QUANTUMGLOBALSUPERFIELDSF( A a)= ID [ F ( A a ) Š f ]= ID [ F ]Š 1 ( ) = ID [ F ]= ID ID e F dx,(6. 2.8) wherew ehavewrittenaninteg ralrepresentationforthefunctional -function.Inthe secondlineof(6.2.8),and intheequationsbelow, F isevaluatedat =0.Toobtain F Š 1wereplace and byreal anticommuting(Faddeev-Popovghost)“elds c ( x )and c( x )(sees ec.3.7).Finally,wecanchooseforthegauge“xingfunctiontheform F ( Aa)=1 2 aA a.Then F =1 2 a a ,andwe have Z = N IDA aIDcIDceSeff, Seff=1 g2 tr d4x [ Linv( A a) Š1 F ( A )2+ ic F c ] =1 g2 tr d4x [ Linv( A a) Š1 4 ( aA a)2+ ic1 2 a ac ].(6.2 .9) (The i isforhermiticity.)Thegauge-“ xingtermcanbew ritteni ntheform1 2 A LA whereL=1 Š Tisthelongitudi nalprojectionoperator(L) a b=1 2 a b Š 1.The totalkineticop eratorbecomes (1+(1 Š 1)L),whichisinvertible:Minusitsinverse (thepropagator)is Š Š 1(1+( Š 1)L).IntheFermi-Feynmangauge, =1,the pr opagatoris Š Š 1. Thegauge-“xedLagrangian,includingghosts,isinvariantunderthe global BRST transformations: A a= i ac c=2 F =1 aA a, c = Š c2,(6. 2.10) withconstantGrassmannparameter .Thesetran sformationsarenilpotent: 2onany

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6.2.Gauge“xingandghosts343“eldvanishes(whentheantighostequatio nsofmoti onareimposed).TheWardidentitiesforthisglobalinvariancearetheSlavnov-Tayloridentitiesofthegaugetheory. b.SupersymmetricYang-Millstheory Fo rs up er sy mmetricYang-Millstheorywequantizefollowingthesameprocedure. Westartwiththef unction alintegralforagaugerealscalarsuper“eld V = VATA,where TAarethegeneratorsofthegaugegroup: Z = IDVeSinv( V ).(6. 2.11) Notethatinsupersymmetrictheoriesthenormalizationfactorof(6.2.3) N=1(see sec. 3.8.b).Theactionis Sinv=1 g2 tr d4xd2 W2= Š1 2 g2 tr d4xd4 ( eŠ VDeV) D2( eŠ VDeV) =1 2 g2 tr d4xd4 [ VD D2DV + higher Š orderterms ].(6.2 .12) Itisinvariantunderthegaugetransformations eV= ei eVeŠ i ,( 6.2.13a) or ,f or i n“nitesimal(see(4.2.28)), V = L1 2 V[ Š i ( +)+ cothL1 2 Vi ( Š )], LXY =[ X Y ].(6.2 .13b) Intheabeliancase,thisis V = i ( Š ).Thekineticoperatoris 1 2 withthesuperspin1 2 projectionoperator1 2 = Š Š 1D D2D,andis notinvertiblebecauseitannihilatesthechiralandantichiralsuperspinzeropartsof V : V0=0V = Š 1( D2 D2+ D2D2) V Wemustnowc hoose gauge-“xingfunctions.Correspondingtothechiralgauge parameterweneeda gauge-variantfunctionthatcanbemadetovanishbyasuitable gaugetransformation.Thereforeitmusthavethesamespinandsuperspinasthegauge

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3446.QUANTUMGLOBALSUPERFIELDSparameterandhenceshouldbechosenachiralscalar.Thegauge-variantquantity F = D2V isasuitablegauge-“ xingf unction.Foranychiralfunction f ( x ),weverify thatgaugetransformationscanbefoundtomake F = f .Forex ample,intheabelian case,underagaugetransformation F ( V ) F ( V)= D2V + i D2 ;ifwechoose i = Š 1D2( f Š D2V ),we “nd F= f Wede “nethefunctionaldeterminant ( V )= ID ID [ F ( V ,, ) Š f ] [ F ( V ,, ) Š f ].(6.2 .14) We“rstwrite( cf.(6.2.5)) Z = IDV Š 1( V ) [ D2V Š f ] [ D2V Š f ]eSinv.(6. 2.15) Asin(6.2.7),weaverageover f and f withaweightingfactorIDfID fexp ( Š1 g2 tr d4xd4 ff ),andobtaintheform Z = IDV Š 1( V )eSinv+ SGF,(6. 2.16) where SGF= Š1 g2 tr d4xd4 ( D2V )( D2V ).(6.2 .17) Wewrite ( V )= ID ID ID ID ed4xd2 F + F +d4xd2 F + F (6.2.18) wherewehavereplacedthe -functionsinvolving(anti)chiralquantitiesbyintegralrepresentationsinvolving(anti)chiralparamet ersandintegrationmeasures.Thevariational derivati vesof F F ,areevaluatedat= =0.Inth ef unction alintegral,where Š 1( V )appears,wer eplacetheparameters,byanticommutingc hiralghost“elds c c.Fina lly,we“nd Z = IDVIDcIDcID cID ceSinv+ SGF+ SFP,(6. 2.19) where

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6.2.Gauge“xingandghosts345SFP= itr d4xd2 c D2( V )+ itr d4xd2 cD2( V ) = tr d4xd4 ( c+ c) L1 2 V[( c + c )+ cothL1 2 V( c Š c )].(6.2.20) Integratingbyparts,wecanwrite( D2V )( D2V )=1 2 V ( D2 D2+ D2D2) V =1 2 V 0V Thequadraticpartofthegauge“eldactionhasnowtheform Š1 2 V (1 2 + Š 10) V = Š1 2 V [1+( Š 1Š 1)0] V ,(6. 2.21) andtheoperatorisinvertible.Toavoid Š 2termsinthepropagatorandthusbad infraredbehavior,wechoosethesupersymmetricFermi-Feynmangauge =1,which leadstoasimple Š 1pr opagator.(The Š signin(6 .2.21)leadstotheusualkinetic termforthecomponentgauge“eld: Š d4 V V A a A a.) Thequadraticpartoftheghostactionhastheform S(2) FP= tr d4xd4 ( c+ c)( c Š c )= tr d4xd4 ( cc Š c c ).(6.2 .22) Thechiralandantichiral cc and c c termsvanishwhenintegratedwith d4 andhave b eendropped.(Suchtermscannotbedroppedinthepresenceofsupergravity“elds: see,forexample,(5.5.16)). Thetotalactionisinvariantundersuper“eldBRSTtransformations.Thesetake theform V = V |= i c= LV[( c + c )+ cothL1 2 V( c Š c )], c=1 D2 F =1 D2D2V c=1 D2F =1 D2 D2V c = Š c2, c = Š c2,(6. 2.23) andtheinvariancecanbeusedtoderivetheSlavnov-Tayloridentitiesofthetheory. Beforeperformingperturba tionexpa nsions,werescale V gV .Thenall quadratictermsare O ( g0),c ubictermsare O ( g ),et c.Werescaleback gV V inthe eectiveaction.Alternatively,wesimplyprovideeachgraphwithafactor( g2)L Š 1,

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3466.QUANTUMGLOBALSUPERFIELDSwhere L isthenumbe rofloops. c.Othergaugemultiplets Wegivetwoother examplesofthegauge-“xingprocedure:Forachiralsuper“eld ,thesolutionofthechiralityconstraint,= D2,givesthekineticaction Sinv= d4xd4 D2 D2,(6.2 .24) andintroducesthegaugeinvariance = D€ €, = D,(6. 2.25) foranarbitraryspinorparameter (see(4.5.1-4)).Suitablegauge“xingfunctionsare thelinears pinorsuper“elds F= D, F€= D€ .(6.2 .26) Toobtainac onvenientgauge“xingtermweaveragewith f€M€f,where M€= D D€+3 4 D€D.(6. 2.27) Thisleadsto Sinv+ SGF= d4xd4 ,(6.2 .28) andastandard pŠ 2pr opagator. As econdexampleisfortheactionofthechiralspinorsuper“eldthatdescribes thetensormultiplet(4.4.46): Sinv= Š1 2 d4xd4 G2= Š1 8 d4xd4 ( D+ D€ €)2,(6. 2.29) withgaugeinvarianceunder = i D2DK K = K .(6. 2.30) Asuita blegauge“xingfunctionis F = Š i1 2 ( DŠ D€ €),(6.2 .31)

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6.2.Gauge“xingandghosts347where F islinear.Thegauge-“xedaction Sinv+ SGF=1 2 d4xd4 [ Š G2+1 F2] = Š1 4 d4xd2 [1 2 (1+ K K )+1 1 2 (1 Š K K )]+ h c =1 2 d4xd4 [ Š1 2 (1+ )(1 2 D2+ h c .)+1 2 (1 Š ) €i €] (6.2.32) (with K K asinsec3.11)takestwoconvenientforms: For =1, Sinv+ SGF= Š1 4 d4xd4 (D2+ € D2 €);(6.2 .33) for = Š 1, Sinv+ SGF=1 2 d4xd4 €i €(6.2.34) (cf.(3.8.36)).

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3486.QUANTUMGLOBALSUPERFIELDS6.3.Supergraphrules Givenanaction S (),wede “nethegeneratingfunctionalforGreenfunctions Z ( J )= ID eS ()+ J ,(6. 3.1) where J isasourceofthesametypeasthe“eld (gen eralifisgeneral,chiralifis chiral,etc .).Thege neratingfunctionalof connected Greenfunctionsis W ( J )= lnZ ( J ).(6.3 .2) Theexpectationvalueofthe“eldortheclassical“eld inthe presenceofthe sourceis ( J )= W J .(6. 3.3) Thisrelationcanbeinvertedtogive J ( ).Theeectiveaction( ),thegenerating functionalofoneparticleirreduciblegraphs,isde“nedbyafunctionalLegendretransform ( )= W [ J ( )] Š J ( ) .(6.3 .4) InthissectionwederivetheFeynmanrulesforthepertubativeexpansionofthe eectiveaction.ThederivationoftheFeynmanrulesforunconstrainedsuper“eldspresentsfewsurprises.Insteadofhaving d4x integralswehave d4xd4 integrals.Propagatorsareobtainedfro mtheinversesofthek ineticoperators,an dverti cescanberead directlyfromtheinteractionterms.However,forchiralsuper“eldstheFeynmanrules re”ectthechiralityconstraints. a.Derivati onofFeynmanrules Webeginbyderiv ingtherulesfortherealscalargaugesuper“eld.Thegauge “xedacti on(intheFermi-Feynmangauge, =1)reads SV= tr d4xd4 [ Š1 2 V V +1 2 [ V ,( DV )]( D2DV )+ ].(6.3 .5) TheFeynmanrulescanbereaddirectlyfromthisexpression:Thepropagatorisminus

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6.3.Supergraphrules349theinverseofthekineticoperator, Š 14( x Š x) 4( Š )or,in momentumspace, Š pŠ 24( Š ).Si ncethespinorderivatives D containe xplicit s,wedo not Fourier transformwithrespecttothe variables.(Ifone doesFouriertransformwithrespectto ,thereis littlecha ngeintheFeynmanrules.)Vertices canbereadfromtheinteraction terms.Thus,thecubicterm1 2 tr [ V ,( DV )]( D2DV )leadstoat hree-pointvertexwith factorsof Dand D2Dactingontwoofthelin es,andagrouptheoryfactor.Inadditionweintegrateover x sand sateachvertexor,equival ently,overloopmomentaand over sateach vertex. Theserulescanalsobeobtainedbystartingwiththefunctionalintegral: Z ( J )= IDVe[ Š1 2 V V + ILint( V )+ JV ]=eILint J IDVe[ Š1 2 V V + JV ]=eILint J e1 2 J Š 1J,(6. 3.6) whereinthelaststepwehaveperformedtheGaussianintegralover V .TheFeynman rulescanbeobtainedusing J ( x ) J ( x, ) = 4( x Š x) 4( Š )andex pandingtheexponentialsinpowerse ries.Thus,weobtainfactorsof ILint( J )co rrespondingtovertices,and the J operators,whenactingonthefactorsof J Š 1J removethe J sandp roduce pr opagators Š 1connectingthevertices.Theresultisexactlyasforordinary“eldtheory,withtheadditionalfeatureof d4 integralsateachvertex,andadditional 4( Š ) fa ctorsineachpropagator. Chiralscalarsuper“eldsusuallyhavea kinetic action(withchiralsources j j )of theform S(2)= d4xd4 Š1 2 d4xd2 m 2Š1 2 d4xd2 m 2+ d4xd2 j + d4xd2 j .(6.3 .7) ToperformtheGa ussianinte gration,werewritechiralintegralsasintegralsoverfull

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3506.QUANTUMGLOBALSUPERFIELDSsuperspace.Achiralintegral Ic= d4xd2 FG ,(6. 3.8) where F and G are arbitrary chiralexpr essions,canberewrittenas Ic= d4xd4 F Š 1D2G ,(6. 3.9) using Š 1 D2D2G = G (3.4.10)and d4xd4 = d4xd2 D2.TheGa ussianin tegral canberewrittenaseW0( j ) ID ID exp d4xd4 [1 2 ( )O O +( ) Š 1D2j Š 1 D2 j ](6. 3.10) where O O = Š mD2 1 1 Š m D2 .(6. 3.11) Theinverseof O O is O OŠ 1= m D2 Š m2 1+ m2D2 D2 ( Š m2) 1+ m2 D2D2 ( Š m2) mD2 Š m2 .(6. 3.12) Perfo rmingtheintegralweobtain W0( j )= d4xd4 [ Š j 1 Š m2 j Š1 2 ( j mD2 ( Š m2) j + h c .)].(6 .3.13) Forageneralinter actionLagrangian ILint(, )wecanwrite Z ( j )=ed4xd4 ILint j j eW0( j ),(6. 3.14) andtheFeynma nrules canbeobtainedfromthisexpression.Since j ( x ) j ( x, ) = D24( Š ) 4( x Š x)(3. 8.10),thereisanoperator D2actingoneachchiral

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6.3.Supergraphrules351“eldlineleavingavertex.Sim ilarly,thereisanoperator D2actingoneachantichiral lineleavingav ertex.However,atapurelychiralvertex,e.g.d2 n,weuseoneof thesefactorstoconvertthe d2 integraltoa d4 integral.Thereforeatsuchverticeswe omitonefactorof D2. Wenows ummarizetheFeynmanrulesforintera ctinggaugeandchiralsuper“elds. (a)Propagators: VV : Š 1 p2 4( Š ), (6.3.15a) : 1 p2+ m2 4( Š ),(6.3 .15b) : Š mD2p2( p2+ m2) 4( Š ),(6.3 .15c) : Š m D2p2( p2+ m2) 4( Š ).(6.3 .15d) Inthemassivecase,the pŠ 2factorsint heand p ropagatorsarealwayscanceled bynumeratorfacto rs(e.g.,fortheverticesgive D2factorsand,aswediscusslater, weobtain D2D2 D2= Š p2 D2).Inthemasslesscasethesepropagatorsareabsent. (b)Vertices:ThesearereaddirectlyfromtheinteractionLagrangian,withthe additionalfeaturethatforeachchiralorantichirallineleavingavertexthereisafactor D2or D2actingonthecorrespondingpropagator,andtherulethatatpurelychiralor antichiralverticesweomitone D2or D2factorfrom amongtheonesactingonthepropagators. (c)Weintegrateover d4 ateachvertex,andinmomentumspacewehaveloopmomentumintegralsd4p (2 )Š 4foreachloop, andanoverallfactor(2 )4 ( kext). (d)Toobtaint heeectiveaction,wecomputeone-particle-irreduciblegraphs. Foreachexterna l linewithoutgoi ngmomentum ki,wemulti plyb yafactord4ki(2 )Š 4( ki)wheresta ndsforanyofthe“eldsintheeectiveaction.Foreach

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3526.QUANTUMGLOBALSUPERFIELDSexternalchiralorantichiralline,wehaveaor facto r,butno D2or D2factors. (e)Finally,theremaybesymmetryfactorsassociatedwithcertaingraphs. AnalternativederivationoftheFeyn manrulesforchiralsuper“eldscanbe obtainedbysolvingthechiralityconstraintsintermsofanunconstrained“eld(seesec. 4.5a): = D2, = D2 ,(6.3 .16) whereisageneral,complexscalarsuper“eld.Theaction,includingsourceterms, b ecomes S = d4xd4 [( D2 )( D2)+ ILint( D2, D2 )] + d4xd2 ( D2)( Š1 2 m D2+ j )+ h c ..(6.3 .17) Chiralintegralscanberewrittenasfullintegralsbyusingupa D2factor.Werecall thatintermsofwehaveanabeliangaugeinvariance, + D€ €(4.5.4).Consequently,thekineticoperatorappearingintheaction, D2 D2,isnotinvertible.Asdiscussedinsec.6.2,wecan“xthegaugeandarriveataninvertiblequadraticaction(the ghostsdecouple) S(2)= d4xd41 2 Š mD2 Š m D2 .(6. 3.18) TheFeynmanrulesarenowthenaiveonesandareidenticaltotheoneswehave obtainedbefore(afterusingthe D2, D2fa ctorsattheverticestosimplifythepropagators,obtainedfrom(6.3.12)).Inparticular,from ILint( D2, D2 ) ,w e again“ndfactors of D2, D2actingonthepropagators,excepttha to ne suchfactorismissingatpurely (anti)chiralvertices,since weconv erteverywheretofull d4 integrals. Itissimpletoobtainthesupergraphrulesforthetensormultiplet,withgaugeinvariantaction S = d4xd4 f ( G ), f ( G )= Š1 2 G2+ ... ,(6. 3.19) withpropagator Š 2 pŠ 4 D24( Š )(andthe hermitianco njugat efor )from

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6.3.Supergraphrules353(6.2.33).However,thereisamuchsimpler formoftheruleswhichresemblestherules forthescal armultiplet(towhichthetensormultipletison-shellequivalentbyaduality transformation:seesec.4.4.c.2).We“rs tnotethatthevertex ateitherendofa pr opagatorhasa D atthevertex(from f ( G )with G = D + D ),andnexttoita D2(asoccursattheendofanychiralpropagator,rule(b)above;thiskillsthe D partof thevertex).Alsonotethatt hespinorindexatthevertexcontractsdirectlywiththe correspondingspinorindexofthepropagator(becausethevertexisafunctionofonly G =1 2 D+ h c .).Contractingtheses pinorindices,andintegratingbypartsall D s fr omtheverticesontothepropagators,weobtainthesameexpressionfortheand p ropagators(withthesamevertices),whichcannowbeaddedtogether(i.e.,the totalcontributionfromgraphswithbothtypesofpropagatorsisthesameasthatfrom onlyonetype,butwithanoverallfactorof2foreachpropagator).Therulesarethus castintothefollowingform:Allverticesarenowsimply consta nts, readfromtheexpansionof f ( G )in G .T he reisonlyonetypeofpropagator,withnospinorindices,whichis Š 1 p2 D D2D4( Š )= Š 1 2 4( Š ).(6.3 .20) (Thealgebrafromthevariouscontributingfactorsis D D2D2 D2D = D D2D .)Each externallinegetsafactorof G .Ifweweretop erformthesamerearrangementofvertex factorsforthesupergraphsofthedualscalar-multiplettheory,wewouldobtain0insteadof1 2 ,theexternal linefactorswouldbe+ ,andtheconstantsatthevertices wouldbeo btainedfrom f(+ )intermsofthefunction fdualto f (seeagainsec. 4.4.c.2).Theon-shellequivalencethenfollowsfromthefactthatthecombinatorics resultingfromusing1 2 =1 Š 0,w ithapropagator1collapsingtoapoint(in andx ), perfo rmstheduality,wherefortheexternallines G =+ ons he ll. b.Asamplecalculation Wenowgiveanexamp leinatheoryofamasslesschi ralsuper“eldinteracting withagaugesuper“eld V .Wecom putetheone-loopcontributionfromthechiralsuper“eldto the V two-pointf unction.Therelevantinteractionisobtainedfrom eV= + V + ... .We “ndacontributiontotheeectiveaction,accordingto ourrulesandFig.6.3.1,

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3546.QUANTUMGLOBALSUPERFIELDS p k + p V ( Š k 2) V ( k 1) Fig.6.3.11 2 d4k (2 )4 d41d42V ( Š k 2) V (+ k 1) d4p (2 )4 D1 24( 1Š 2) D 2 2p2 D2 24( 2Š 1) D 1 2( p + k )2 .(6. 3.21) Notethatintheaboveexpression D1 = 1 +1 2 1€p€, D1€= 1€ Š1 2 1 ( k + p )€, D2 = 2 +1 2 2€( k + p )€, D2€= 2€ Š1 2 2 p€.(6. 3.22) Althoughwedonotindicatethemomentumdep endenceexplic itly,itisimplicitthatthe momentumisthat leaving thevertexthroughthepropagatoronwhichtheoperatorsact. (Fromthe V2interacti ontermwealsoobtainatadpole-typediagram;itscontributioncancelsasimilarcontributionfromthediagramweareconsidering,orvanishesifwe usedimensi onalregularization). The D scanbemanipulatedlikeordinaryderivatives.TheyobeyaLeibnitzrule, andatransferrule 4( 1Š 2) D€ 2( p )= Š D€ 1( Š p ) 4( 1Š 2),(6.3 .23) whichcanbecheckedbyexaminingtheexplic itformoftheoperators.Anotherexample ofthetrans ferruleis

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6.3.Supergraphrules355D1 24( 1Š 2) D 2 2= D1 2 D1 24( 1Š 2) = D1 2D1 24( 1Š 2).(6.3 .24) Insideintegralsthe D scanbeint egratedbyparts(seesec.3.7).Thus d4 [ D( p ) f ( p )] g ( Š p )= Š d4 f ( p ) D( Š p ) g ( Š p ).(6.3 .25) Thiscanbemosteasilyunderstoodin x -space.Since D= +1 2 i €€weared oing integrationbypartsin andin €. Armedwiththesefactswereturntotheevaluationoftheexpressionin(6.3.21). Wecon centrateonthe dependenceandwritetherelevantpartas d41d42V ( Š k 2)[ D1 2 D1 212][ D1 2D1 212] V ( k 1).(6.3 .26) Wehave a bbreviated 4( 1Š 2)= 12.Wenowinte gratebypartsand“nd“rstofall [ D2 D2 ][ D2D2 ] V = D2 D2[( D2D2 ) V ] = D2[( D2 D2D2 ) V +( D D2D2 ) DV +( D2D2 ) D2V ] = D2[ Š p2( D2 ) V + p€( D€D2 ) DV +( D2D2 ) D2V ],(6.3 .27) whereweh aveused( D )3=0andthe anticommutationrelations { D, D€} = p€when actingonthepropagatorwithmomentum p Be foreproceedingwemakethefollowingimportantobservation:Since 4( )= 2 2,multi plyingtwoidentical -functionstogether,ormultiplyingoneby giveszero.Wehavethereforethefollowingrelations: 2121= 2112=0, 21D21=0, 21D221=0, 21D D€21=0, 21D D221=0,

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3566.QUANTUMGLOBALSUPERFIELDS21D2 D221= 21 D2D221= 211 2 D D2D21= 21, 21D D2D21= C21.(6. 3.28) Intheserelations,weobtainanonzeroresultonlyifallthe sinthesecond -function areremovedbydierentiation.Hencetwo D sandtwo D saren eededandonlytheir momentumindependentpartscontribute.Expressionsofthiskind,butwithmore D s, canbereducedtooneoftheaboveformsbyusingtheanticommutationrelations.In theexpressionswithfour D stheorderisirrelevant(exceptforproducingsomeminus signs). Returningtoour calculation(6.3.27),andletting D2actonthefactorstoitsright, weseethato utoftheaprioripossiblesixterms,onlythreesurvive: [ Š p2( D2D2 ) V Š p€( D€ D€D2 ) D€DV +( D2D2 )( D2D2V )].(6.3.29) Finally,using D€ D€= €€ D2we “nd 4( 1Š 2)[ Š p2Š p€ D€D+ D2D2] V ( k 1).(6.3 .30) Insertingthisresultintotheoriginalintegral(6.3.21),weusetheremaining -functionto dothe 2integral,and“nallyobtain1 2 d4k (2 )4 d4 V ( Š k )[ d4p (2 )4 Š p2Š p€ D€D+ D2D2p2( k + p )2 ]V ( k ).(6.3.31) Theresultconsistsofanordinaryloopmomentumintegral,withusualpropagatorsand somemomentumfactorsinthenumerator,andoperators D D ,actingont heexternal super“el ds( i.e., D and D dependon k ,not p ).The p2termiscan celedbythetadpole di agrammentionedabove,(orgiveszeroindimensionalregularization)sothatthe“nal contributiontotheone-loopself-energyislogarithmicallydivergent.Thisisaconsequenceofgaugeinvariance.Theremainingtermsinthenumerator,inagauge-invariant regularization(suchasdimensional),combinetoform1 2 D D2D,giv ingaresultproportionalto W2.

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6.3.Supergraphrules357c.Theeectiveaction Intheexampleabove,the -functionhasreducedtheexpressiontoonewhich involvesasingle .Thisi sagenerala ndimportantresultofsuper“eldperturbationtheory.Theeectiveactionisasumoftermsinvolvingproductsof“eldsevaluatedatdifferentpoints,andaGreenfunctionwhichisanonlocalfunctionofitsarguments: =n d4x1... d4xnd41... d4nG ( x1... xn; 1... n)( x1, 1) ... DV ( xi, i) ... (6.3.32) Itturnsout,however,thatbymanipulationofthecontributionsfromanygraph,wecan reduceittoanexpressionthatislocalin i.e. =n d4x1... d4xnd4 G( x1... xn)( x1, ) ... DV ( xi, ) ... .(6. 3.33) Wedothisasfo llows:Consideranarbitrary L -loopcontributiontotheeective action.Itconsistsofpropagators,withfactors 4( iŠ i +1)and D operatorsactingon them,ext ernalsuper“eldfactors,and d4iintegrals.Wechooseanypropagatorfroma particularvertex v toanothervertex v,andintegr atebypartstoremoveallthe D s fromits -function.Theoriginalcontributionnowbecomesasumofterms.Ifthereare otherpropagators,eachofwhichconnects v and v,weusetherelatio ns(6.3.28):The termsvani shunless each oftheother -functionshasexactlytwo D sandtwo D sacting onit,inwhichcasetheycanbereplacedby1.Wenowusethefree -functiontodothe -integralat vandshrinkallthepropagatorsbetweenthetwoverticestoapointin -space.Werepeattheprocedure,choosingapropagatorleadingtoanewvertex v, untilweh averemovedall -functionsandperformedall -integralsexcepttheoriginal oneat v .Whe neverweh avemorethantwo D sandtwo D sonalineweusetheanticommutationrelationstoreplace D D pairsbymomenta.Wea releftwithasumof terms,allwithasingle integral,andvariousfactorsofloop-momentacomingfromthe anticommutatorsof D s,asw ellas D factorsactingontheexternalsuper“elds,coming fromtheintegra tionbyparts. InthecourseofevaluatingFeynmandiagrams,wemayencounterloop-momentum ultravioletdivergences,andasuitableregularizationprocedureisneededtohandle them.Wediscussregularizationissueslateron.Forthetimebeingweassumethat

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3586.QUANTUMGLOBALSUPERFIELDSthereexistsapro cedurethatallowsustocarryoutthemanipulationswehavedescribed aboveinsidemomentumintegrals. Theexpressionfortheeectiveactionin(6.3.33)revealsoneimportantfact:We haveendedupwitha d4 integral,eventhoughintheoriginalclassicalactionwemay havehad d2 integrals.ThisisaconsequenceofourFeynmanrules:Allourvertices carry d4 integrals,andnowhereinourmanipulationsdoesa d2 appear.Inparticular, iftheoriginalactio nhad purelychiral d2 massorcubicinteractionterms2or3, radiativecorrectionsdonotinduce“niteorin“nitemodi“cationsoftheseterms.Thisis the no-renormalizationtheorem forchiralsuper“elds.Massesandcouplingconstants are renormalized,butonlyasaconsequenceofwavefunctionrenormalization.(Any d4 integralcanbewrittenasa d2 integralanda D2operatoractingontheintegrand; however,thiswillnotproducetheabovete rms.)Thistheoremisvalidinperturbation th eory.Sofarnoonehassucceededingivingexamples,infourdimensions,whereit mightfailnonperturbatively,butaproofofitsgeneralvaliditydoesnotexist.Even withinperturbationtheorythereexiststhepossibilityofapathologicalinfrared-type behaviorwhichm ightinvalidateit.Forexample,ifinthecourseofevaluatingtheeectiveactionatermd4 2D2 werepro duced,the D2operatorwhichcomesfromconvertingthe integraltochiralform,whenactingonthechiral“eld,wouldgive D2D2 Š 1=andwe woulde ndupwithacontributiontothechiralcubicvertex. Whethersuchpathologicalbeh aviorcanbe obtainedinanycalculationwithasensible infraredregularizationisdoubtful. d.Divergences Wenowdiscussth edivergencestr uctureoftheeectiveaction.Therearetwo issuesinvolved:Wem ustdeterminewhichtermsintheeectiveactionaredivergent (powercounting),andwhichtermsinthecla ssicalactionleadonlytodivergencesthat canbeabsorbedinarenormalizationofthepa rameters(renormaliz ableinteractions). Werest rictourdiscussiontointeractinggaugeandchiralscalarsuper“elds(withnonegative-dimensioncouplingconstants). Thepossibledivergencesoftheeectiveactioncanbeunderstoodbystraight powercounting(sees ec.6.6)orsimplybyadimensionalargument:Thedivergentparts ofgraphsthatcontainnosubdivergencesgiverisetolocaltermsintheeectiveactionof

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6.3.Supergraphrules359theform = d4xd4 IP (, V D, ...),(6.3.34) where IP isapolynomialinthe“eldsandtheirderivatives.Sincetheeectiveaction mustbedi mensionless,and d4 hasdimen sion2, IP mustal sohavedimension2.has dimension1, Dhasdimension1 2 ,and V isdime nsionless.Therefore,graphswithmore thantwoexternalsareconvergent.Aor p ropagatorproducesanumerator fact orof m whichcontributestothedimensionof IP andthereforereducesthedegreeof divergence.If IP ismadeupofonlychiralsuper“eldsthe integrationwillgivezero unlesssome D s(atleasttwoofthem,to contractindices)arepresenttomaketheintegrandnonchiral,andagainthe D scontri butetothedimensionof IP reducingthe numberof“eld sthatcan appear.Finally, if gaugeinvariancerequires V toappear throughits“e ldstrength W= i D2DV (oritsnonabeliangeneralization),thislimits thepossibledivergencesinvolving V “elds.(Thisisanoversim p li“cation:Wemustuse thefullmachineryofSlavnov-Tayloridentities,atleastinthenonabeliancase,oruse th eb ac kg round-“eldmethod(seesec.6.5)toanalyzethedivergencesinvolvinggauge super“elds.) Thenetresultoftheanalysisistoestablishthattheonlylocaldivergentterms containatmostoneandone a nd,whiletheycontainanarbitrarynumberof V factors,theseenterinama nnerwhichiscontrolledbytheSlavnov-Tayloridentities.Fora renormalizabletheoryofchiralscalarmultipletsinteractingwithavectormultiplet(we omittheghostterms)therenormalizedclassicalactionhastheform d4xd4 [ RegRVRR+ tr RVRŠ tr R Š 1( D2VR)( D2VR)] +1 gR 2 d4xd2 trWR 2+[ d4xd2 IP (R)+ h c .],(6. 3.35) wherethesubscript R labelsrenormalizedquantities.(Intheexponentialwehavewrittenexplicitlythegaugecouplingconstant g thatwenormallyabsorbinto V .) Since V isdime nsionless, VRisingenera lano nlinearfunctionof V i.e.,thewavefunctionrenormalizationf actormaybeafunctionof V :Wecanhave functionalrenormalizations VR= f ( V ),whereeachcoecientintheTaylorexpansionof f isa

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3606.QUANTUMGLOBALSUPERFIELDSrenormalizationconstant.Sinceallsuchrenormalizationsareproportionaltothe“eld equations( S = S V V ),theyvanishonshell.(Suchnoncovariantrenormalizations areavoidedinthebackground“eldgaugesthatwediscussbelow.) Ghostsaredescribedbychiralsuper“elds whichfollowthesamerules.Thedivergencesofthetheoryarealllogarithmic,exceptthatoftheFayet-Iliopoulosterm,which isquadratic.(Howev er,asweshalldiscussinsec.6.5,thistermisnotproducedby radiativecorrections.) Ingeneral,renormalizableinteractionsareassociatedwithdimensionless(orpositivedime nsion)couplingconstants.ForFeynmangraphs,sinceateachvertexwehavea d4 integralwithdimension2anda d4x integralwithdimension Š 4,wemayallowupto theequivalentoffour D sateachvertex.Thisisindeedthecasewiththegauge“eld self-couplings,andalsotheusualverti cesinvo lvingchiralsuper “elds,wherethe D factorscomefromourFeynmanrules. Ontheotherhandatermsuchas 2,or4, wouldleadtoa nex cessof D sattheverti cesandanonrenormalizabletheory. e.D-algebra Inthenextsectionwegiveanumberofexamplesofevaluationofsupergraphs. Aspreparationwediscussseveralsimpli“cationsthatweuseinperformingthemanipulationofthe D sandthe integration.Thenumerousintegrationsbypartsthathaveto beperfo rmedcanleadtolongintermediateexpre ssions,andalotofeort(andpaper) canbesavedbydoingthemanipulationsdirectlyonthegraphs. Wedrawthes upergraphandindicateonit,adjacenttothevertices,the D factors actingonthepropagatorsintheorderinwhichtheyact.Weignoresignshavingtodo withtheorderingofthe D s:Thesewillbedete rminedlater.Thus,anexpressionsuch as D2D4( Š ) D D€D ,witht helastthree D sacting backwardsonthe argumentofthe -function(andthusintheorder D“rst,then D€next,etc.) ,wouldbe representedonthegraphasshownin“g.6.3.2:

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6.3.Supergraphrules361 D2DD D€DFig.6.3.2 Thetransferrulecanbeimplementedbyslidingthe D stotheleft,keepingtheorder. Thiscorrespondstowriting Š D2DD D€D4( Š )with Dacting “rstonthe argumentofthe -function.Wemustkeeptrackofthe Š signcomingfromtransferring anoddnumberof D s.(Notetheorderintheseexpressions,e.g., D1 12D 2 = D1 D2 12= Š D1 D1 12.) Weuset hecommutationrelationstoreplacetherightmost D D by p€.(Si nce p ishermitian p€= p€butwemainta inthedistinctiontokeeptrackoftheorderin whichthe D sappeared.)When ween counterexpressionssuchas D2 D2D2wereplace themwith Š p2D2. Theintegrationbypartscanalsobecarriedoutdirectlyonthegraphs.Forexample,weshowin“g.6.3.3theintegrationbypartsonavertexcomingfromthe V interaction: D2 D2 D€D2 D2D2 D2D2 D€Fig.6.3.3 Startingfromagiven graphingeneralweobtainseveral,becauseintegrationby partsgivesseveralcontributions.Weremovethe D sfromanygivenlineandusethe -functiontodooneofthe integrals,thuscontractingthelinetoapointin space. Wen eednotindicateexplicitlythiscontract ion:Alinewithoutanyoperatorsonitis understoodtobecontracted.Wheneverseverallinesconnectthesamepairofvertices, ifallthelines(otherthantheonewehaveclearedof D s)haveexactlytwo D sandtwo D seach,weuse(6.3.28)toreplacethemby1.Ifanylinehasfewer D sor D s,the contributionvanishe s.Ifanylinehasmore D sor D s,weusetheanticommutation

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3626.QUANTUMGLOBALSUPERFIELDSrelationstoreducetheirnumber.Intheendwehaveasumofgraphs,withmomentum factorsfromt heanticommut ators,and D sactingontheexternallinesonly. Tomakethispro cedureclear,weredotheexampleconsideredabove(“g.6.3.1, (6.3.21-31)),workingdirectlyonthegraph,asshownin“g.6.3.4: D2 D€DD2D D€D D2 D2 D2 D2 D2 D2 D2 D2 D2 D2 D2 D2D2D2D2D2D2D2D2D2D2D2D2DFig.6.3.4 Inthelaststepwehaveonlyindicatednonzerocontributions(theothersvanishtrivially b ecauseof(6.3.28).) Inthecourseofthemanipulationsonthegraphs,wemustkeeptrackof Š signs comingfromtransfersandfromintegrationbyparts.However,weneednotkeeptrack of Š signsthatcomefrompassinga D pastan other D ,norfr omsignsthatcomefrom raisingorloweringindices.(Onthegraphs,wedonotindicatetherelativeorderof D s ondierentlines,norwhichindicesareupo rdown).T hesesignscanbedeterminedat theendofthecomputationint hefollowingmanner:Ontheoriginalgraph,wehavefactorssuchas D2 D2=1 4 DD D€ D€,andalso,froma vertexsuchas V ( DV )( D2DV ), adjacent factors Dand1 2 D€ D€Dwherewedeterminetheinitialsignbyrequiringthat inanycontractedpair,the“rst D or D hastheupperindex.Thesevariousfactorsmay endupinadierentorderinthe“nalexpression,e.g., D... D€... D... D€,po ssibly actingondierentsuper“elds;however,westillwriteacontractedpairwiththe“rst i ndexraised.Todeterminethe“naloverallsign,wecountthenumberoftranspositions neededinthe“nalexpressiontobringthe D sbacktotheorigina lorder.Thi sistrue evenifsomeofthe D shaveb eenreplacedbymomenta:Anexpressionsuchas p€will

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6.3.Supergraphrules363correctlykeeptrackofthetranspositions.Wedo not counttranspositionsofcontracted pairs,sincetheconventionforraisingandloweringindicescancelssuchsigns: XY=+ YX.Aquickwaytoco untthetranspositionsis todrawlines connecting allcontractedpairswithlines,andcountthenumberofintersections:anoddnumber meansanoddnumberoftranspositions,andhencea Š sign,whereasanevennumber meansno Š sign. Therearemanyothertricksthatonecanusetosimplifythemanipulations.We givethefollowingtwinglingrulewhichisoftenuseful: D2DDDDDD Fig.6.3.5 Wearenowre adytoconsiderfurtherexamplesofgraphevaluation.

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3646.QUANTUMGLOBALSUPERFIELDS6.4.Examples Inthissectionwegiveanumberofexamplesofsupergraphcalculations.Mostof theexamplesw ereencounteredinvariouscalculationsthathavebeenperformed.For morecomplicatedoneswereferthereadertothecalculationsofthe3-loop -functionin N =4 Ya ng -M ills,andthe3-and4-loop -functionintheWess-Zuminomodel. Wedoourm anipulatio nsdirectlyonthegraphsuntilonlyanordinarymomentum integralremains.Fornotationalconveniencewesometimesindicateafactor p2mult iplyingapropagatorwiththesamemomentumbya drawnonthecorrespondingline.In thecaseofalinewithno D sactingonit,wesometimesleaveitinthegraph,whileat othertimes,w henwedraw -spacegraphs,wecontractitout.Toestablishtheprocedurewebeginwithsomesimpleexamples. Forthema ssiveWess-Zuminomodelweconsider“rstsomeselfenergygraphs: (1) D2 D2 Fig.6.4.1 d4 ( Š p )( p ) d4k (2 )4 1 ( k2+ m2)[( k + p )2+ m2] .(6. 4.1)

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6.4.Examples365(2) D2 D2 Š mD2( k + p )2 Š mD2( k + p )2 Š mD2p2 m D2Fig.6.4.2 Wehave used D2D2 D2= Š p2 D2.Atachiralvertexthe D2factorsc anbeputoneither linebyintegra tionbyparts. Theresultisd4xd4 =0b ecausetheintegrandischiral.Weconsidernextatrianglediagramwith3and 3verti ces: (3) m D2Š mD2p2 m D2D2 D2 D2 Fig.6.4.3 Thus,theone-loopcontributionstothisthree-pointfunctionarezerointhemassless caseandnonlocalinthemassivecase.

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3666.QUANTUMGLOBALSUPERFIELDS(4)For m =0,with3and 3verti ces, D2 D2 D2 D2 D2 D2 D2 D2D2D2D2D2D2D2D2D2 Fig.6.4.4 Inthesecondgraph,thesmallloopiscontractedtoapointin -space,afterwhichthe D2D2operatorscanbetransferredacrossitandallactonthesameline.The“nal graphshowstheactualmomentum-spacediagramonewouldhavetoevaluate.Apropagatorhasbeencanceledby (5)Againinthemasslesscase, A B C E F D2D2D2D2D2D2D2 D2 D2 D2 D2 D2 D2D2 Fig.6.4.5a Wehave placedthe D sand D soncertaintwoofthethree linesforconve nience,and havelabeledthevertices.Inthesecondgraphwehavetransferred D2, D2fromC,B,to E,F,respectively.Wenowintegratebypartsthe D2factor atE.Thiswillgenerate threeterms.

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6.4.Examples367 D2 D2 D2 D2 D2 D2 D€ D€D2D2D2D2D2D2D2D2D2D2 Fig.6.4.5b Byexaminingthe -spaceloopsAECBAandEFBCE,itisclearthatineachgraphthe D2sonABandBF,respectively,mustgiveasingletermwhenintegratedbypartsat theirr espectivevertices(AandF).Weobtain k q D D D2 D2 k€q€Fig.6.4.5c Wehave used D€D2 D2D = k€ D D2D = Š k a .(6. 4.2) Ournextexamplehas eVinterac tions:

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3686.QUANTUMGLOBALSUPERFIELDS(6) A B C E F D2D2D2D2 D2 D2 D2 D2Fig.6.4.6a IntheloopBCFwehavejusta D2 D2factor,sow econtr actittoapoint.Similarly,we contracttheAElinetoapoint.Forclaritywedrawa -spacegraphwhere q and h are themomentaoftheABandEFlines: h q A B C E F D2D2D2 D2 D2 D2Fig.6.4.6b Weintegr atethe D2factorothemiddleline.Itcannotgoon soitmustg ooneither thetoporbottomline,orsplit.Becauseof(6.3.28)the D2factormustfo llowit.This isthesam eascom puting D2D2[ ( q ) ( h )]wherethe sarechiral.Theresultissimply Š ( q + h )2 ( q ) ( h ).Ther efore,the D manipulationis“nishedandweobtain Š ( Š p )( p )( q + k )2mu ltiplyingastandardmomentumintegralwiththepropagatorsoftheoriginaldiagram. We givenowanexampleinnonabelianYang-Millstheory:

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6.4.Examples369(7) k V DD D2 D2D2 D2D2 D2D D€D2 D2DD D D€ ( p ) ( p) Fig.6.4.7 V ( Š p Š p) D( p ) D€ ( p) d4k (2 )4 k€k2( k Š p )2( k + p)2 .(6. 4.3) (8) N =4 Ya ng -M illstheory. Insec.4.6.bwehavegiventheclassicalactionofthistheoryintermsof N =1 super“elds.Herewediscusssomeofitsquantumproperties. Thetheoryisdescribedbysuper“elds V ,i( i =1,2,3),allinthe adjointrepresentationofanarbitrarygroup,andwithinteractionsgovernedbyacommoncoupling constant g .Itiscla ssicallyscaleinvariant,andbothcomponentandsuper“eldcalculationshaveestablishedthatits -functionvanishestothreeloops,sothat,perturbatively, thescaleinvariancesurvivesquantization.Proofsexistthatextendthisconclusiontoall ordersofpertur bationtheory.Herewediscusssome oftheexplicitsupergraphcalculationsfores tablis hing ( g )=0, andleavethegeneralargumentstosec.7.7. Weaddtothecl assicalaction(4.6.38)thegauge“xingandghostterms (6.2.17,20-22)withgaugeparameter =1+ O ( g2).To O ( g0)thisc hoiceg ivesthe Fe rmi-Feynmangaugeandapropagator Š 1thatavoidsseriousinfraredproblems. Ho we ve r,thetransversepartoftheself-energyreceives(local)radiativecorrections, whereasthelongitudinalpartdoesnot(asfollowsfromtheWardidentities).Tostayin theFermi-Feynmangauge,wemustmaintaint heequalityofthelongitudinalandtransverseparts,a ndwedothisbyadjustingthe O ( g2)par tsof ineachordero fperturbationtheory(actually,theradiativecorr ectionsvanishatoneloop,andonlyariseat O ( g4)).

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3706.QUANTUMGLOBALSUPERFIELDSAtthecla ssicallevelthe O (4)invarianceofthetheoryrequirestheequalityofthe gaugeand123couplingconstants.Althoughthegauge“xingprocedurebreaksthe O (4)symmetry(thiscouldbeavoidedifwehadan N =4super“eldfo rmalism),gauge in va rianceshouldinsurethatthecouplingconstantsreceiveacommonrenormalization. Therefore,thetheoryhasonlyone -function,whichwecancompute,forexample,by comparingtherenormalizationofthe Cijkijkvertexf unctionandthe iiwave functionrenormalization.However,thevertexbeingchiral,receivesnoradiativecorrections(seesec.6.3),sothattoestablish ( g )=0itissu cienttoshowthatthe iiself-energyis“nite.(Weobservethatif O (4)invarianceofthequantumeectiveaction werenotspo iledbythegauge“xingpro cedure,“nitenesstoallorderswouldfollow immediately:The “nitenessofthe Cijkijkvertexwouldi mplythe“nitenessofall otherlocaltermsintheeectiveaction.Inprinciple,thedesiredresultshouldstillfollowfromthe O (4)Wardidentities,butinpracticethenonlinearityofthetransformations(4.6.39,40)makesthemdiculttoapply.) Tolowordersin V (suci entforthethree-loopcalculation)theactionis S = tr d4xd4 { iiŠ1 2 V V + cc Š c c + g [ i, V ]i+1 2 gV { DV D2DV } +1 2 g ( c+ c)[ V c + c ] +1 2 g2[[ i, V ], V ]i+1 8 g2[ V DV ] D2[ V DV ] +1 6 g2( D2DV )[ V ,[ V DV ]]+1 12 g2( c+ c)[ V ,[ V c Š c ]]1 3! g3[[[ i, V ], V ], V ]iŠ1 2 (1 Š 1) V 0 V + ... } + tr { d4xd2 ig1 3! Cijki[j,k]+ h c } (6.4.4) with V = VATA,i=iATA, c = cATA, [ TA, TB]= ifAB CTC, fAB CfDC B= Š k AD.(6. 4.5)

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6.4.Examples371UsingtheFeynmanrulesitistrivialtoseethatattheone-looplevel,intheFermiFeynman gaugede“neda bove,the selfen ergyisidenticallyzer o.Thecontr i butions fromthetwo graphsbelowcancel: D2D2 D2 D2 Fig.6.4.8a Itiseasytoverifythattheone-loopcorrectionstotheghostandvectorself-energiesalso completelyvanish.Fortheformer,thisistrueinanytheory,butforthelatteritisdue tothemultiplicity(3)ofthechiralmultiplets,whichleadstocancellationsamongthe threegraphsbelow: Fig.6.4.8b Thisresultistriviallytrueinthebackground“eldmethod:(seesec.6.5).Then the“eld V doesnotcontributeandthethreechiral“eldsexactlycancelcontributions from three chiral ghosts.Thisalsooccursforthe V thr ee-pointfunction,andissucienttoestablishinanindependentwaythat (one-loo p)=0.Wereferthereaderto theliteratureforotherone-andhigher-loopcalculationsandsummarizethesupergraph results:(a)One-loopthree-pointfunctionsare“nite.The V -“eldfour-andhigher-point functionshaveadivergencethatcanberemovedbyanonlinear V -“eldreno rmalization. (The divergenceneverarisesinthebackground“eldmethod.)(b)Atthetwo-looplevel theghost self-energ iesarestillzero,wh ereasthosefor V andiareonly “nite.(Consequently,thehigher-ordercontributionstothegaugeparameterare O ( g4).)(c)Atthe three-loopleveltheiself-energyis“nite,thusensuringthevanishingofthe -function. Wepres entargumentsforprovingtheresultstoallordersinsec.7.7.

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3726.QUANTUMGLOBALSUPERFIELDSThevanishingofthe -functiontoallordersofperturbationtheoryleadstothe conclusionthat N =4 Ya ng -M illsisa“nitefour-dimensional“eldtheory(uptogauge artifacts;e.g.,exceptinsupersymmetricba ckgr oundorlight-conegauges,divergences arepresent,butonlyingauge-dependentquantities). Anothert heorywithinteresting“nitenesspropertiesis N =2 Ya ng -M illstheory. Ithasone-loopdivergences,butis“nitetoallhigherordersofperturbationtheory(as exp licitlyveri“edattwoandthreeloops),makingitsuperrenormalizable.Bycoupling anappropriatenumberof N =2hypermul tipletsto N =2 Ya ng -M illstheory,onecan arrangefortheone-loopdivergencestoca ncel,andthusconstructacompletely“nite theory(inperturbationtheory).(Inthespecialcaseofoneadjoint-representation hypermul tiplet,w eobtain N =4 Ya ng -M illstheory.)

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6.5.Thebackground“eldmethod3736.5.Thebackgro und“eldmethod a.OrdinaryYang-Mills Thebackground“eldmethodisextremelyusefulinsupersymmetricYang-Mills theory,and essentialinthequantumtheoryofsup ergravity.In thissectionwereview thebackground“eldmethodforordinaryYang-Mills,andthenextendittothesupersymmetriccase.Theextensioninvolvessomesubtleties,primarilybecauseofthenonlinearityofthegaugetransformations. Ingaugetheorieswestartwiththegauge-“xedfunctionalintegral(6.2.9)and intro ducesourcescoupledtothe“elds,de“ning Z ( J )= IDAIDcIDceSeff+ JA, JA d4xJ aA a.(6. 5.1) Weintr o duce W ( J )= lnZ ( J )and de“netheeectiveactionbyaLegendretransform ( A )= W ( J ) Š J A A a= W J a .(6. 5.2) Thisquantityisnotgaugeinvariantingeneral.Physicalquantitiescomputedfromit aregaugeinvariant,andtheGreenfunctionssatisfySlavnov-Tayloridentitiesthat expresstheunderlyinggaugeinvariance,butmanifestgaugeinvarianceislostbecauseof thegauge“xingprocedure. Ontheotherhand,theeectiveactioncomputedinthe background“eldmethodismanifestlygaugein variant.Itisequiva lenttotheusualone, butismoreconvenienttohandle. Inthebackground“eldquantizationofYang-Millstheorieswefollowaprocedure similartothatofsec.6.2a.Westartwiththegauge-invariantLagrangian ILinv( A a) (other“eldsmaybepres entbutwedonotindicatethemexplicitly)and split the“eld intoabackgroundan dquant umpart: ILinv( A A a+ A a). Theactionisinvariantundertwo kindsoftransformationsthatgivethesame ( A A a+ A a): Quantum: A A a=0, A a= a + i [ A a],(6.5 .3) a = a + i [ A A a],

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3746.QUANTUMGLOBALSUPERFIELDSBa ck gr o und: A A a= a A a= i [ A a].(6.5 .4) Weconsid ernowthefunctional Z Z ( A A )= IDA aeILinv( A a+ A A a),(6. 5.5) andquantizeasbeforeto“xthe quantum gaugeinvarianceexceptthat,tomaintain manifestinvariancewithrespecttothe ba ckground gaugetransformations,wechoosethe gauge-“xingfunctionsothatittransforms covariantly underthesetransformations. Thisrequiresinparticularthatwecovariantizethederivativesthatappeartherewith respecttothebackground“eld: aA a aA a= aA aŠ i [ A A a, A a].Therem ai nderof thequantizationpro cedureisthesame.WerequiretheFaddeev-Popovghoststotransformcovariantlyunderbackgroundgaugetransformations,andwechoosetheweighting function exp ( Š1 g2 tr f2)tobeinvar iant.Wet husobtai nthefo llowing expression: Z Z ( A A )= IDA aIDcIDceSinv( A + A A ) Š1 4 g2 tr ( A )2+ SFP.(6. 5.6) Z Z is manifestlyinvariantunderbackgroundgaugetransformationsbutitssigni“canceis notobvious.Toelucidateitsmeaningweconsideranobjectde“nedexactlylike Z Z exceptthatwealsocouplethequantum“eldtoasource: Z( J A A )= IDA aIDcIDceSeff( A a, A A a)+ JA.(6. 5.7) Wecannowpassfrom Z( J A A )to( A, A A )byaLege ndretransformationinthepresence ofthe“xed“eld A A .Ontheot herhand,returningto Zitself,wecanmakeachangeof variables A Š > A Š A A whichgives Z( J A A )= eŠ J A A IDA aIDcIDce[ ILinv( A a)+ ILGF+ ILFP+ JA ]= eŠ J A AZ ( J A A ).(6.5 .8) Itcontainstheusual ILinv( A a) butunusualgauge“xingandghosttermsthathaveadditionaldepe ndenceon A A a.There fore, Z ( J A A )isthe usualgeneratingfunctionalbut

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6.5.Thebackground“eldmethod375with A A -dependentgauge“xingandghostterms.Nowwehave W( J A A )= lnZ( J A A )= lnZ ( J A A ) Š J A A = W ( J A A ) Š J A A ,(6. 5.9) and ( A, A A )= W( J A A ) Š JA=[ W ( J A A ) Š J A A ] Š JA= W ( J A A ) Š J ( A+ A A ), A= W J = W J Š A A .(6. 5.10) Therefore ( A, A A )=( A+ A A A A ),(6.5 .11) istheusual eectiveaction( A ),evaluatedinanunusual A A -dependentgauge,andat A = A+ A A .Inparticula r,ifintheevaluationof( A, A A )werestricto urselvesto graphswith noexternal Alines(vacuumgraphs),i.e.,set A=0,wew illobtain( A A ), theusual eectiveaction.Butthese A-vacuumgraphsaresimplytheone-particleirreducible subsetofthegraphsobtainedfrom Z(0, A A )= Z Z ( A A ).(Actually,thisisan oversimp li“cation.Whatoneobtainsisnotex actlytheeectiveaction,because A A lines fromthe gauge-“xingtermgiveadditionalcontributions.However,becauseofgauge invariance,it canbeshownthatthesehavenoeectonS-matrixelementssothatthe identi“cation,thoughstrict lyspeaking notcorrect,canbeusedwhencomputingphysicalquant ities.) Ourconclusionisthattheeectiveactionisobtainedfrom Z Z ( A A )byevalu atingin pertur bationtheoryone-particle-irre duciblegraphswi thonlyinternal A alinesande xternal A A alines(asw ellasghost,andothernon-gauge“eldlines).Inparticular,ifwe expand ILinv( A A + A )= IL ( A A )+ IL( A A ) A + IL( A A ) A2+ ... ,the “rsttermdoesnotcontri butetoloopgraphs(itistheclassicalcontributionto),andthesecondcanbe droppedbecauseitdoesnotcontributetoone-particle-irreduciblegraphswithnoexternal A lines.The A2termgivesthecompletecontribution(fromthegauge“eld)tooneloopgraphs.Forhigher-loopgraphs,inter nalverticesarereadfromthehigherorder expansion,andallthetermscontributetoverticesthatinvolvetheexternal A A lines.

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3766.QUANTUMGLOBALSUPERFIELDSThereisanadditionalfeatureofthebackground-“eldquantizationthatisnotusuallyencounteredintheYang-Millscasebutthatisimportant.Thisistheappearanceof theNielsen-Kalloshghost.Inthegauge-ave ragingprocedureweusedthesimplestexponentialfactortoproducethegauge-“xingte rmintheeectiveLagrangian.However,a morecomplicatedaveragingfunctioncouldbeused,e.g., exp ( fMf )where M isany operator(matrix).Toproperlynormalizet heaveragingprocedure,wemustdivideby detM .If M is“eldindependent,thisisatrivialfactor.However,if M isafunctionof thebackground“eld,wenormalizethegauge aver agingbyintr o ducingintothefunctionalintegralafactor IDfIDbefMfebMb,(6. 5.12) where b isa ghost “eld,w ithoppositestatisticsto f .Whenweca rryoutthe f integrationus ingthe -functionofsec.6.2.a,weareleftwiththe b “eld.Thus,the“nalformis Z Z ( A A a)= IDA aIDcIDcIDbe[ ILinv( A + A A )+ ILGF( A A A )+ ILFP( c c, A A A )+ ILNK( b A A )](6.5.13) where ILNK= bMb .If M isindepe ndentofthebackground“eld,theadditionalghost givestrivialcontributionsandcanbedropped;otherwise,sincetheghost“eld b hasno interactionswithothe rquant um“eldsandsinceitentersquadratically,itonlycontributesattheo ne-looplevel. Tomotiva tetheprocedureweuseinthebackground“eldquantizationofsupersymmetricYang-Millstheory,wepointoutt wo aspectsofthebackground-quantum spli tting A a A A a+ A aofordinaryYang-Mills.Thissplittinghasthevirtuethatthe transformations ( A A a+ A a)=( a Š i [ A A a, ])+( Š i [ A a, ]),whichleavetheaction invariant,canbeinterpretedasordinarygaugetransformationsofthebackground“eld accompaniedbycovariantgaugerotations(linearandhomogeneous)ofthequantum “eld.Furthermore,inanexpansion IL ( A + A A )= [IL( n )( A A )]( A )n,(6. 5.14) eachterminthepowerseriesisseparatelyinvariantunderthesetransformations,since A atransformslinearlyandhomogeneously,asdothefunctionalderivativesof IL ( A A ).

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6.5.Thebackground“eldmethod377Thus,ifwetruncatetheseries,aswedoinaperturbativeloop-by-loopevaluationofthe eectiveaction,wema intainthebackgroundgaugeinvariance. Thiswouldnotbetrueif thetransformationofthequantum“eldwerenonlinear. b.SupersymmetricYang-Mills InsupersymmetricYang-Millstheoryth ec lassicalactionisinvariantunder nonlinear gaugetransformations eV eV= ei eVeŠ i ,andthes p litting V V + V V is unsuitable.Tomotivatethesubsequentprocedure,we“rstreexaminethebackgroundquantumsplittingofordinaryYang-Millstheoryfromadierentpointofview.Westart withtheoriginalgaugeandmatteraction,invariantunderthelocaltransformations Aa= a Š i [ A a, ]a nd,forsomematter“eld, = i [ ].U nder global transformationswithconstant west illhaveinvariance,withthegauge“eldsrotatinglikethe matter“elds.Forlocal ,wecanintr o duceanewinvariancebykeepingthecovariant transformations i [ A a]and i [ ]for all“elds,a ndintroducingaseparategauge“eld, thebackg round“eld A A atocovariantizetheder ivatives.Sinceintheoriginalactionall derivati vesenter edintheform a= aŠ i [ A a,],thiscovar iantizationamountstothe replacement a= aŠ i [ A a,] aŠ i [ A a,]= aŠ i [ A A a+ A a,],(6.5.15) whichisequivalenttotheordinaryquantum-backgroundsplitting.Wenowhavetwo in va riances:theoriginalonewherethebackground“eldisinert,andthenewone,under whichallthe“eldstransform. WeobtainalinearsplittingA A + A A becausethe gauge“eldenterslinearlyinthecovariantderivative. InsupersymmetricYang-Mills theorythisissointheabeliancase,butnoti nthe nonabeliancase.However,thephilosophyisthesame.Westartwiththelocallyinvariantgaugetheory,observethatitis invariantforglobaltransformations(with= = aconstant matrix,notasuper“eld),underwhichthegauge“eldstransformcovariantly(linearlyandhomogeneously, since eV ei eVeŠ i implies V ei VeŠ i )and now gaugethistransformationby covariantizingwith theaidofabackground “eld.Thisamountstothereplacement DA Awhere Aisabackgroundcovariantderivative.Wealsohavetotreatthe covariantlychiralsuper“eldsproperly. We recallthatsupersymmetricYang-Millstheorycanbeformulatedintermsof constrainedcovariantderivatives.Thereasonforsolvingtheconstraintsandintroducing

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3786.QUANTUMGLOBALSUPERFIELDSthegaugeprepotentialsisthatonlytheseunconstrainedobjectsaresuitableforquantization.Insolvingtheconstra intswehavethechoiceofwor kinginth ev ectorrepresentationorinthechiralrepresentation.Thelatterismoreconvenientforquantization, expressingthetheoryintermsoftherealsuper“eld V ,rathert hansuper“elds, with ar edundantgaugeinvariance.Wewanttomaintainthisadvantageinthebackground “eldmethod andworkwithaquantum V .Ontheot herhand,whenweintroducethe backgroundcovariantderivatives,itisusefultothinkoftheminthevectorrepresentation.Infactitispossibletoexpressallour resultsintermsofthe(constrained)backgroundcovariantderivativesthemselves,withouteverintroducingexplicitlythebackgroundgaugesuper“elds,i.e.w ithoutsolvingthe constraints,andinthatcasetheonly representationthatisavailableisthevectorrepresentation.Theadvantageofworking withthebackgroundderivativesdirectlyisthatbackgroundcovarianceismanifestand weobtainsi gni“cantsimpli“cationsandimprovementinthepowercountingrulesfor Feynmang raphs. Wealsoexpr esscovariant lychiralsuper“eldsin termsoftheq uantum“eld V and background-covariantlychiralsuper“elds.Thelatterthereforedependimplicitlyonthe background“eldsandwouldseemnottobesuitableforquantization.However,thisis notalwaysthecase:Atmorethanoneloop,a ndevenatonel oopforre alrepresentationsofthegaugegroup,weformulatecovariantFeynmanrulesdirectlyforcovariantly chiralsuper“eldsth atleadtoconsiderableimprov ementovertheordinaryones. Startingwith theordinarycovariantderivativesweperformthesplittingbywritingthem,inthe quantum-chiralbutbackground-vector representation,as = eŠ V eV, €= €, a= Š i {, €} ;(6. 5.16) where and €arebackgroundcovariantderivativessatisfyingtheusualconstraints. The s tr an sformcovariantlyundertwosetsoftransformations: (a)Qua ntum: eV ei eVeŠ i A A,(6. 5.17) withbackgroundcovariantlychiralparameters = €=0, i.e.,

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6.5.Thebackground“eldmethod379A ei AeŠ i .(6. 5.18) (b )B ac kg round: eV eiKeVeŠ iK, A eiK AeŠ iK,(6. 5.19) witharealparameter K = K i.e., A eiKAeŠ iK.(6. 5.20) Thebackground“eldtransformationsof V canberewrittenas V eiKVeŠ iK,(6. 5.21) i.e., V transformscovariantly. *** Whilethisprocedurehasgivenusacorrectquantum-backgroundsplitting,incontrasttothecomponentYang-Millscaseitresultsindierenttransformationsof Aunderquantumandbackgroundtransformations.However,thetransformationofthe unsplitgauge“eld is th es am e.Tounderstandthesplittingofthegauge“eldwesolve theconstraintsonthebackgroundcovariantderivatives: = eŠ De €= e D€eŠ .(6. 5.22) Hencet hesplittingof thefullderivativesis = eŠ VeŠ De eV, €= e D€eŠ .(6. 5.23) Wetransf ormtoabackgroundchiralrepresentationbypre-andpost-multiplyingall quantitiesby eŠ and e ,resp ectively.Then eŠ eŠ VeŠ De eVe € D€,(6. 5.24) andthespli ttingiseq uivalenttoreplacing eVby eV ( split )= e eVe .(6. 5.25) Inotherwords,wesplitthefull V intoaq uantum V andbackground and ina

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3806.QUANTUMGLOBALSUPERFIELDSparticular,non linearfashion.(Intheabeliancasethisreducesto V V + + which isjusttheordinarysplittingsinceby(4.2.72) + = V V .) Theusualchiralrepresentationtransformationsof(6.5.25) ( e eVe )= ei 0( e eVe ) eŠ i 0,(6. 5.26) (where0isordi narychiral, D€0=0),canbew ritteni ntwo ways: (a) ( e eVe )= e [( eŠ ei 0e ) eV( e eŠ i 0eŠ )] e = e ( ei eVeŠ i ) e ,( 6.5.27a) i.e.,thequantumtransformations(6.5.17), withbackgroundcovariantlychiral ,or (b) ( e eVe )=( ei 0e eŠ iK)( eiKeVeŠ iK)( eiKe eŠ i 0),(6.5 .27b) i.e.,thebackgroundtransformations(6.5.19)(cf.(4.2.70-71);recallthatthe0partof thetransformationof doesnot aectthetransformationofthebackgroundcovariant de rivatives).ThisisverysimilartothesituationincomponentYang-Mills. ThegaugeLagrangianhastheform trW2= Š tr (1 2 [ €, { €, } ])2.(6. 5.28) Whenwesubsti tute(6.5.16)into(6.5.28),weobtainasplittingoftheactioninto exp licitquantum V sandbackgroundcovariantderivatives.Sincethe stransform covariantly,theLagrangianwillbeinvariantunderbothbackgroundandquantumtransformations.Furthermore,since V transformshomogeneously,expandingtheLagrangian inpowersof V willmaintainthebackgroundinvari anceterm-by-term,whichisoneof therequiredpropertiesofagoodsplitting. Whencovariantlychiralsuper“elds, €=0arepr esent,we“rstexpressthem intermsofbackgroundcovarian tlychirals uper“eldsby=,= eV(inthe quantum chiralrepres enta tion) €= =0, andthen linearly sp litthemintoasumof backgroundandquantum“elds.Thequantum“eldstransformunder

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6.5.Thebackground“eldmethod381(a)Quantumtransformations: = ei = eŠ i .(6. 5.29) (b)Backgroundtransformations: = eiK, = eŠ iK.(6. 5.30) Thechiral“eldactionisinvariantunderbothquantumandbackgroundtransformations. We examinenowthebackground“eldquantization.Weproceedasintheconventionalapproac h,butcompute Z Z = IDVIDcIDcID cID c ( 2V Š f ) ( 2V Š f )eSinv+ SFP.(6. 5.31) We havechosenbackground-covariantlychiralgauge“xingfunctions,andthismeans thattheFaddeev-Popovghosts,introducedasinsec.6.2,arealsobackgroundcovariantlychiral.Finally,wegaugeaveragewith IDfID fIDbID beŠd4xd4 [ ff + bb ],(6. 5.32) wherethebackgroundcovariantlychiralNielsen-Kalloshghosts b b havebeenintroducedtonormalizeto1theaveragingover f f .Thislea dstothe“nalform Z Z = IDVIDcIDcID cID cIDbID beSeff, Seff= Sinv+ SGF+ SFP+ bb ;(6. 5.33) which,exceptfortheNielsen-Kalloshghosts,islike(6.2.19),butwithbackground covariantderivativesandcovariantlychiralsuper“elds.TheN.-K.ghostsinteractwith thebackground“eld,andonlygiveone-loopco ntri butions.Ifwecoupleexternalsources tothequantum “elds,e.g.,d4xd4 JV ,thege neratingfunctional Z Z ( J )willstillbe in va riantunderbackgroundtransformations,providedwerequirethesourcesto

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3826.QUANTUMGLOBALSUPERFIELDStransformcovariantly, J = i [ K J ]. Whatwemustdonowisarguethatthebackground“eldfunctional,obtainedby settingsourcestozeroandcomputingone-particle-irreduciblegraphswithonlyinternal quantumlinesandexternalbackgroundlines,i sequivalentt othe usualeectiveaction, exceptforbeingcomputedinadierentgauge.Thisislessdirectthanintheordinary casebecausethesplittingishighlynonlinear.Wepresentthefollowingargument: Thesplitting(6.5.25)is V V + + + nonlinearterms .(6. 5.34) Inagaugeforthebackground“eldswhere = =1 2 V V V V wewritethisas V f ( V V V ) where f (0, V V )= V V and f ( V ,0)= V .Ifnowintheori ginalfunctionalintegralweadda sourcetermd4xd4 J [ f ( V V V ) Š f (0, V V )]tode “nea Z( J V V )wew illhavea JV coup ling,andcouplingtohigherordertermsin V and V V (whichareirrelevantwhencomputingtheS-matrix),butnolinearcoupling J V V .Whenweset V V =0weobta inthe conventional Z ( J ).Asin(6.5.8)wemakeachangeofvariablesofintegrationwhich involvesthe inverse ofthefunction f .U nderthistransformati ontheinvariantgauge actiongoestoitsusualformintermsof V ,the gauge“xingandghosttermschangeina complicated,butphysicallyirrelevantmanner,andthecouplingtothesourcebecomes simply JV Š J V V .Furthe rmore,theJacobianofthetrans formationis1(se es ec.3.8.b). We no wh av et he sa meformasinordinaryYang-Millstheory,andweconcludethatthe background“eldfunctionalcomputedbysetting J =0, i.e.,evaluating graphswithonly internalquantumlines,doesgivetheusualeectiveactionasafunctionofthebackground“eld,albeitinanunconventionalgauge.Thereforeallphysicallyrelevantquantumcorr ectionscanbeobtainedfromthebackground“eldfunctional.Wenowdiscuss howtoevaluateitinp erturbationtheory. c.CovariantFeynmanrules Weconsid er“rstcontributionsfromonlythequantumgauge“eld V .The eectiveLagrangianis Š1 2 g2 tr [( eŠ V eV) 2( eŠ V eV)+ V ( 2 2+ 2 2) V ].(6.5 .35) Allthe dependenceonthebackground“eldsisthroughtheconnectioncoecientsand

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6.5.Thebackground“eldmethod383neverthrough thegauge“eldsthemselves. Thequadraticactionhastheform Š1 2 g2 trV [ Š 2 Š i W W +1 2 i ( W W)+ 2 2+ 2 2] V .(6. 5.36) Usingthecommutationrelations [ €, €]= C€€W W,(6. 5.37) thiscanberewrittenas Š1 2 g2 trV [ Š i W W Š i W W€ €] V ,(6. 5.38) where =1 2 a aisthebackgroundcovariantdAlembertianand Wisthebackground “eldstrength.Introducingconnectioncoecients(dependingonthebackground “elds)by A= DAŠ i A,wecanseparate outafreekineticterm,andinteractionswith th eb ac kg round: Š1 2 g2 trV [ 0Š i a aŠ1 2 i ( a a) Š1 2 a aŠ i W( DŠ i ) Š i W€( D€Š i €)] V .(6. 5.39) Thisexpressionissucientfordoingone-loopcalculationsusingconventionalpropagatorsforrea lscalarsup er“eldsandtheusual D -manipulations.Sincetheinteraction withthebackground“eldsisatmostlinearin D s,andatleastfour D saren eededina loop, the“rstnonvanishingone-loopcontribut ionfromVisinthefour-pointfunction. Self-interactionsforcomputinghigher-loopcontributionscanbeobtainedfromthe higher-orderin V termsintheLagrangian(6.5.35). Wenowt urntocontributionsfrom(fully)cov ariantlychiralphysicalsuper“elds andbackgroundcovariantlyc hiralghostsuper“elds.Inprinciplewehavetosolvethe chira lityconstraint €=0(bywr iting= e 0intermsofanordinarychiralsuper“eld),butthisintroducesexplicitdependenceonthebackgroundgaugeprepotentials whichwewishtoavoidifpossible.Instead,wereexaminethederivationoftheFeynman rulesforchiralsuper“elds.

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3846.QUANTUMGLOBALSUPERFIELDSWeconsid erthegeneratingfunctionaloftheform Z ( j j )= ID ID eS +(d4xd2 j + h c .),(6. 5.40) whereand j arecovariantlychiral €= €j =0.Forthet imebeingwe neednot specifywhetherthesearefullcovariantderivativesorjustbackgroundcovariantderivatives.Inprin ciplewede“neID astheintegr aloverthecorrespondingchiral-representation“eld0(antichiralfor integration ), butinpracticewesimplyde“neitby theGaussianintegral ID ed4xd21 2 2=1.(6. 5.41) Additional “eldsmaybepresentbutweneednotindicatethemexplicitly. We de“ne covariantfunctionaldierentiationby ( z ) ( z) = 28( z Š z).(6.5 .42) Thisformcanbederivedfrom(3.8.3),orbywriting= 2,intermsofageneral super“eld,andcovariantizing(3.8.13).Manifestlycovariantrulesforchiralsuper“elds cannowbefoundbyadirectcovariantizationoftheusualmethod.Thecovariantizationoftheidentity D2D2= 0(where 0denotesnowthefre edA lembertian) b ecomes 22= +, += Š iWŠ1 2 i ( W),(6.5 .43) withthecovariant .Weconsider“rstt hemasslesscase. Weca rryoutthefunctionalintegrationoverbyseparatingouttheinteraction termsanddoingtheGaussianintegral,andobtain Z = eSint( j j )eŠd4xd4 j + Š 1j,(6. 5.44) whereisthefunctionaldeterminant = ID ID eS0, S0= d4xd4 .(6. 5.45)

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6.5.Thebackground“eldmethod385Ingeneraltheaboveexpressionmuststillbeintegratedoverotherquantum“eldsthat maybepresent. Beforeweevaluate,whichwillgiveaseparate,one-loopcontributiontothe eectiveactionfromthe“eld,weexaminetherestofthecontributions.Theexpressionfor Z isidenticaltotheonein(6.3.14),exceptforthepresenceofcovariantderivativesandco variantlychi ralsources,andthefactor.Theperturbationexpansiontakes thesameform,exceptthatfromthefunctionaldierentiationwegetfactorsof 2or 2actingonchiralandantichirallines.Thepropagatorsaregivenby Š + Š 1, butinaperturbativecalculationweseparate +intoafreepart,whichleadsto pŠ 2pr opagators, andtheremainder,whichgivesadditionalinteractionvertices.However,atnostagedo ween counterexp licitgauge“elds,onlyconnectionsand“eldstrengths.(Theexplicit dependenceonthequantumgauge“eldswillbeneededonlywhenwefunctionallyintegrateoverthem.) Wenowevaluate.I tgivesthecom pleteone-loopcontributionofthechiral super“eldtographswithonlyexternal V linesandcouldbeevaluatedbyusingstandard Feynmanrules, butthiswewishtoavoid.Thisturnsouttobepossibleonlyfor real representationsoftheYang-Millsgroup.Ofcourse,realrepresentationsarefrequently theonesofinterest:e.g.,theYang-Millsghostsareintheadjointrepresentation,which isalwaysreal.Wethereforeconsider“rstth ecaseofrealreprese ntations,a ndreturn latertothecomplicationscausedbycomplexrepresentations.Wearestillconsidering thema sslesscase. Theaction S0leadstotheequationsofmotion(inthepresenceofsources) O + j j =0, O O 0 2 20 .(6. 5.46) Wede “nean actionwhoseequationsofmotionare O O2 Š j j =0, O O2= 220 0 2 2 ,(6. 5.47) intermsofthesquareof O O .Thisactio nisgivenby S 0+ S 0,where S 0= d4xd21 2 += d4xd41 2 2.(6.5 .48) Intermsofitwecanwritethefunctionalintegral

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3866.QUANTUMGLOBALSUPERFIELDS2= ID ID eS 0+ S 0= | ID eS o|2=( ID eS o)2.(6. 5.49) Wehave usedthefactthat S 0anditshermitianconjugatecontributeequallyto,as canbeseen,forexample,byexaminingtheresultingFeynmanrulesbelow.(Thisprocedur ei sa na logoustothedoublingtrickinQED,wheretheanalogueof O O is €and of +is C + f,with ftheel ectromagnetic“eldstrength.) Wenowintegrate S 0bysepara tingout D2D2from 22andtreating ( 22Š D2D2)asaninterac tionterm.Theresultis =ed4xd21 2 j [ 22Š D2D2] j eŠd4xd21 2 j oŠ 1j.(6. 5.50) (Writinginstead 22= D2eŠ VD2eVinthechiralrepresentationgivestherulesforthe one-loopexpressionintheusualnoncovariantformalism.)Therefore,acalculationofthe one-loopcontributionconsistsinevaluatinggraphswithpropagators pŠ 24( Š )and vert i ces 22Š D2D2givingrise t oastring ... [ 22Š D2D2]i4( iŠ i +1)[ 22Š D2D2]i +1... ,(6. 5.51) withd4iintegralsateachvertexandoneloop-momentumintegral.Wecarryoutthe evaluation inthechiralrep resent ation, sothat €= D€.Wecon centrateonthe i vertex,andfromthenextvertexwetemporarilytransferthe D2= 2factor acrossthe -function.Wenowusetheidentity( 22Š D2D2) D2=( +Š 0) D2(inthechiral representation).Havingperformedthismaneuverwereturnthe D2toitsoriginalplace, andproceedtomanipulatethenextvertexinthesameway.Thisprocedurecanbecarriedoutatallverticesbutone,whichretainsitsoriginalform.Theresultingrulesfor theevaluationofare,withtheusualpropagator, onevertex: D2( 2Š D2),(6.5 .52) othervertices: +Š 0,(6. 5.53) withthecovariantderivativesinchiralrepresentation.Nowonlyonevertexcontributes any D sandasaconsequencetheevaluationofone-loopgraphcontributionsfromchiral

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6.5.Thebackground“eldmethod387super“eldsisconsiderablysimpli“ed.Higherloopsaregivenbytherestoftheexpressionin(6 .5.44). Uptonowwehavenotsp eci“edwhetherthe sarefullorbackgroundcovariant derivatives.Ifbothquantumandbackgroundgauge“eldsarepresent,itismoreconvenienttoca rryouttheaboveprocedureatanearlystage,beforewewrite= eV, i.e.,workwithfullycovariantlychiralsuper“elds(butnotfortheghosts,whichareonly backgroundcovariantlychiral).Theresultofthecalculationisexpressibleintermsof thefullcovariant derivatives,andonlyatthatstage,havingintegratedoutthechiral super“elds,doweneedtomakethebackgro undquantumsplitting onthegauge“elds. Thedoublingtrickcannotbeappliedcovariantlywhenthescalarmultipletisin ac omplexrepresentationoftheYang-Millsgroup.Ifwewritethecovariantlychiralin termsofordinarychiral0( D€0=0)inthev ectorrepresentation = e 0,(6. 5.54) wehave = e 0,(6. 5.55) and 2 = eŠ D2e*e 0,(6. 5.56) isnotint hesamerepresentationas(doesnotsatisfythesamechiralitycondition) exceptwhentherepresentationisreal(inwhichcase*= Š ).Therefore,theoperator O O in(6.5.46)cannotbesquared,sinceitisnotrepresentation-preserving.Asa result,wemustuserules atoneloop whicharenotexpressedman ifestlyint ermsofconnectionsA, butinvo lveexplicitgauge“elds. In(6.5.45)weexpressintermsof0,andintro duceordinarychiralsources j0( D€j0=0).Wehaveinsteadof(6. 5.46)thefollowingequationsofmotioninthepresen ce ofexternalsuper-Yang-Mills: O O 0 0 + j0 j0 =0, O O = 0 D2eV D2eV *0 .(6. 5.57)

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3886.QUANTUMGLOBALSUPERFIELDSThenoncovariantobject O O canbesquared,sinceitpreserves( D€-)chir ality: O O2= D2eV *D2eV0 0 D2eV D2eV .(6. 5.58) Theaction S0 obtainedfrom O O2againgivesacontributionequaltothatofitshermitianconjugate.Asin(6.5.50)weseparatea D2from eV *D2eVandtreattherestasan interaction.Thepropagatorisasbefore,butthevertexisnow D2( eV *D2eVŠ D2).(6.5 .59) Notethat,for real representations, V *= Š V = Š V ,sothisver texisjust D2( 2Š D2), andtherulesof(6.5.52,53)canbeobtained.Ingeneral,foragroupcontainingfactors forwhichisinarealrepr esentation,wecanwrite V = V1+ V2,where V1*= Š V1, but V2* = Š V2([ V1, V2]=0),andwrit ethevertexas D2( eV2*1 2eV2Š D2), 1 = eŠ V1DeV1= D+1 .(6. 5.60) Then V1appearsintherulesonlyas1 Awhile V2appearsexplicitly. Thenetresultisthattheeectiveactionisexpressedmanifestlyintermsof Afo rY ang-Millsfactorsthatoccurcoupledonlytorealrepresentations,and always for hi gher-loopcontributions.However,atoneloop,andonlyforYang-Millsfactorscoupled tocomplexr epresentations,thecontributionmustbecalculatedinawaywherethe covarianceisnotmanifest. Ourmethodscanalsobeappliedtomassivechiralsuper“elds.Inthatcasethe term j + Š 1j of(6.5.44)isreplacedwith j 1 +Š m2 j +1 2 [ j m 2 +( +Š m2) j + h c .],(6. 5.61) adir ectcovariantizationoftheresult(6.3.13) .Inperfo rmingthedoublingtrick,weuse ( m )=( Š m ).Wereplac etheki neticope rator O O ( m )= m 2 2m (6.5.62) by

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6.5.Thebackground“eldmethod389O O ( Š m ) O O ( m )= 22Š m20 0 2 2Š m2 .(6. 5.63) Aftermakingthecorresp o ndingreplacementsin(6.5.50),weobtain Š ( 0Š m2)Š 1for thepropagator,whiletheverticesarethesameasbefore. Wesu mmarizetheprocedureforevaluatingtheeectiveactioninthebackground “eldforma lism:One-loopgraphswithonlyexternalgauge“eldlinesareobtainedfrom thequadraticLagrangianfor V in(6.5.39),andbyevaluatingforeachchiralsuper“eld.Higherloopsareobtainedwithverti cesinvo lvinginteractionsofthequantum “elds,eitherfromthehighe r-orderexpansionofthe V Lagrangian,orfromtheperturbativeevaluationof(6.5.44).Therulesforloopswith(some)externalchirallinesfollow fr om(6.5.44)andaretheusualonesbutwithcovariantpropagatorsandvertices. d.Examples Wenowpresent someresults.Webeginbyinvestigatingtheradiativegeneration ofaFayet-Iliopoulosterm(4.3.3)foranabeliangauge“eld,andconsider“rsttheonelooptadpolegraphwithachiral“eldinside(Fig.6.5.1). Fig.6.5.1 Ifthechiral“eldismassless,wecandropitwhenusingdimensionalregularization.In thema ssivecase,accordingtotheusualrules,itwouldseemtocontributebutgauge invariancerequiresthattherebetwochiral“eldsofoppositecharges,andtheircontributionscancel.Therefore,gauge-invariantPauli-Villarsregulatorscannotcontributeto thisgrapheither.Asaresult,thegraphmu stbede“nedtovanishinthemasslesscase inanygauge-invariantsupersymmetricregu larizationpro cedure,dimensionalorPauliVillars. However,inthecaseofrealr epresentations,withthecovariantrules,thereisno needforsuchanargument.Atthevertexwehaveacontribution(see(6.5.52);to

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3906.QUANTUMGLOBALSUPERFIELDSlineari zedorderweneednotdistinguishbetweenfullandbackgroundderivatives) D2( 2Š D2)= D2[ Š i DŠ i (1 2 D)](6.5.64) (tolineari zedorder),andwedonothavetwo D sintheloop.Thus,f orrealrepresentations,thegraph vanishes justby D algebra. Actually,eventhiscalculationisunnecessary,becausewecangiveasimpleproof thattheFayet-Iliopoulostermisnevergener atedinperturbationtheoryforrealrepresentations:Thistermcorrespondstoacontributiontotheeectiveactionoftheformd4xd4 V .Howev er,accordingtoourcovariantFeynmanrules,a V neverappearsata vertex,onlyco nnectionsand“eldstrengths,sothatnosuchtermcanbeproduced. Wenextcalc ulatetheone-loopcontributiontothe V self-energyfroma ma ssive chiralsuper“eldi nareal representation.Ifweusetheordinarynon-backgroundrules therearethreegraphstocompute(becausewehavemassivepropagators),andthey havetobecombinedtoexhibitthegaugeinvarianceofthe“nalresult.Also,thereare some D manipulationstobeperformed.Here,thereisessentiallynothingtodo.We consideragaintherelevantgraph,showninFig.6.5.2. Fig.6.5.2 On ev ertexisgivenby(6.5.64),whileattheothervertexwehave(againtolinearorder) +Š 0=[ Š i a aŠ i (1 2 a a)]+[ Š iWDŠ i (1 2 DW)].(6.5.65) Sincewerequiretwo D sandtwo D sintheloop,th ereisauniquetermwithcontributions Š i D2Dfrom onevertex,and Š iWDfromtheo ther.The answeris1 4 ktr d4 W( p )( Š p ) d4k (2 )4 1 ( k2+ m2)(( k + p )2+ m2) .(6. 5.66) (Thefactor k wasde “nedin(6. 4.5).)Weobservethatwiththecovariantrulesthereis noseagull-tadpolecontribution.Thiswouldhavetocomefromthenonlinearpartofthe

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6.5.Thebackground“eldmethod391one-vertexformula(6.5.64) butitdoesnothaveenough D stocont ribute. Wecanobt ainthe V se lfenergyinnonabelianYang-Millstheorywithoutanycalculation.Accordingtothedi scussionfollowing (6.5.39),ifwelookatgraphswithtwo externallines,therearenotenough D sintheloop.Theonlysourceof D sarethe W terms,andea chfact orof W bringswith itjustone D .Thusthewholec ontri butionto theselfenergycomesfromthethreechiralghosts,andthereforeweobtainananswer whichisjust Š 3timesthatfr omthechiral“eldw econsideredabove (withthe “elds nowbeingbackground).Thisisageneralfeature:Asalreadymentioned, V sstartcontributingatonelooponlybe gi nningwiththefour-pointfunction.Toseetheimplicationsofthisremark,wegivenowacomputationoftheone-loop,four-particleS-matrix in N =4 Ya ng -M illstheory. Theone-loopcontributionswithexternal V lines comefroma V loop,fromthe threechiral“elds,orfromthe threechiralghosts. B ecauseofthestatisticsoftheghosts thechiralcontributionscancelexactly.Thisistrueforagraphwithanarbitrarynumber ofexternalvectorlines.Inparticular,itimpliesthatthetwo-andthree-pointfunctions areidenticallyzeroattheone-looplevel.Therefore,weneedonlycomputethe V -loop contribution.Wehavejustaboxdiagram,withfactors Š i ( WD+ W€ D€)ateach vertex,andwemust k eeptermswi thtwo D sandtwo D s.The D -algeb raistrivial, andweobtainforthefourV amp litude =1 2 tr d4p1... d4p4(2 )16 d4 (2 )4 ( pi) Go( p1... p4) [ W( p1) W( p2) W€( p3) W€( p4) Š1 2 W( p1) W€( p2) W( p3) W€( p4)],(6.5.67) where G0isthecontributionfromthefour-pointscalarboxdiagram G0= d4k (2 )4 1 k2( k Š p1)2( k Š p1Š p2)2( k + p4)2 .(6. 5.68) Thetraceisoverinternalsymmetryindices,andallthesuper“eldshavethesame argument.

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3926.QUANTUMGLOBALSUPERFIELDSThisresultisvalido-shellandisultraviolet“nite.On-shellitgivestheone-loop S-matrix,butitisinfrareddivergent.(ToobtaintheS-matrixwedropthe piintegrals andsumover pipermutat ions .The W sgivekinematicalfactorsproportionalto momentaandpolarizations).Thesimplicityofthecalculationisdueinlargeparttothe absenceofc hiralsuper“eldcontributions.Intheparticulargaugeweareusingthereare noself-energyortrianglegraphstoconsider,andthewholeS-matrixisgivenbythebox graph.Wewillencounterasimilarsituationinsupergravity(seesec.7.8).

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6.6.Regularization3936.6.Regularization a.General Theperturbativerenormalizationofsuper“eldtheoriesisinprinciplenodierent fromthatofordinary“eldtheories.Weneedaprocedureforregularizingdivergentintegrals,andaprescriptionforsubtractingultravioletdivergences.Wemustdealwith renormalizablenon-polynomialLagrangians(e.g.,supersymmetricYang-Millsinasupersymmetricgauge),andusethesupersymmetryWardidentitiesinthecourseofrenormalizationor,alternatively,us earegula rizationschemethatmanifestlypreservessupersymmetry.Wedonothavemuchtosayaboutrenormalization.Forrenormalizable models,weintroducerenormalizationconstantsintheclassicalactionandusethemto cancel,orderbyorderinperturbationtheory,thedivergencesweencounter. Aswehavealreadymentioned,insupersy mmetrictheoriestherearefewerdivergencespresentthaninnonsupersymmetricones.Ingeneral,thedegreeofdivergenceof anysupergraphcanbedetermi nedbythedimensionalargumentofsec.6.3orbythefollowingpowercountingrules:Inrenormalizab letheori esallsupersymmetricverticeshave four D s(eitherfromthe D2and D2ofchiralsuper“elds,orthe D, D2Dofgauge super“elds).Innonrenormalizabletheoriesthereareadditionalfactors,butwe“rstconsidertherenormalizableca se.Allvert i ceshavea d4 factor. Wecons ideran L -loopgraphwith V verti ces, P pr opagatorsofwhich C areor m assivechiralpropagators,and E external linesofwhich Ecarechiralora ntichi ral. Fromthev erticesthereare V factorsof D2 D2 q2.T he pr opagatorsproduce qŠ 2factors, butor p ropagatorsgiveanadditional D2qŠ 2 qŠ 1factor.Eachloopproducesa d4q q4anduse supa D2 D2 q2factorfrom D2 D2 = .Eachexterna lchira l line accounts forone D2 q missingatthecorrespondingvertex.The super“cial degr eeof divergenceis(using L Š P + V =1) D=4 L Š 2 L Š 2 P +2 V Š C Š Ec=2 Š C Š EcTherefore,forgraphswithonlyexternal V sthesuper“cialdegreeofdivergenceistwo (butgaugeinvarianceimprovesthis),andzeroiftherearetwoexternalchirallines.Furthermore,iftheexternallinesareallchiralanadditional D2mustcomeouto ftheloop:d4 n=0soonemusth aveatleastd4 n Š 1D2foran onzeroresult.Thereforethe

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3946.QUANTUMGLOBALSUPERFIELDSconvergenceisimprovedandonlygraphswithoneandone linemaybedivergent. Forreno rmalizabletheoriesweobtaintheresultsofsec.6.3.Insupergravityonthe otherhand,whereateachvertexwehavetheequivalentofsixfactorsof D ,the degree ofdivergenceofagraphis2 Š C Š Ec+ V .Thisres ultcan alsobeobtainedbya dimensionalargument(seesec.7.7). Regularizati onisanimportantpartofanyrenormalizationscheme.Althoughin principleanyregularizationmaybeused,inpracticeitispreferabletouseaschemethat iscomputationallysimpleandmaintainsasmanypropertiesoftheclassicaltheoryas po ssible.Thissimpli“estheren ormalizationprocedure,whichmustnotonlymakethe quantumtheory “nitebysubtractionofdivergences,butalsomustmaintainunitarityby po ssiblesubtractionofadditional“nitequanti ties.Fortheorieswith(globalorlocal) symmetries,suchadditionalsubtractionsaredeterminedbytherequirementthatrenormalizedGreenfunctionssatisfyWard-Takahashiidentities,andinthecaseofnonabelian gaugetheories,Slavnov-Tayloridentities.However,itispreferabletoemployaregularizationschemethatmanifestlypreservesallsymmetries;thisallowsarenormalization schemethatrequiresthesubtractionofonlythedivergentparts,sotheapplicationof Ward-Tak ahashi-Slavnov-Tayloridentitiesisunnecessary. b.Dimensionalreduction Dimensionalregularizationhasproven tobethemostpracticalmethodofregularizationincomp onent“eldtheoriesbecauseithasthreeproperties:(1)Itmanifestly preserves(almost)allsymmetries,thusbypassingtheWard-TakahashiorSlavnov-Taylor identities;(2)theregularizedgraphsarenohardertocalculatethantheunregularized onesandrequireonlyoneregulator,thedimensionalityofspacetime;(3)renormalization isasimpleprocedure,requiringonlyminimal subtraction.Theprescriptionfordimensionalregularizationis:(1)Writetheactioninaformwhichisvalidforanydimension Dofspa cetime;(2)calculateFeynmangraphsfo rmallyinarbitraryspacetimedimensions,integratingoverDcomponentsofeachloopmomentum,giving“eldsofany LorentzrepresentationthenumberofcomponentsappropriatetothatvalueofD,and perfo rminganyalgebraicmanipulationsthatwouldbevalidfor“niteintegrals(i.e.,performingtheintegralindimensionsDforwhi chit is“niteandanalyticallycontinuingin D);(3)renormalizebysubtractingf romdivergentcontributions(asD 4)onlytheir

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6.6.Regularization395polepart s(prop ortionalto1 D Š 4 ),andnoadditional “niteparts,“rstinsubdivergences andthe nforthesuper “cialdivergence(inamputatedone-particle-irreduciblegraphs, i.e.,theeectiveaction). Thisprocedurehastwodrawbacks:(1)Itmustbesupplementedbyaprescription forhandlingsymmetrieswhichdonotc ommutewithparity,i.e.,involving 5or a b c d, andinparticularforcorrectlyobtainingchiralanomaliesforthosecaseswheretheyare present.(2)Itdoesnotmaintainsupersymme try:Theprescriptio nfor giving “eldsof anyLorentzrepresentationthenumberofc omponentsappropriat etoDdi mensionsdoes notkeepFermiandBosedegreesoffreedombalanced.Amodi“cationoftheprescription,whichwouldcontinueafour-dimensionaltheorytoatheorysupersymmetricinD dimensions,isnotpossibleeither.Forexa mple,ifDisincreasedpast10,aglobally supersymmetrictheorywouldhavetobecon tinuedtoalocallysupersymmetricone.We wouldhav espi ns 2b ecausethenumberofsupersymmetrygeneratorsincreaseswith incr easingD.Wenowdescribeamodi“cationofdimensionalregularizationintendedto preservesupersymmetry,andreturnlatertothe“rstdiculty. Sincethechangeinstructureofsupersymmetrictheoriesasthenumberofsupersymmetrygenerators(4 N )isincreasedis notuniform,weconsiderkeepingthisnumber “xed.Forregularizingultr avio letdivergencesitisonlynecessarytocontinueto lower dimensions,anditisthenpossibletokeepthenumberofsupersymmetrygenerators “xedattheirfour-dimensionalvalue.Ingeneral,ourprescriptionforcontinuingtolower dimensionsistocontinue only thedimensionalityofspacetime,butkeeptherangeofall Lorentzindicesthesame,asiftheywereinternalsymmetryindices.AswereduceD,an N -extendedsupersymmetryc anbereinterpretedasan N-extendedsupersymmetry, N> N .Forex ample, N =1inD=4d imensionscanberegardedas N =2inD=3 dimensions.Inthisway,thenumberofbosonicandfermionicvariablesstayequal.Such acontinu ationiscalleddimensionalreduction.Hereweconsidercontinuationonly fromD= 4toD < 4. OurrulesforapplyingdimensionalreductiontoregularizecomponentFeynman graphsare:(1)Allindiceson the“elds,a ndcorrespondingmatrices,comingfromthe actionaretreatedas4-dimensionalindices;(2) asinordina rydimensionalregularization, allmomentumintegralsareintegratedoverD-componentmomenta,andallresulting Kronecker sareD-dimensional;(3)sinceD < 4always,any4 -dimensionalKronecker

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3966.QUANTUMGLOBALSUPERFIELDScontractedwithamomentumequalsthatmomentum a bp b= p a,andany4-d imensional contractedwithaD-dimensionalonegivestheD-dimensionalone a b b c= a c,where thei ndicatesD-dimensionalquantities.The“rstruleisnecessarytopreservesupersymmetry,sinceitkeepsthenumberofcomponentsthesame;thesecondrulepreserves alltheusefulpropertiesofdimensionalregularization(e.g.,gaugeinvariance);thelast rulede“nestheregularizationasdimensionalreduction. Unlikeinordinarydimensionalregularization,both4-dimensionalandD-dimensionalquantitiesoccur.Therefore,whenappliedtocomponents,dimensionalreduction requireshandlingmoretypesof“elds:e.g.,a4-dimensionalvectorbecomesaD-dimensionalvectorand4-Dscalars.Thiscancause dicultiesinnonsupersymmetrictheories, sincealargervarietyofdivergencescanoccur,butinsupersymmetrictheoriessupersymmetryallowsonlydivergencescontainingthefullsetof4-dimensional“elds.Forexample,([ A a, A b])2 ZA 2([ A a, A b])2+ ZAZ([ A a, i])2+ Z 2([ i, j])2, butinsupersymmetrictheoriestheD-dimensional extended supersymmetrythatresultsfromreducing 4-dime nsionalsupersymmetryensuresthat ZA= Z.(Intheori eswithonlyscalarsand spinors,theonlydierencefromusualdimensi onalregularizationisinthenormalization ofthespinortrace,andhencetheseproblemsdonotarise.) Whenappliedtosuper“elds,dimensionalreductionistheuniqueformofdimensionalregularizationthatallowsthenaivealgebraicmanipulationofthe4-dimensional spinorderi vatives Dindivergentasw ellasconvergentsupergraphs.Thisrequirement leadstothefollowing de“nitionofregularizationbydimensionalreductiononsupergraphs:(1)PerformallalgebraasinD=4,obtainingaformwhereall -integrationhas b eenperformedi.e.,thegraphisexpressedasanintegraloverasingle d4 N ofpro ducts ofsuper“eldsofvariousmomentatimesanordinarymomentumintegral andistherefore man ifestlysupersymmetric; (2)performtheremainingmo mentumintegralinD-dimensions.Instep(1),weusethe4-dimensionalidentity p€p€= p2(r ecall p21 2 p ap a(3.1.16,18)).NoD-dime nsionalKroneckerdeltasariseatthisstage.Instep(2),symmetricintegrationsgenerateD-dimensionalKroneckerdeltas,e.g., p€p€2 D p2 € €,(6. 6.1) where € €= a bhastheproperties

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6.6.Regularization397 € € =D 2 €€, € € € €= € €,(6. 6.2) andspi norindicesarestillmanipulatedasin4dimensions. Ontheotherhand,adimensi onalregularizationscheme,whichlikeordinary dimensionalregularizationscontinuedspinorindices(includingtheoneon )toDdimensionaloneswith k =2D 2 Š 1components,wouldha veproble ms:e.g.,dk dk = Dk Dkwouldnolon gerhavewellde“nedstatistics,andwouldintroduce higherderivatives(for k > 2)intotheaction,requiringsomenonminimalsubtraction scheme(suchasanalyticregularization). Unfortunately,althoughitpreservessupers ymmetry,regularizationbydimensional reductionleadstoambiguities.Forexample,letusconsidertheexpression [ a fp bq cr ds e ]=0(6.6 .3) whichvanishesinD < 5b ecauseitistotallyantisymmetr icin5indiceswhichtakeless than5values.(Thisalsofollowsifwewritethevectorindicesintermsofspinorindices anduse4-dimensio nalspinormanipulations.)Ifwenowcontractwith f aweobtain (DŠ 4) p[ aq br cs d ]=0(6.6 .4) Since p[ aq br cs d ]doesnotvanishinD=4,andwemustrequireitnottovanishinD =4to avoidgen eratingarbitrarycoecientsforsuchtermsuponcontinuation,wehavean inconsistency.Thiscanalsobeviewedasanambiguity:Byevaluatingasupergraphin twodier entways,wemayobtainresultsthatdierbythelefthandsideof(6.6.4).If thesupergraphisconvergent,thisambiguitydisappearsinthelimitD 4.However,if itisdivergent, a “nitedierencebetweenthetwowaysofevaluatingthegraphmay result. Thesameambiguityispresentincomponenttheories(supersymmetricorotherwise)thathavechiralanomalies,wherethecorrespondingexpressionis 0= f a[tr ( 5 f ap / q / r / s /)+ tr ( 5 ap / q / r / s / f)]=(DŠ 4) tr ( 5p / q / r / s /)(6.6.5) Toderivethis result,wehaveusedtheidentities

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3986.QUANTUMGLOBALSUPERFIELDS{ a, b} =2 a b, { 5, a} =0, a a=D,(6. 6.6) whichareequivalenttotheprescription(6.6.2)combinedwiththe4-dimensionalspinor algebra. Thisproblemarisesbecausewehaverequiredthatourregularizationrespectslocal gaugeinvariance:Whenweconsidertheorieswithaxialcouplings,wemustuseaprescriptionsuchas(6.6.6)thatrespectschirali nvarian ce.Thismakesitimpossibletocalculate(unambiguously)anomaliesthatshouldbethere.Modi“cationsof(6.6.6)exist thatgivethecorrectanomalies,butunfortunately,thesealsogivespuriousanomalies thatmustbeeliminatedbyusingWard-Takahashi-Slavnov-Tayloridentities,whichis justwhatwe weretrying toavoid. c.Othermethods Theredoexistalternativeschemesforsupersymmetricregularization,atleastfor specialsystems.Fortheoriesthatallowtheintroductionofmassterms,wecanuse su pe rsymmetricPauli-Villarsregularization.Thisisthecase,forexample,intheWessZuminomodel(ormodelswithseveralchiralscalarsuper“elds)whereonecanworkwith theregularizedLagrangian ILR= d8z ( + ci ii) + d6z [1 2 ( m 2+ Mi2 i)+1 3! (+ i)3]+ h c .,(6.6 .7) orinmodelsofchiralmultipletscoupledtoaYang-Millsmultiplet, forregularizingchiral lo ops, ILR= d8z ( eV+ ci ieVi) +1 2 d6z ( m 2+ Mi2 i)+ h c ..(6.6 .8) Heretheiarechiralregulator“elds,andthelimit Mi istobetakenattheend ofthecalculations.

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6.6.Regularization399Fo rs up er sy mmetricYang-Millstheorieswecanusehigherderivativeregularization.Forexample,theusualcovariantactioncanbemodi“edtoread tr d4xd21 2 W(1+ Š 2 +) W,(6. 6.9) andsimilarmodi“cationscanbemadeinthegauge“xingandghosttermstoproduce pr opagatorswith kŠ 4behavior forl arge k .A si no rdinaryYang-Mills,allmultiloopdiagr am sa resuper“ciallyconvergent.Howeverthisproceduremustbesupplementedbya dierentoneloopregularization. St raightforwardPauli-VillarsregularizationcannotbeusedforYang-Millstheories b ecauseitdestroysgaugeinvariance.However,inthebackground“eldmethoditseems pe rfectlyacceptable.Inthismethodtheeectiveactionismanifestlycovariant,and sincethequantum“eldstransformcovariantly(rotateliketensors),onecanaddamass term,andthereforemassiveregulators,wit houtdestroyingthe gaugeinvariance.What isnotentirelyclearisthatthiscanbedoneingeneralinamanifestlyBRSinvariant way, i.e.,withoutdestroyingtheunitarityoftheS-matrix.Butthereseemtobeno problemsattheone-looplevel,sothatacombinationofhigher-derivativeandone-loop Pa uli-Villarsregularizationis ap erfectlyacceptableprocedur ef ormaintainingmanifest supersymmetryinthebackground“eldformalism.Arelatedprocedureforone-loop graphscanbeusede veninano n-backgroundformalism. Anotherregularizationp rocedure,whichhasbeenusedforone-loopgraphs,is pointsp litting.We“rstcon siderthenonsupersymmetriccase.Theregularizationis appliedtoone-loopgraphsbyexpressingthemastracesofpropagatorsinexternal“elds, andseparatingthecoincidentendpointsofthepropagator: d4xG ( x x ) d4xG ( x x + ),(6.6 .10) where is an i n“nitesimalregulator.WritingtheGreenfunction G asafunctionalaverageof“elds, G ( x y )= < ( x ) ( y ) > = ID eS ( ) ( x ) ( y ),(6.6 .11) wecanexpressthepoint spli ttinginthefollowingformintermsofanexplicitoperator: G ( x x + )= < ( x ) e ( x ) > .(6. 6.12))

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4006.QUANTUMGLOBALSUPERFIELDSThisproceduremustbemodi“edinordertopreservegaugeinvariances.However,for theform(6.6.12),gaugecovariantizationistrivial:Wereplacethepartialderivative withacovariantone = Š iA : < ( x ) e ( x ) > .(6. 6.13) Thisisequivalenttotheform: < ( x ) IP { exp [ Š ix + x dx A ( x)] } ( x + ) > ,(6. 6.14) wherethelineintegralisalongastraightlineand IP meanspathordering.Theequivalencecanbe proven,evenfor“nite ,bywriting theexponentialin(6.6.13)asaproduct of exponentialsofin“nitesimals,andthenreorderingallthe stotheright(which translatesthe A s).Incalculations,itismoreconve nienttohavethemanifestlycovariantform(6.6.13)intermsof s. Thesupersymmetricgeneralizationisstraightforward:For ausethesuperspace covariantderivative.(Inprincipleonecouldalsotranslatein with butthe integrationisalready“niteanddoesntneedregularization.)Theaboveequivalencetothe path-orderedexpressionalsoholdsinsuperspace. Whilesuchregularizationme thodsmaintainsupersymmetry,theyarecumbersome. Someformofdimensionalregularizationispreferable,forallthereasonswegaveearlier. Aswehavealreadydiscussed,atthesupergraphlevelthisamountstodoing“rstallthe D -algebrainfourdimensions,andthendimensionallycontinuingthemomentumintegrals.Theresultsaremanifestlysupersymmetric,butpresumablytheinconsistencieswe havediscussedearlierwillgiverisetosomeambiguitiesintheresults.

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6.7.AnomaliesinYang-Millscurrents4016.7.AnomaliesinYang-Millscurrents Asanexampleofourrulesforchiralsuper“eldsandregularizationmethods,we calculatethesupersymmetricversionofthe Adler-Bell-Jackiwanomaly.Weconsidera chiralmu ltipletcoupledtobothpolarandaxialvectorgaugemultiplets.Theonlyphysicalcomponentintheanomalymultipletis theanomalyinthechiralsymmetrycurrent correspondingtophasetransformationsofthechiralsuper“elds.Thissymmetrycommuteswithsupersy mmetrytransformationsandshould bedistingu ishedfromR-symmetry,whichdoesnotcommutewithsupersy mmetry.TheR-symmetrychiralanomaly appearsinthemultipletofsuperconformala nomalies,whichalsoincludesthetraceand supersymmetrycurrentanomalies.Wewilldiscussthisinsec.7.10. Weconsid ertheactionforscalar mult ipletscoupledtovectormultiplets: S = d4xd4 eV.(6.7 .1) Forsimp licity,weassumeanevennumberofscalarmultiplets,inpairsofopposite char gewithrespecttopolarvectorgauge“elds.ThetwoWeylspinorsinsuchapair formaDiracspinor,withtheusualtransformationunderparity.Thecolumnvector isthusinarealrepresentationofthesymmetrieswhichthepolarvectorsgauge.Wecan alsoconsiderthecouplingofaxialvectorgauge“elds,withrespecttowhichthetwo membersofapairhavethesamecharge.TheDiracspinorsofthepairscoupletothese axialvectorswitha 5.Toi ndicatethesetwotypesofvectors,andthecorresponding twotypesof v ectormult iplets,weseparate V intopolarandaxialparts: V = V++ VŠ; V+= Š V+*, VŠ= VŠ*.(6.7 .2) The*referstocomplexconjugationinthesenseof(3.1.9)( V *= Vt,since V = V), but canrefertomatrixcomplexconjugationifanappropriaterepresentationischosen.This isasp ecialcaseofthesituationdiscussedafter(6.5.59).Sinceisinarealrepresentationofthegroupof V+,thepolarv ectormult iplet,wecanuseimprovedruleswith respecttoit,butmustusetheunimprovedform(atoneloop)for VŠ,theaxialvector mult iplet. Weconsid ertheone-loopgraphswithtwoexternalpolarvectorsandoneexternal axialvector.Dependingonthegroupstruct ure,theanomalyinth e lineoftheaxialvectormayormaynotcancel.Iftheaxialanomalyisnonvanishing,axialgaugeinvariance

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4026.QUANTUMGLOBALSUPERFIELDSislost,andtheaxialvectorcannotbeconsi deredphysical.Tobegeneral,weconsider theaxialvectorasmerelyadeviceforde“ningtheappropriateaxialcurrent.Wethen evaluatethedivergenceofthatcurrentintermsofthepolarvector“eldsappearingat theothertwolegs. Varyingtheacti onwithrespecttoanyoftheaxialvectormultiplets VŠ,weobtain theaxialcurrentsuper“elds JA= TA,(6.7 .3) wherew ehavew ritten VŠ= VŠ ATA.Gau geinvarianceoftheactionrequirestheonshellconservationlaw 2JA=0(asfo llowsfromsubstitutingthetransformationlaw eVŠ= ei eVŠeŠ i intotheactionandvaryingwithrespecttothechiralgaugeparameter).Thereforewe de“netheanomaly AAby 2JA= AA.(6. 7.4) Wew ill“ndthatth eanomaly 2J isproportionalto W2.Thecompon ent(ax ial) currentisgivenby j€=1 2 [ €, ] J | .Itsdivergen ceisthereforegivenby €j€ [ 2, 2] J | ( 2W2Š 2 W2) | a b c df a bf c d,w hi ch isthefamiliarcomponent result. Theanomalycanbecalculatedbyevaluatingthematrixelement 2< TA > ,(6. 7.5) whereisnowcovariantlychiralwithrespectto V+.Inthecalc ulationsbelowweomit thegrouptheoryfactor.Wec omputethematrixelement < > andattheendwe musttake thetraceofitsproductwith TA. Wew illevaluatetheanomalybythreemetho ds:(1)theAdler-Rosenbergmethod, (2)withaPauli-Villarsregulator,and(3)withpoint-splittingregularization. IntheAdler-Rosenbergmethodweneedonlycomputeatrianglegraphwithone axialandtwopolarvectorsatthevertices.Other,self-energy-typegraphs,withonevectoratonevertexandtwoattheotheralsoco ntri bute.However,theircontribution merelycovariantizesthatfromthetrianglegraphandtherefore,byimposinggauge in va riance,thefullresultcanbeextractedfromthisgraph.Weusethebackground-“eld formalismofsec.6.5.Attheaxialvertexwehave,from < > itself,

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6.7.AnomaliesinYang-Millscurrents403D2 D2,(6. 7.6) whileattheothertwoweusethelinearizedexpression(6.5.65)with on-shell, Landaugaugepolarvectors( a a= DW=0): Š i ( a a+ WD).(6.7 .7) Thesupergraphisshownin“g.6.7.1: p Š qapaŠ iWDbpbŠ iWD D2p + q p D2Fig.6.7.1 Itiseasytocheck,byintegrationbyparts,thatthe WD, WDtermsdo not contributeonshell.Thereforethe D manipulationistrivialandwemustevaluatean ordinarygraph,asinscalarQED,with1atonevertex,and Š i a aattheothers.The Feynmanint egralis d4p (2 )4 p ap bp2( p Š q )2( p + q)2 a( q ) b( q).(6.7 .8) Thisdirectlygivesthecontributiontothematrixelement < > .(Ifconsid eredasan ordinarytrianglegraph,withexternalvectorsattachedafterwards,thefactorof2from functionallydierentiatingthetwo ascorrespondstothisgraphplusthatwithcrossed v ectorlines.) Accordingtooursupersymmetricdimensio nalregularizationprescriptiontherest oftheevaluationshouldbecarriedinDdimensionsandtheothergraphsshouldbe included.However,gaugeinvariancerequiresthat a( q )enterth eresulti ntheform F a b= Š iq[ a b ],andatermoft hisformc anonlybeobtainedfromthetrianglegraph,by extrac tingfromtheintegralthe( “nite)partproportionalto q aq b.Afterintr o ducing Feynmanpar ametersandshiftingtheloopmomentumthispartcanbeeasilyextracted,

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4046.QUANTUMGLOBALSUPERFIELDSandweobtainforthecompletecontributionto < > Š1 4 1 (4 )2 1 ( q + q)2 F a b( q ) F a b( q)(6. 7.9) or,in x -space < > =1 4 1 (4 )2 1 ( F a bF a b).(6.7 .10) Here F a b= [ a b ]=1 2 C (€ W€ )+1 2 C€€( W )(6.7.11) andhence F a bF a b=1 2 ( ( W ))( ( W ))+ h c = Š 4 2W2+ h c .,(6.7 .12) wherewehaveusedthe“eldequations W= 2W=0. Theanomalyisgivenby 2[1 4 1 (4 )2 1 F a bF a b]= Š1 (4 )2 1 22W2= Š1 (4 )2 W2.(6. 7.13) Thismustbemultipliedbythegroupgenerator TAandatracetaken(with W= WBTB). InthePauli-Villarsregularizationmethodwecomputem lim 2( < > Š < m m> )(6. 7.14) wheremisamassiveregulator“eld.Inthisre gularizedexpressionwecanusethe equationsofmotion 2 =0, 2 m= m m(6.7.15) sothattherelevantquantitytocomputeis Šm lim m < mm> (6.7.16)

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6.7.AnomaliesinYang-Millscurrents405Using(6.5.44,61)wehave < mm> = j j W ( j ) = 2Š m 2 +( +Š m2) 2= Š m 2 +Š m2 .(6. 7.17) Wemustther eforecomputealoopgraph,with 2replacedby D2(thisistheonly sourceof D s),and( +Š m2)Š 1expandedinpowersofthebackground“eld(weneed atleasttwo D s): 1 +Š m2 1 0Š m2 ( Š iWD) 1 0Š m2 ( Š iWD) 1 0Š m2 + ... (6.7.18) Theonlynonzerocontributioninthe m limitcomesfromthetermexplicitly written.Inmomentumspaceitco rrespondstoatrianglegraphwith D2atonevertex and WD, WDattheothertwo.Theanomalyisthereforegivenby Šm lim m2 d4p (2 )4 1 [ p2+ m2][( p Š q )2+ m2][( p + q)2+ m2] (2 W2) = Š 1 (4 )2 W2(6.7.19) asbefore. Inthepoint-splittingmethodwecompute D2< e > = < 2e > = < e (eŠ e )2 > .(6. 7.20) Usingthecommutationrelations(4.2.90)ofthecovariantderivatives,we“ndeŠ €e = €Š €W+1 2 €€( €W)+ O ( 3).(6.7 .21) Wethen expandtheremaininge =e [1 Š i + O ( 2)],expressev erythingin termsof < ( x )e ( x ) > = < ( x ) ( x + ) > ,andtake thelimit 0.

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4066.QUANTUMGLOBALSUPERFIELDSHowever,wecanlimitthenumberoft ermsweneedconsiderbyevaluating < ( x ) ( x + ) > “rst.Theonlytermswhicha redivergentinthelimit 0(from po we rc ounting)aregivenbythetadpoleandpropagatorgraphs,asshownin“g.6.7.2: Š i (aaŠ iWD) D2 D2D2D2 Fig.6.7.2 (Asbefor e,wehavetwofactors D2and D2fromand ,and +Š 0Š i ( a a+ WD).)The WDtermdoes notcontribute.Thegraphsare evaluatedas d4p (2 )4 eŠ i p1 p2 = 1 (4 )2 1 2 Š a d4p (2 )4 eŠ i pp ap2( p + k )2 = 1 (4 )2 1 2 i (6. 7.22) sothat < ( x ) ( x + ) > = 1 (4 )2 1 2 [1+ i + O ( 2)].(6.7.23) Wethusk eepfactorsmultiplying < > onlyto O ( 2),andalsod ropfactors O ( 2) whichhavea D€actingon < > .There sult is D2< e > ( Š €W D€Š1 2 €W€W) < e > ( Š €W D€+ 2W2) 1 (4 )2 1 2 (1+ i ) 1 (4 )2 ( Š i 1 2 €€W D€€+ W2).(6.7 .24) (Noteinparticu larthato nlythee partoftheremaininge contributes,andonly

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6.7.AnomaliesinYang-Millscurrents407upto O ( )ineŠ €e .) Usingthechiralrepresentationrelation D€€= iC€€W,(6. 7.25) weobtain thesameresultasb ythe previoustwomethods.(NotethatinordinaryQED thecalculationisslightlysimplerbecausethepoint-splitpropagatorgoesonlyas Š 1.) Athigherlo ops,andalsoatoneloopforrealrepresentations,ourcovariantFeynmanrulesapply.Consequentlythetrianglegraphcontributiontotheeectiveaction dependsontheconnectionsand“eldstrengt hsandn otonthegauge“eldsthemselves and,bysimplepowercounting,itisthereforesuper“ciallyconvergent.Wedrawtwo co nc lusions:Therearenoone-loopchiralanomaliesfortheYang-Millsmultipletitself (thechiralghostsareinarealreprese ntationoft hegroup),andthereare nohigher-loop chiralanomalies foranymultiplet:Forthechiralcurrentde“nedby(6.7.3)theAdlerBard eentheoremholds. TheAdler-Bardeentheoremisnotincon”ictwiththeexistenceofhigher-order contributionstothe -function.Aswementionedatthebeginningofthissection,the chiralcu rrentthatisinthesamemultipletwiththeenergy-momentumtensorisnotthe onewehavediscussedhere,buttheR-symmetryaxialcurrent.Itisamemberofthe superc urrent de“nedbycouplingtosupergravity,andingeneralitsanomalydoesreceive higher-orderradiativecorrectionsasdoth eanoma liesofitssupersymmetricpartners.

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Contentsof 7.QUANTUMN=1 SUPERGRAVITY 7.1.Introduction408 7. 2. Ba ck gr o und-quantumsplitting410 a.Formalism410 b.Expandingtheaction415 7.3.Ghosts 420 a.Ghostcounting420 b.Hiddenghosts424 c.Morecompensators426 d.Choice ofgaugeparameters429 7.4.Quantization431 7.5.Supergravitysupergraphs438 a.Feynmanrules438 b.Thetransve rsegauge440 c.Linearizedexpressions441 d.Exam ples 443 7.6.CovariantFeynmanrules446 7.7.Generalpropertiesoftheeectiveaction452 a.N=1 452 b.GeneralN455 7.8.Examples460 7.9.Locallysupersymmetricdimensionalregularization469 7.10Anomalies473 a.Introduction473 b.Conformalanomalies474 c.Classicalsupercurrents480 d.Superconformalanomalies484 e.Localsupersymmetryanomalies489 f.NottheAdler-Bardeentheorem495

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7.QUANTUMN=1 SUPERGRAVITY 7.1.Introduction Thequantizationofsuper“eldsupergravitypresentsanumberofnewfeatures andcomplicationsthatwedi scussinthischapter.Oncethegauge-“xingprocedureand ghoststructurehave b eendeterminedFeynmanrulescanbeobtained.Usefulassupergraphsareinglobalsupersymmetry,theirpowerisawesomewhenitcomestodoingperturbationtheorycalculationsinsupergravity.Muchofthesimplicityofsuper“eldcalculationsinsupergravity,ascomparedwithco mponentcalculations,occursbecause,asin globalsupersymmetry,wedealwithobjectshavingfewerLorentzindices.Thesupergravitysuper“eldisaLorentzvector,asc omparedtot heLorentzsecond-ranktensor andvector-spinorofcomponentsupergravity.ConsequentlytheinteractionLagrangians havefewerterms,andthetensoralgebraismu chsimple r.Forexample,thethree-gravitonvertexcontains171terms,whilethecorre spondingthree-vertex insupergravityconsistsofonly27.Asaresult,itispossibletodocalculationsinsuper“eldsupergravity thathavenotevenbeenattemptedinordin aryquantumgravityorcomponentsupergravity. Theinvestigationofthedivergencestructureofquantumsupergravityisalsovery mu ch fa c ilitatedbytheuseofsuper“elds.Manycancellationsduetosupersymmetry happenautomatically,anditismucheasiertolistandunderstandthepossiblecounterterms.Incomponentcalculations,withnon-supersymmetricgauge-“xingtermsforthe gravitonandgravitino,thecorrespondingcancellationsdonotoccurautomatically,and it is mu ch morediculttodeterminewhatin“nitiesmightbepresentorabsent. Background “eldmethodsplayacrucialrolehere.Sincethecalculationsarenever veryeasy,andthealg ebradoesgetcomplicated,itisessentialtokeepsomecontrolof thegaugeinvarianceofthetheory,andthisisbestaccomplishedbyworkinginthemanifestlygauge-invariantbackground“eld formalism.Inparticular,wehavetheusual propertyth atalldivergencesaregaugeinvariant:Theformalismavoidsthenoncovariantdivergencesofgravitationaltheoriesquantizedinnonbackgroundgauges.Weshall se et ha t,justasinYang-Millstheory,thebackground“eldquantizationhasthefurther virtueofsimplifyingsomeofthevertices.Italsoallowsustoworkwithonlybackgroundcovariantderivativesratherthanprepotentials,andconsequentlyleadstosome

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7.1.Introduction409improvem entinthepowercountingrulesforthetheory. Thenewfeatureofthequantizationprocedureistheappearanceoflargenumbers andnewtypesofghosts,besidestheFaddeev-PopovandNielsen-Kalloshghosts.They ariseeitherbecauseofcertainconstraintst hatthegauge-“xingfunctionssatisfy,orare introducedtoremovecertainnonlocalitie sthathaveb eenproducedbythegauge-“xing procedure.Inaddition,we“ndnumerousghosts-for-ghosts. Wedisc uss“rstthebackground“eldquantizationprocedure.OrdinaryquantizationFeynmanrulescaneasilybeobtainedfromtheoneswederivebysettingthebackground “eldstozero,butinpuresupergravitythereislittleadvantagetousingthem: Ingeneral, L -loopcalculationsinordinary“eldtheorypresentaboutthesamelevelof dicultyas L +1-loopcalculati onsinthe background“eldmethod.Inthefollowing sectionswediscussthebackground-quantumsplitting,whichwepatternaftertheonein Ya ng -M illstheory,thenumberandkindsofghostsonemayencounter,andthechoiceof gauge-“xingfunction.OncewehavetheLagrangian,wecandiscussgeneralproperties oftheeectiveaction,derivesuperg raphrules,anddoloopcalculations. Weconsid eronly n = Š1 3 supergravity.Insec.7.10.eweshallarguethat N =1, n = Š1 3 theoriesareinconsistentatthequantumlevelduetoanomaliesintheWard identitiesoflo calsupersymmetry(exceptwhenpartofanextendedtheory).

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4107.QUANTUMN=1SUPERGRAVITY7. 2. Ba ck gr o und-quantumsplitting Wedivi dethediscussionintwoparts:thesp littingitself,andtheexpansionof theaction.AsintheYang-Millscase,thesplittingintoquantumandbackground“elds is nonlinearanddoneintermsofexponentials.Thesimplestwaytounderstanditisas anexpansionofthe(constrained)covariantderivativesintermsofunconstrainedquantumprepotentials(neededforquantization) andconstrainedbackgroundderivatives; thesimplestwaytoobtainitisbyre-solvingtheconstraints,asinsec.5.3.,butusing backgroundcovariantinsteadof”atsupersp acederivatives.Exceptforsomesmallmodi“cationsexplainedbelow,theresultscanbewrittenalmostimmediately. Theexpansionoftheactionisalgebraicallylengthy,butstraightforward.Wegive th ep ar tq uadraticinthequantum“elds,buttheprocedurecanbeextendedfor“nding higher orderterms. a.Formalism Thequantum-backgroundsplittinginsupe rgravityfollowsapatternverysimilar tothatofYang-Mills,andwesimplyrepeatithere,referringthereaderbacktosec.6.5 formotiv ationandanexplanationoftheprocedure.Westartwiththeconventional derivati ves(with de gauged U (1);seesec.5.3.b.8) A= EA MDM+(A M +A€€ M€€), [ A, B} = TAB CC+( RAB M + RAB€€ M€€);(7.2 .1) covariantunderthe(vector-representation)transformations: A = eiKAeŠ iK, K = K ; K = KMiDM+( K iM + K€€i M€€).(7.2 .2) Thesolutiontotheconstraintsexpressesthederivativesintermsoftheunconstrained prepotentials H€and andordinary”at-spacederivatives DM,inthechiral representation,asinsec.5.3.Forthetimebeingweuse achiraldensi tycomp ensator.Weachieve ourquant um-backgroundsplittingbysubstitutingintothesolutionbackgroundcovariantderivativesforthe”atderivatives.The“elds H€and arethequantum“elds,

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7.2.Background-quantumsplitting411whilethebackground“eldsappearimplicitlyinthebackgroundderivatives. Wetake thebackgroundcovariantderivatives Ainthev ectorrepresentation, €=( ), i a=( i a),andrequiret hemtosatisfythes ameconstraintsas A. Thisimmediatelyallowsustosolvetherepresentationpreservingconstraints: {, } = T + R( M )(7. 2.3) anditshermitianconjugateareobviouslysatis“edby(cf.sec.5.3.b.2) €=( €+ € M + €€€ M€€),(7.2 .4a) = eŠ H ( + M + €€ M€€) eH,(7. 2.4b) where H = HAi Aintroducesth equant um“eld HAwhichalsoappearsinandthe quantumconnection .Wehavewri ttenthederivativesinachiralrepresentationwith respectto HA.Repla cing eH= ee andmulti plyingallquantitiesby e fromtheleft and eŠ fromtherightwouldtakeustoaquantumvectorrepresentation.However,for quantization,itissimplertoworkwith H Itisalsoconvenienttode“ne ba ckgroundcovariant ha ttedobjectsasin(5.2.23), butfrombackgroundcova riantderivativesand H : €= €, = eŠ H eH, €= Š i { €} ;(7. 2.5) A= EA B B+( A M + A€€ M€€)= Š ( Š 1)AeŠ H ( A) eH;(7. 2.6) andd e“ne TAB Cand RABintermsofthem.Wealsode“nethesuperdeterminants,with theirappropriatehe rmiticityconditions: E = sdetEA M, E E = sdet E EA M, E = sdet EA B; EŠ 1=( EŠ 1)eŠ H, E EŠ 1=( E EŠ 1), EŠ 1E EŠ 1=( EŠ 1)E EŠ 1eŠ H.(7. 2.7) Wehave usedtheidentity,foranyfunction f f E EŠ 1HE E = Hf + fi ( Š 1)A AHA(7.2.8) (dro ppingtheterm iHA( E EŠ 1 AE E )= Š iHA( Š 1)BT T T TAB B=0(see (5.3.42)).The

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4127.QUANTUMN=1SUPERGRAVITYbackgroundcovarianti zationofth eoperator eŠ Histhus eŠ E EŠ 1HE E= E EŠ 1eŠ HE E .Itresults inthehermiticityconditions(from(5.3.51b)and(7.2.9)) (1 eŠ E EŠ 1HE E)Š 1= eŠ H(1 eE EŠ 1HE E), EŠ 1=( EŠ 1)eŠ E EŠ 1HE E.(7. 2.9) Weuseaconv entionalconstrainttodeterminethevectorcovariantderivative €= Š i {, €} ,(7. 2.10) andconventionalconstraints T€€€=0(orequiv alently T€ (€ € )=0)and T€ ( € )€=0to determinethespinorconnections: €€€= Š € (€ € )ln Š ( ), € = Š1 2 T€ ,( € )€Š ( €€);(7. 2.11) (compareto(5.3.55)and(5.3.25)afterdegauging). Finally,weimposethe n = Š1 3 conformal-breakingconstraint,whichdetermines byaproce duresim ilartothatofsec5.3: = Š 1( eŠ H )1 2 (1 eŠ E EŠ 1HE E)1 6 EŠ1 6 ,(7. 2.12) where isbackgro undcovariantlychiral: € = =0.Forqu antumcalculations, wehaveto eitherexpress exp licitlyintermsofanordinarychiralsuper“eldandthe backgroundgauge“eldor,whatispreferablei ngen eral,derivecovariantFeynmanrules thatallowustoworkwithitdirectly. Thefullderivatives Atr an sformcovariantlyundertwosetsoftransformations: (a)Backgroundtransformations: A = eiK AeŠ iK, H= eiKHeŠ iK, = eiK eŠ iK, ( M )= eiK ( M ) eŠ iK, A = eiKAeŠ iK, K = K = KAi A+( K iM + K€€i M€€).(7.2 .13)

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7.2.Background-quantumsplitting413Thesetransformationsfollowfromthe requirementthatthefullderivative Atransform covariantlyand bebackgr oundcovariantlychiral:From(7.2.4a)or(7.2.6,12)itfollowsthatand transformcovariantly,andtherefore,from(7.2.4b) H transforms covariantly. (b)Quantumtransformations: A = A;( 7.2.14a) eH= ei eHeŠ i eX (), =ei Š1 3 ( a aŠ Š i G G a a) ,(7. 2.14b) A = LA B() ei BeŠ i ;(7. 2.14c) where =Ai A = ,[ €,] =0;(7. 2.14d) foranybackgroundcovariantlychiral ( € =0).Weha vetakenthest a ndardchiral representationtransformations(5.2.16,67)forthequantumsuper“elds H and except forcertainmodi“cationsrequiredbecauseofthe Asin H and.Since [ H ,]=( H AŠ HA) i AŠ BHA( T TAB C C+ R RAB( M )),(7.2.15) ei eHeŠ i generatesLorentztransformationterms.Weintroduce X ()= X M + X€€ M€€tocancelthem.Similarly,theusualtransformationlawof hastheadditional G G a atermb ecausewerequire tobebackgroundchiraland €( a aŠ Š i G G a a)=0.Thetrans formation (7.2.14c)of Afollowsfrom (7.2.14a,b).The-dependentLorentztransformation LA B()correspondstothe A Btermin(5.2.21)withadditionalcontributionsfrom X (). Inpractice,weneverneedtocompute X exp licitly.Using(7.2.15)andtheBakerHausdortheorem, ei eHeŠ i = eH+ Y ( M )(7.2.16) forsomeL orentztransformation Y ( M ).Si nce Hisascalaroperator,

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4147.QUANTUMN=1SUPERGRAVITY[ H, Y ( M )]= Y( M )( 7.2.17a) forsomeL orentzrotation Yandhence eH+ Y= eHeŠ X,(7. 2.17b) thus de“ning X .Intro ducing X into(7.2.14b)isequivalenttotheprescriptionofdroppingatanyst ageofthecalculationLorentzterms other thanthoseimplicitin A.For convenience,weintroducetheoperation <> thatremovesexplicitLo rentzgeneratorsas fo llows:Forany A = AA A+( A M + A€€ M€€), (7.2.18a) wede “ne < A > AA A, < eA> e< A >.(7. 2.18b) Thetransformationlawfor H canberewritten eH= < ei eHeŠ i > .(7. 2.19) Thequantum-backgroundsplittingwehavedescribedisequivalenttothefollowing spli ttingoftheprepotentialintermsofaquantumchiral H andbackgroundvector : eH ( split )= < e eHe > ,(7. 2.20) whichisanalogoustotheYang-Millscase(6.5.25).Theusualchiralrepresentation transformationlaw < e eHe >= ei 0< e eHe > eŠ i 0(7.2.21) canberewrittenaseitherbackground(7.2.13)orquantum(7.2.14)transformations analogoustothoseof(6.5.27). Asinthenonbackgroundcase ,thequantumt ransformationsmustpreservechirality(7.2.14d).Therefore,takesthefollowingform,expressingitintermsoftheunconstrainedsupergravitygaugeparameter L(cf.(5.2.14)): €= Š i €Š 3L,= 2Š 3L.(7. 2.22) (Wehav emadethere de“nition L Š 3Ltosimplifyquantization,aswillbe explainedinsec.7.4.)Furthermore,wechoosethe(quantum)€-gauge H= H€=0

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7.2.Background-quantumsplitting415(see(5.2.18)),whichdetermines€intermsof L: €= eŠ H 2 Š 3 L€+ O O ( R R G G W W ).(7.2 .23) Thissupersymmetricgaugechoicedoesnotintroduceanyghosts. Wenowtaket heclassicalsupergravityaction( 5.2.48)intermsoffullsuper“elds andexpressitintermsofthequantumgauge“eldsandthebackgroundcovariant derivati ves,using( 7.2.4-12): SC= Š32 d4xd4 EŠ 1, EŠ 1= E EŠ 1 EŠ1 3 (1 eŠ E EŠ 1HE E)1 3 eŠ H ,(7. 2.24) Thisexpressionisthedirectbackgroundcovari antizationof(5.2.72),includingafactor of E EŠ 1tomakeitadensity.Thequantum“eldsappearexplicitlyandin E ,wh ilethe background“eldsappearimplicit lyincovariantderivativesand E E b.Expandingtheaction Ournexttaskistoe xpandtheactioninpowersofthequantum“elds.Thisisa tedious buthealthyexerciseandweoutlinethestepsneededtogetthequadraticpart, whichweneedfordiscussinggauge-“xing,andfordoingone-loopcalculations.Cubic andhigher-ordertermsareneededforhighe r-loopcalculations,butwedonotderive themhere. Wemustexpa ndtheexponen tialsandthedeterminant EŠ1 3 in(7.2.24)inpowers of H .We“rstd e“neAby EA= < A> = EA B B ( A B+A B) B= A+A.(7. 2.25) Theexpansionof EŠ1 3 isthen,toq uadraticorderin(and H ): EŠ1 3 =[ sdet (1+)]Š1 3 = exp ( Š1 3 strln (1+)) =1 Š1 3 str +1 18 ( str )2+1 6 str (2)

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4167.QUANTUMN=1SUPERGRAVITY=1 Š1 3 ( Š 1)AA A+1 18 [( Š 1)AA A]2+1 6 ( Š 1)AA BB A.(7. 2.26) To “ndtheexplic itformofA B,wereturnt o(7. 2.5,6)andwriteAexp licitly.Since < €> = €= €, €=0 .( 7.2.27a) From < > = < eŠ H eH> = + < [ H ] > +1 2 < [[ H ], H ] > + < O ( H3) > (7.2.27b) weobtain B(+ B)fromtheco ecientof Bontherighthandsideof(7.2.27b). Since H isascalaroperator,wecandropLorentzrotationtermsproducedbythecommuta torsateachstage,becausetheycanpr o duceonlymoreLor entz terms(see (7.2.17a)).Weobtain a Bfromtherighthandsideof < a> = Š i < { €} > = aŠ i < { €,[ H ] } > Š i1 2 < { €,[[ H ], H ] } > .(7. 2.27c) AgainwecandropLorentztermsatintermedi atestagesofthecalc ulation:Theycontri buteonlyto a€,andsi nce€ B=0 (toallorders),theyd onotcontri butetothe determinant E .We“rs t “nd Btolowestorderin H : [ H ]=[ H bi b]= i [ H b] b+ iH b[ b] = i ( H b) bŠ H€( R R €Š G G€ )+ Lorentzterms andhence = Š H€G G€, b= i H b.(7. 2.28) (Againwecanignore€.) Pr o ceedinginthismannerwethen“nd,totheorderin H€necessaryforthe quadraticaction( H H€ €):

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7.2.Background-quantumsplitting417 = Š (1 Š1 2 iH ) H€G G€Š1 2 ( H€) H€[1 2 (€G G€ )+ €€( W Š1 2 ( )R R )] +1 2 H€G G€H€G G€Š1 2 R R R R H2, b= i H b, a = Š i ( Š €H€G G€+ R R H€), a b= € H b+ €€ .(7. 2.29) Wehave usedthefactthat,forthepartoftheactionquadraticin H ,wen eedonlythe linearpartsof €and€ ,andwecanalsodro panyto talderivativesofquadratic terms(but notifweweretocomputehigher-ordertermsintheaction).Aftersubstituting(7.2.29)into(7.2.26)we “nd,againdroppingirrelevantterms, EŠ1 3 =1 Š1 3 { € H€Š (1 Š1 2 iH ) G G H Š1 4 ( H€) H€[ (€G G€ )+ €€(2 W Š ( )R R )] +1 2 [( G G H )2Š 2 G G2H2] Š R R R R H2}+1 18 ( € H€Š G G H )2+1 6 {( € H€Š €€H€G G€)( € H€Š €€H€G G€) Š 2( €H€G G€Š R R H€) H€Š [( G G H )2Š 2 G G2H2]}.(7. 2.30) Wealsohavether elevanttermsof(using(7.2.10),andexpanding =1+ ): (1 eŠ E EŠ 1HE E)1 3 =1 Š1 3 i H +1 9 ( H )2, eŠ H =1+( + )+ Š iH .(7. 2.31) Finally,weobtainthequadraticpartoftheLagrangianbymultiplyingtogetherthe

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4187.QUANTUMN=1SUPERGRAVITYvariousc ontri butions,toobtain: E E EŠ 1=1+( + +1 3 G G H )+ Š1 3 i ( Š ) H +1 3 ( + ) G G H +1 3 R R R R H2+1 18 ( G G H )2+1 12 ( H )2Š1 36 ([ €, ] H€)2Š1 18 ( G G H )[ €, ] H€+1 3 R R ( H€)( H€) +1 12 ( H€) H€[ (€G G€ )+ €€(2 W Š ( )R R )] Š1 6 H€( € € + € € ) H€.(7. 2.32) Byusingtheidentity(with =1 2 € €) € € + € € = CC€€(Š + { 2, 2}Š1 2 [ Š R R €+ G G€ +( G G€) M+ W€€€ M€€, €])+2 R R R R M M€€Š ( ( R R ) ) M€€,(7. 2.33) wecanr ewrite(7.2.32)as E E EŠ 1=1+{ + +1 3 H€G G€}+{ +1 3 i ( Š ) €H€+1 6 H€ H€+1 12 ( €H€)2Š1 36 ([ €, ] H€)2Š1 3 [( 2+3 2 R R ) H€][( 2+3 2 R R ) H€]}+{1 3 ( + ) H€G G€+1 18 ( H€G G€)2+2 3 R R R R H€H€+1 12 ( 2R R + 2 R R ) H€H€+1 6 H€( R R 2+ R R 2) H€Š1 12 H€G G€[ €, ] H€+1 12 H€[( ( G G )€) €H€+( (€G G€ )) H€] Š1 18 ( H€G G€)[ €, ] H€+1 6 H€( W H€+ W€€€ €H€)},(7. 2.34)

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7.2.Background-quantumsplitting419wherewehavealsousedtheBianchiidentities(5.4.16,17,18).Theexpressioninthe “rstsetofbracesislinearinthequantum“elds.Ifsourcesarecoupledtothem,variationwithrespectto H and gives R R = J and G G€= J€,using d4xd4 E EŠ 1 = d4xd2 eŠ 3( 2+ R R ) = d4xd2 eŠ 3R R (7.2.35) (seesec.5.5.e;weareinthebackgroundvectorrepresentation).Theexpressioninthe secondsetofbracesisthedirectcovariantizationofthefreeLagrangian,whichis obtainedbysettingallbackground“eldstozero: Š1 3 IL(2)= +1 3 i ( Š ) €H€+1 6 H€[ Š D2 D2Š D2D2] H€+1 12 ( €H€)2Š1 36 ([ D€, D] H€)2.(7. 2.36)

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4207.QUANTUMN=1SUPERGRAVITY7.3.Ghosts Inthenextsectionweshalldiscussindetailthequantizationofsuper“eldsupergravity.Theprocedureisnotentirelystra ightforwarda ndrunsintoanumberofsubtletiesnotnormally encounteredinsimplertheories,soitisdesirabletoknowaheadof timethegeneralghoststructureofthequantizedtheory.Thisisthetopicofthepresent section.Wediscussthefollowingsubjects:(a)howthelinearizedghoststructurecanbe easilydeterminedbeforeperformingthequantization;(b)themodi“cationstotheFaddeev-Popovprocedurenecessarywhenusingconstrainedgauge-“xingfunctionsinthe pr esenceofbackground“elds;(c)howtoobtainonlypropagatorsthatgoas pŠ 2,and avoidinfra reddiculties,whilestillkeepingtheactionlocal,bytheintroductionof additional“elds;and(d)thenecessityforappropriateparametrizationofthegauge transformations(forwhich(density)compensatorsarecrucial)sothattheFaddeevPopovproce dureisapplicable,andsothatweonlyusethetypesofsuper“eldsallowed in an ar bitrarysupergravitybackground. a.Ghostc ounting We“rstgiveasim pleruleforcountingallthegh ostsinanygaugetheory.In ordinarygaugessomeoftheseghostsmaydecouple,butinbackground“eldgaugesall th eg hostscoupletothebackground.Webeginbyderivingtherulesforageneralcomponent-“eld gaugetheory.Tostreamlinenotation,wedropallindicesandindicate abnormal-statistics“eldsbyprimes.ThegeneralquadraticLagrangianforanygauge “eldcanbewri tteni ntheform A n A ,whereisa projectionoperatorand n isan integer(when A isatensor)orhalf -integer(when A isaspi nor: 1 2 /).Forphysical “elds n =1or1 2 .(Theope rators /,etc.maybecovariantw ithres p ecttobackgr o und“elds,andmayincludenonminimalcouplingstothebackground.)Thegauge invarianceis expressedas A = ,with =0.After gauge“xing,theLagrangian b ecomes A nA but,inordertocancelthe A = mode,whichdidnotoccurinthe originalLagrangian,wemustintroduceaghost B.ItsL agrangianisobtainedthrough thesubstitution A Š > A= Binthegauge-“xedLagrangian.Wethusobtain IL = A nA +( B) n( B)= A nA + B n +1B(7.3.1) whereisanewproj ectionoperator.Inthesimplestcases=1(e.g .,if A isthe

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7.3.Ghosts421photon“eld)andwearethrough.Moregenerally(e.g.,if A isanantisymmetrictensor gauge“eld) Bhasagaugeinvariance,andwemustcontinuetheprocedureuntilno gaugeinvarianceremains.(Thisisa“nitepro cedure,sinceeachghosthasonelessvector i ndexthanitspredecessor). The“nalLagrangianthushastheform IL = A nA + B n +1B+ C n +2C + ... .(7. 3.2) Fortenso r“elds,a “eldwithkine ticoperator mrepresents m “eldsofthattypewith kineticoperator ;forsp inors, mrepresents2 m “eldsw itha /.(If incl udesbackgroundinteractionstheircontributiontotheeectiveactionis lndet m= mlndet =2 mlndet /.)Thus,for physicaltensor“elds A ( n =1), the numberofsu ccessive“eld sgoesas1, Š 2,3, Š 4,...,whileforphysicalspinor“elds ( n =1 2 ),theygoas1, Š 3,5, Š 7,...,wheretheminussignsindicateabnormalstatistics. Thesenumbersrepresentthenetnumberofnormal-statisticsminusabnormal-statistics quantum“eldsinthelinearizedLagrangian,allcouplingtothebackground“elds.Furthermore,aswewillseebelow,allthe“elds inthiscounti ngdecoupleathigherloops (andatoneloopwheno neisquantizinginordinarygaugesratherthanbackground“eld gauges)exceptforthephysical“eldsand(fornonabeliantheories)theFaddeev-Popov ghostsofthephysical“elds.Theremayalsobeadditionalcompensating“eldscoming inpairsofoppositestatistics(catalysts:seebelow)whichcancelinthiscounting,and whichalsocancelinone-loopbackground“eldcalculations(butmaycontributefor higherloops).Exampleswherethiscountingincludesmorethanjustthephysicaland Fadd eev-Popov“eldsare:(1)thegravitino,whichhas1, Š 3inste adof1, Š 2,duetothe appearanceoftheNielsen-Kalloshghost(seebelow);(2) p -forms,whichhave 1, Š 2,3 ,...,( Š 1)p( p +1)inste adof1, Š 2,4,...,( Š 1)p2p, duetoNielsen-Kalloshghosts andhidde nghosts(seealsobelow). Generalizationofthecountingrulestosuper“eldsisstraightforward,thougheach casehastobetreatedseparatelybecauseofthegreatervarietyofsuper“eldgaugetransformations.We“rstgeneralizetosuper“eld stheresultof thepreviousparagraphfor obtainingthenumberofsuper“eldswithst a ndardkinetictermcorrespondingtoone withahigher-derivativekineticterm.When misthekineticoper atorforgeneral unconstrainedsuper“elds,itisequivalentto m generaltensorsuper“eldswithkinetic

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4227.QUANTUMN=1SUPERGRAVITYoperator ,or2 m generals pinorsuper“eldswithkineticoperator /.(Note that,asdiscussedinsec.3.8, sdet =1 unless hasnontr ivial -dependence.)However,forchiral super“eldsthesituati onisslightlymoresubtle(see(3.8.28-36)): d4xd2 m m i =1 d4xd2 i i 2 m i =1 d4xd4 ii,(7. 3.3a) d4xd4 m 2 m +1 i =1 d4xd4 ii,(7. 3.3b) d4xd4 € mi € m +1 i =1 [1 2 d4xd2 + h c .].(7. 3.3c) The formof(7.3 .3a)givestheresultforchirals calarsuper“elds,whereasthe formisapplicabletochiral(undotted)spinorsuper“elds.Thislatterresultcanalsobe relatedto(7.3.3c)bynotingthat(7.3.3a)for m =1and( 7.3.3c)for m =0arem erely dierent gaugechoicesforthegauge-“xedactionforthetensormultiplet(cf. (6.2.32-34)). Wenowconsi dersomeexamplesofsuper“eldgh ostcounting.Thesimplestisthe v ectormultiplet.TheclassicalLagrangianis V 1 2 V ,with V = i ( Š ),whereis chiral and1 2 isthesuperspin1 2 projectionoperator.Aftergauge“xing(whichremoves 1 2 )andth es ubsti tution V Š > i ( Š )withchira lghosttocan celthegauge modes,weobtain IL = V V + .(7. 3.4) (With d4 integrationthe and termsgivezero,modulononminimalcoup lingswhichweincorporateintothede“nitionof .)TheghostLagrangianis equivalenttothreeoftheusualterms (see(7.3.3b)).Wethusexpectthreechiral ghosts,whichagreeswithourresultsfr omexplicitquantizationinsec.6.5. As econdexampleisthatofthetensormultiplet,withclassicalactiond4xd2 1 2 +andgaugeinvariance = i D2DK K = K .After gauge“xing (whichremovest heprojector1 2 +),substitutionleadstoa“rstgenerationghost Lagrangian V 21 2 V,andits gaugeinvariance V= i ( Š )leadst oasecond

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7.3.Ghosts423generationghostLagrangian 2 .(The gaugeinvarianceof Visthesam easthe gaugeinvarianceofthe variation of : ( K )= ( K + i Š i ).)Ween dupwith one term,two V Vterms,a nd“ve terms. Athirdexam pleisthatofthegeneralspinorsuper“eld ,whicht ogetherwitha compensatingchiralscalardescribesthe(3 2 ,1)multi plet(seesec.4.6;weareusingthe secondformof(4.6.42),butwiththecompensator V gaugedtozero).TheLagrangian hastheform IL =1 2 €i €+ h c Š 2 + crossterms (7.3.5) whereisasumof projectionoperatorsand IL hasthegau geinvariance =+ iDK = Š D2K ,withchiralandreal K .Inthe gauge“xed Lagrangianisabsentandwehaveachiralspinorghost andarealscalarghost V(correspo ndingtoand K ,resp ectively)withoutanyfurthergaugeinvariance(the variation ,is not in va riantunderanychangesofor K ): IL = €i €+ + €i € + V V(7.3.6) ( crosstermscanbeeliminatedbyasuitablechoiceofgauge“xingfunction). TheotherformofthetheoryhastheLagrangian1 2 €i €+ h c .+ ... witha dierent,and gaugeinvariance = i D2DK1+ iDK2.Afterin cludingghosts,it b ecomes IL = €i €+ V 2 V 2+ V 1 2V 1+ 2.(7.3 .7) Thechiralscalar“eldisasecond-generationghost,arisingfromtheinvariance V 1= i ( Š )(duetotheinvarianceof under K1 K1+ i ( Š )).Wethus obtaintheequivalentofthreerealscalargh ostsand“vechiralscalarsecond-generation (normalstatistics)ghosts. Weconsid ernow n = Š1 3 supergravityitself,w ithkineticLagrangian H H + (+ H crossterms),whereisasumof projectionoperators.Wehave the(linearized)gaugeinvariance H€= D L€Š D€L, = D2DL,withg eneral spinorgaugeparameter L.After gauge“xingwehaveaLagrangian H H + + € i € ,wheretheg hosttermis obtainedbysubstitutioninthe

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4247.QUANTUMN=1SUPERGRAVITYgauge-“xedLagrangianwiththegaugeparameter Lreplacedbytheghost .We haveanew gaugeinvariance, =whereischiral.(Thisre”ectstheinvariance of th eo riginalgaugetransformationsunder L=.)Wethusintro duceasecondgenerationchiralghost andare“nallyledtotheform IL = H H + + /+ / .(7. 3.8) Weobtain theequivalentofthree“rst-generationgeneralspinorghosts,withLagrangian / ,andtwos econd-generationchiralspinorghosts.Inthenextsectionwewillderive theseresultsfromthegauge“xingprocedure,andgivetheresultsforavariantformof the n = Š1 3 compensatorwhichleadstoadierentsetofghosts. b.Hiddenghosts InadditiontotheNielsen-Kalloshghost,whichemergesfromacarefulapplicationofthegauge-averagingprocedure(see( 6.5.12,13)),thereisasecondsubtletythat mayoccur,andwhichmustbehandledcorrectlyinordertoarriveatthecorrectsetof ghosts.Thishastodowiththeoccurrenceofgauge-“xingfunctionswhichsatisfyconstraints.Inthenonsupersymmetriccasethesimplestexampleisgivenbythe2-form A a binabackgroundgravitational“eld.ThenaivetHooftgaugeaveraging IDf a ( bA a bŠ f a) exp ( Š d4xg1 2 f2)= exp [ Š d4xg1 2 ( bA a b)2](7. 3.9) wouldgivea nincorr ectresult,sinceth econstraintinthe functi onalimplies f =0, andintroducesextraneousde pe ndenceontheexternalgravitational“eld.Wewould ther eforeliketoputjustthetransversepartof f inthe functional,and inthegaugeaver agingfunctionas f21 2 f a( a bŠ1 2 a Š 1 b) f b.Howev er,sincethenonlythe transver separtof f appearsinthefunctionalintegral,theintegrandhasagaugeinvariance f a= a ,sowemustint roduceappropriategauge-“xingand(Faddeev-Popovand Nielsen-Kallo sh)ghostterms.Theintermediatestepsvarydependingonthechoiceof gauge-“xingfunction(e.g., f vs. Š 1 f ),butthenetresult isthatoneobtains Š 1 additionalscalar“elds,(a hiddenghost)andthusthetotalsetof“eldsconsistsof1 2-form, Š 21-fo rms,and+3scalars(vs.the+4expectedfromconsideringjusttheFaddeev-Popov ghostsofthevectorghostsofthe2-form),inagreementwithourgeneral countingargumentgivenabove.Similarargumentsapplytohigher-rankforms.

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7.3.Ghosts425Asimplerformofthea rgumentforthenecessityofthesehiddenghostscanbe giveninsupersymmetrictheories.Consideragainthechiralspinorgaugesuper“eld, withgaugeinvariance = i D2DK andgauge-“xingfunction F = F =1 2 i ( DŠ D€ €).Becauseofthechiralityof, F satis“estheconstraint D2F =0,sothat F isalinearsuper“eld.T heusualgauge-“xingprocedureinvolves introducingi nthef unction alintegral ( F Š f );however,thelinearnatureof F would implyt hat f isalsolinear,anunfortunatefeaturesinceitisimpossibletofunctionally integrateordierentiatewit hresp ecttolinearsuper“elds. Thisdicultycanbeavoidedbycompleting F t oagen eralsuper“eld,bythe additionofchiralandantichiralpiecestoit.Wedothisbyreplacing F inthe functionalwiththeexpression F = F +( D2 Š 1 + D2 Š 1 ).(7.3 .10) andfunctionallyintegratingover aswell.Thechi ralsuper“eld isthehidde nghost. Now D2 F = is unconstrained,andso f isalso. To understandtheprocedureweexamineitscomponentform.The -function ( F Š f )isapro ductof -functionsfortheindividualcomponentsof F Š f (see (3.8.17a)).Since F isthe1 2 partof F andtherestisthe0part,thecomponentsof thetwotermsin(7.3.10)appearindierentcomponent -functions.The D2f | D D2f | and D2 D2f | componentsaresetequaltocomponentsof withoutspacetimederivatives (whichiswhyw eincl udedthe D2 Š 1factor), andwitho utany F =1 2 i ( DŠ D€ €) contributions.Averagingwith expd4xd4 f2producesanac tionforthese componentsthatdoesnotcontributetothefunctionalintegralupon integration(trivial kineticterms).The f | Df | componentsproducestandardgauge-“xingtermsforthe gaugecomponentsofwhichareabsentintheWess-Zuminogauge(namely and B (4.5.30)),andwhosecontribut ionisthereforecan celedbycorrespondingghosts.Finally, the[ D€, D] f | componentgivesthegauge-“xingfunction bAa b+ a Š 1G ,where G = ImD2 | and A a bistheantisymmetrictensorcomponentof.Aver agingofthis componentof f givesaterm( bA a b)2Š G Š 1G :the usual A a bgauge-“xingtermasin (7.3.9)plusthehiddencomponentghost(+1scalarwith Š 1,countingas Š 1scalar with ),inagreementwiththeabovediscussioninthecomponenttheory.

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4267.QUANTUMN=1SUPERGRAVITYAsim ilarsituationoccursinsupergravity duetotheappearanceofthegauge-“xingf unction F= D€H€satisfying D2F=0.Thisw illbediscu ssedinmoredetailin thenexts ection. Notethatthesehiddengho stscoupleonlytobackground“elds,andthuscontri buteonlyatoneloop.Itwouldbedesirabletohaveageneralderivationofthese ghostsbasedonBRSTinvarianceofthegauge-“xedaction,fromwhichSlavnov-Taylor identitiescouldbederived.TheappropriateBRSTtransformationswouldbethose whoseSlavnov-Tayloridentitiesimpliedgauge-independenceoftheeectiveaction.At presentthisapproachhasnotbeenworkedout. c.Morecompensators UnwantedtermsintheLagrangian,suchasthoseleadingto pŠ 4termsinthe pr opagatorornonlocalvertices,cansometimesbecanceledbyintroducingadditional “eldsandgauge-“xingthemconveniently.Sinceonlytheghostsdiscussedintheprecedingsectionsaren eededtopreserveunitarity,contributionsofthesecatalyst“eldsmust themselv esbecanceledbytheirownghosts,andindeedthishappensattheone-loop level.Thecatalystsmayingeneralinteractwiththeotherquantum“elds,andhence contributeathigherloops,whereastheirghostsdont.Ifoneweretointegrateoutthe catalysts,theirhigher-loopcontributionswouldsimplyreproducetheunwantedterms thatthecatalystseliminatedinthe“rstplace. Catalystsarejustatypeof(tensor)compensator.Forexample,thecompensator intheStueckelbergformalism(sec.3.10.a)isintroducedsimplytoimproveultraviolet be ha viorofthepropagator,anddecouplesduetogaugeinvarianceoftheinteraction term.Inourcase,thesecompensatorsimproveinfraredbehavior,anddonotdecouple. Furthe rmore,catalystsgenerallyareintroducedbyghost“elds,whereaspreviouslywe discussedcompensat orsrelatedtoonlyclassical“elds. Asanexample, considerthelinearizedLagrangian IL = A [(1 Š )+ ] A ,with 2=and =0,1.Ifweread ierentialoperator,e.g., Š 1 ,theabove Lagrangianwouldleadto pŠ 4pr opagators.ToobtainthesimplerLagrangian IL = A A onecouldmakea“eldrede“nition A=[(1 Š )+ Š1 2 ] A butifwere nonlocalthiswouldintroducenonlo ca litiesintheinteractionterms.

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7.3.Ghosts427Insteadweintroduceacatalyst“eld B intheLagrangian,eitherwithatrivial kinetictermortogetherwithaghostwhichcancelsit.Wethenmakeshifts A A + OB B B + OA thatcanceltheunwantedterms,andgiveno AB cross terms.Forinstance,intheexampleabove,ifisamatrix,wechoose B sothat B = B (andintroducealsoaghost“eldwithoppositestatisticsand B= B)andwe addittotheLagrangiantoobtain IL= IL +(1 Š ) B B + B B.Firstmakingthe shift B B + A ,thent heshift A A Š (1 Š ) B ,weobtain IL= A A +(1 Š ) B B + B B.Forback groundinte ractions B and Bwillcancelatone loop, butclearlythe A shiftcanleadtoquantuminteractionsof B ( butnot B).An equivalentprocedureconsistsofmakingthesubstitution A A + B intheoriginal Lagrangian,andgoingthroughthegauge“xingprocedureforthenewgaugeinvariance thathasbeenintr o duced,namely A = B = Š ( = ).Ingeneral,thisisthe simplestprocedure. Asasuper “eldexample,weconsiderarealscalar V inthepresenceo fanon-shell backgroundsupergravi ty“eld,w ithLagrangian IL0= V ( Š 2+ a {2, 2} ) V (7.3.11) with a =0,1.Thesuper“eld V has pŠ 4termsinitspropagatorandcomplicatedvertices(couplingtot hebackgroundgravitational“eld),butitcanbeshownthattheresult fortheeectiveactionisindependentof a .Toshowt hisusingcatalyst“eldsweintroducethem,forexample,bymakingtheshift V V +( + ), € =0.(7. 3.12) Wehave nowthegau geinvariance V =+ = Š ,withachiralparameter.We choosethe gauge“xingfunction F = 2( V Š a 1 Š a )and gauge“xingterm 2(1 Š a ) F F ,andareledto theLagrangian IL = V V +2 a 1 Š a .(7. 3.13) The Lagrangianisequivalenttothatforthreeordinarychiral“elds i, i =1,2,3.The gauge-“xingpro cedurealsointroducesthreechiralghosts,justasfortheusual V super“eld(twoFaddeev-PopovandaNielsen-Kalloshghost).Theyexactlycancelthethree ordinarychiral“eldsattheone-looplevel,andleaveuswith V V .

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4287.QUANTUMN=1SUPERGRAVITYIfwehadconsideredasystemsimilartotheabove,butwhere V hadquantum interactions,the swouldalsohavesuchinterac tions.Thentheeectofthe swould betorepro duce,if integratedout,thenonlocalitiesthatwouldhavebeenintroducedif, insteadoffollowingthea boveproce dure,wehadmadeanonlocalrede“nitionof V to casttheoriginalLagrangianinthe V V form.Inthiscase,the(o-shell)Greenfunctionshav egen uine a -dependence. Thegeneralprocedureisthefollowing:ConsidertheLagrangianofanarbitrary super“eld oftheform n(0+i cii) ,(7. 3.14) where0+i i=1and0isaparticularsuperspinchos enforconvenience,e.g.,the highestsuperspinin orthesuperspinthatoccursmostfrequentlyininteractionterms. Ifsomeoftheconstants ciareequal,wemaycombinethe correspondingprojection operatorsiintoasingleone (incl uding0,whichisme relyawith c0=1).Also, some cimayvanish,whichimpliesacorrespondinggaugeinvariance.Wenowintroduce catalystsbytheshifts +i Oi i,iOj= ijOj,(7. 3.15) where Oiareoperatorsthatmaybenonlocal,butonlytotheextentthatallnonlocalitiesintheinteractiontermscaneventually bere moved.Thenwe“xthecorresponding gaugeinvariances =i Oii, i= Š i,(7. 3.16) insuchawaythattheLagrangianfor b ecomessimply n ,andallc rossterms betw een and iarecanceled:Thegauge-“xingfunctions Fi= Oi ( Š ci1 Š ci Oi i)(7. 3.17) withgauge-“xingterms (1 Š ci) Fi n( Oi Oi)Š 1Fi(7.3.18)

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7.3.Ghosts429givetheLagrangian n +i ci1 Š ci iOi  nOi i.(7. 3.19) ( Oi Oiisinvertibleon Fi.Also, itcanbeshownthat Oi( Oi Oi)Š 1Oi =i.)Notethat thisproce dureincludes“xingoftheordinarygaugeinvariance.WethenaddtheFaddeev-Popov andNielsen-Kalloshghostsasi [ 2 i( Oi Oi) 1 i+ 3 i n( Oi Oi)Š 1 3 i+ h c .].(7. 3.20) Intheinteractingcase, Oimustbechosen sothatanybackgrounddependencecomes outlocal,inc l udingextratermswhichmayresultfrommanipulationsofthebackground dependent and Oi.Itmaybe necessarytochoose Oisuchthat ihasitsow n gauge invariancei ndependentof ,ortocombi neseveraliinsuchawaythattheabove Lagrangianfor alsoneeds“xing,inwhichcasetheentireproceduremustberepeated forthose s.Howev er,the salwayshave fewercomponentsthantheir s(atleast for N =0or1supersy mmetry),sotheseriesmusteventuallyterminate. d.Choice ofgaugeparameters Inanygaugetheory,somecareisrequiredtoensurethattheFaddeev-Popov quantizationprocedurewillleadtocorrect,unitaryresults.Onewaytocheckunitarity istocompareresultswiththoseobtainedina ghost-free(e.g.,axial)gauge.InsupersymmetrictheoriessuchagaugeistheWess-Zuminogauge,andonewaytoinsureunitarityisbymakingcertainthatonecanpasssmoothlyfromcovariantgaugestothe physicalgaugewithoutintroducinganyextraunphysicaldegreesoffreedom.Thiswill certainlybethecaseifthesuper“eldgaugetransformationsaresuchthattheyallowthe gau gi ngtozerooftheunphysicalcomponentsbyalgebraic,non-derivativetransformations( A = andnot,e.g., or a a).Forexample,inordinarycomponentYangM illstheorythegaugetransformationcanbewritteneitheras A a= a oras A a= a .Howev er,thelatterchoicewouldgiveaFaddeev-PopovghostLagrangian c c (insteadofjust c c ),andtheextra wouldgiv eanontrivial contribution whichwoulddestroyunitarity.ItispossibletomodifytheFaddeev-Popovprescription tocorr ectlyhandlethesituation,butthesimplestprocedureistochoosethegauge parametersinsuchawayastoavoidtheproblem.

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4307.QUANTUMN=1SUPERGRAVITYInthecaseofsupergravity,itcanbeveri“edbytheprocedurejustdescribed(cf. 5.2.10)thatthe Lparametrizationistheonlycorrectone.Aswesawinsec.5.2.cthis parametrizationcanbeused onlyifwealsohavethecompensator(s)inthetheory. Eliminationofthecompensatorwouldintroduceconstraintsonthegaugeparameter,the solutionofwhichwouldexpress Lintermsofderivativesofoth ersuper“eldparameters. However,asintheexampleabove,thiswo uldintroducespuriousextraghostsinthe naiveFaddeev-Popovprocedureandunitaritywouldbelost,unlesstheprocedurewere modi“ed. Thereisonemorerestrictionwhich mustbeob servedinchoosinggauge parametrizations(andthusghosts):Ingeneral,notallsuper“eldswhicharerepresentationsofglobalsupersymmetryarealsorepres entationsoflocalsupersymmetry.Inparticular,for n = Š1 3 th eo nlytypeofchiralsuper“eldsallowedareoneswithonlyundottedspi norindices:Theexistenceofadottedchiralspinorwouldimply 0= { €, €} €= Š 2 RM€€€= Š RC€ (€€ ) =0.(7. 3.21) Generally,thechoiceofgaugeparametersmu stberestrictedtothosewhichcanexistin an ar bitrarybackground.

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7.4.Quantization4317.4.Quantization Inthissectionwepresentthedetailsofthequantizationprocedureforsupergravity.Thisinvolveschoosinggauge-“xingfunctionswhichallowallkinetictermstotake simpleforms,and“ndingtheresultingghosts(Faddeev-Popov,Nielsen-Kallosh,andhidden).Suchsimpli“cationsoftenrequiretheuseofappropriatecompensatorsand/orcatalysts.Thisproce dureis“rstappliedtothephysical“elds,thentotheresultingghosts, theghostsghosts,etc.(Theg hostsreduceinsizeateachstep,sotheprocedurequickly terminates.)Fornowweworkwit hon-shellbac kgro und“elds( R R = G G =0),so thepart oftheactionquadraticinthequantum“eldsbecomes(see(7.2.24,34)),inunits =1 (ormakingt heusualrescaling( H ) ( H )), S = d4xd4 E EŠ 1[ Š 3 + i ( Š ) H Š1 2 H H Š1 4 ( H )2+1 12 ([ €, ] H€)2+( 2H ) ( 2H ) Š1 2 H€( W H€+ W€€€ €H€)].(7.4.1) Thequantizationwithon-shellbackground“eldsissucientforcomputingphysical quantities(S-matrixelements)inpure N =1andexte ndedsupergravity.Wewilldiscussthegeneralsituationlater. We havethefollowing(o-shell)gaugeinvarianceunderthequantumtransformations(from(7.2.14)and(7.2.22,23)): H€=( L€Š €L)+ O ( H )+ O ( ), = Š ( 2+ R R ) Š 3L Š1 3 [( 2+ R R ) Š 3L] .(7. 4.2a) Notethatthesecondequationcanberewrittenas 3=( 2+ R R ) L.(7. 4.2b) (Onshell weca nset R R =0.) Tocan celthe H crossterms,wechoosethefollowinggauge-“xingfunction: F= €( H€+ i a € Š 1 Š 3)(7. 4.3)

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4327.QUANTUMN=1SUPERGRAVITY(forsomeconstant a tobedeterminedbelow).Thisisthemostconvenientgaugechoice. Itcorrespondstoamodi“cationofthetransversegauge €H€=0(sees ec.7.5.b).We havede“nedthechiraldAlembertians += + W M Š= + W€€€ € M€€,(7. 4.4a) on ar bitrarychiralsuper“elds(i.e.,withanynumberofundottedspinorindices)by + ... = 2 2 ... Š € ...€= 2 2 € ...€.(7. 4.4b) Wehave used 3=(1+ )3in(7.4.3)insteadofjust sothattheFaddeev-Popovprocedurewillcontributenonlocaltermsto onlythekineticterms oftheghosts,andnotto theirquant uminteractions,duetotheformof(7.4.2b).(Thisisthereasonforour introductionof Š 3intothetransformationlaws(7.2.22).)Nonlocalkinetictermscan bemadelocalbyuse ofcatalystghosts,sothisisaharmlessnonlocality,whereasnonlocalinteractiontermswouldbeaproblem. We“xthe gaugeby“rstcompletingthelinea rsuper “eldgauge“xingfunction Ftoagen eralsuper“eld,therebyintroducinghiddenghosts ,ands ubsequentlyaveragingovergaugesbyusingaweightingfunctionthatleadstosomeofthedesiredsimpli“cations:Wewishtocancelall H termsintheLagrangianwhichwouldcontributeto pr opagatorsexceptfor H H ,incl udingthe H crossterms.Thisgaugeischosenby introducinginthefunctionalintegralthefactor ID ID €ID ID € ( F+ 2 Š 1 +Š ) ( F€+ 2 Š 1 Š €Š €) exp{ d4xd4 E EŠ 1 [ Š1 4 ( Š € €)2Š1 12 ( + € €)2+( €)( €)]},(7. 4.5) andcarryingouttheintegralsover .Thisgivesthe gauge-“xingterms andthe hidden ghostaction SGF= d4xd4 E EŠ 1{[1 4 ( H )2Š1 12 ([ €, ] H€)2Š ( 2H ) ( 2H )]

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7.4.Quantization433+[ Š5 3 i ( H )( a 3Š a 3) Š4 3 ( a26+ a2 6)+10 3 a a 3 3] + €i € + Š 1}.(7. 4.6a) Uponlineari zation(andusingtheon-shellcondition R R =0), the termsbecome Š 5 i ( H )( a Š a )+30 a a ,sowechoose a =1 5 (7.4.6b) tocan celthe H crossterms in(7.4.1). Wecanshow, however,thatthehiddenghostLagr angiangivesnocontributions. (Notethatithasnoquantuminteractionsandcontributesatmostattheone-loop level.)Weperformtwosuccessiverotations(withunitJacobian) (1) + ib € 2 Š 1 Š €, € €;(7. 4.7a) (2) € €+ ic € 2 Š 1 +;(7. 4.7b) choose b and c tocan celthe €crossterms,andrewritethehidden-ghostactioninchiralform S = d2 eŠ 3+ h c .,(7.4 .8a) (recallthatthebackgroundisinvectorrepresentation)or,withthe“eldrede“nition Š3 2 S = d2 eŠ + h c .= d2 + h c .,(7.4 .8b) intermsofanordinarychiralsuper“eld = eŠ .Thusthe hiddenghostdecouples fromthebackground“eldandgivesnocontributiontotheeectiveaction.(Thisisjust thecovariantizationof(7.3.3c)for m = Š 1.) TheFaddeev-Popovghosts areobtainedins ta ndardfashionfromthegauge“xingfunctions.Forthetimebeingwewrite onlythekineticterms(arisingfromthe H and independentpartofthegaugetransformation;theremaindergivesrisetoghostquantum“eldinteractions).Aftersomealgebraweobtain

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4347.QUANTUMN=1SUPERGRAVITYŠ ( € Š €)( €Š €) Š1 5 [( 2 ) Š Š 1( 2 € €)+( 2 € €) + Š 1( 2 )].(7.4.9) Wenowwishtos implifytheghostLagrangianbyputtingitinthestandardform € €.Wetherefo reintroducecatalystswiththeshifts + ( V1+ iV2), + ( V 1+ iV 2).(7.4 .10) Inadditiontotheinvarianceduetotheseshifts,theLagrangianhasalsotheinvariance duetothefactthatthe“eldsappearonlyas €, € .Wethushav ethe gauge transformations =+ L ( V1+ iV2)= Š L €=0; = + L, ( V 1+ iV 2)= Š L, € =0;(7. 4.11) withthechiralspinorparameters, .These parameterswillintroducesecondgenerationchiralspinorghosts ,andwealso havetherealscalarghosts V 1,2,3,4associatedwiththeinvariancesparametrizedbythecomplex L L.After gauge“xingand somechangesofvariablesthekineticLagrangiancanbeputinstandardform(see (7.4.14a)).Theone-loopcontributionof Vi, V icancelsthatof V i, but Viand V ihavequantuminter actions(because and do,and Viand V ienterthroughthe shifts in(7.4.10)). Theaveragingin(7.4.5)hastobenormalizedbyintroducingaNielsen-Kallosh ghost3 ,tocompens atethecont ributionsfromthe “elds.ItsL agrangianis Š1 4 ( 3 Š € 3€)2Š1 12 ( 3 + € 3€)2+( 3€)( €3 ).(7.4 .12) Wecannowa pplytheusualprocedure(asdescribedinsec.7.3)ofthecatalyststoplace theLagrangianinthestandardform:Weshift 3 bytherepresentationswhosecoecientsin(7.4.12)arenot1,andthen“xthegaugetomakethem1(see(7.3.11-13)for anexample).Inthiscase,wemaketheshift 3 3 + 2 3+ 3, €3=0.(7. 4.13) Thisallowsusto“xthesuperspin0( 3)andtwooft he(four)superspin1 2 ( 3)par tsof

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7.4.Quantization4353 skineticterm.Wethenchoosethemostgeneralgauge-“xingfunctionsandweightingsforthenewinvariances(correspondingtoarbitraryvariationsof 3and 3,andthe correspondingvariationsof 3 ),introducetheappropriatenewghosts,makeshifts,etc. Thenetresultisthatallthecatalystscancelasintheexample(7.3.11),leavinguswith justtheNie lsen-KalloshghostwithconventionalLagrangian.(Thisghosthasnoquantuminteractions.) Infact,theformofthe(quantum-quadratic)Lagrangianwaspredictableforall “eldsexcept H ,sincebydime nsionalanalysisandLorentzinvariance(and,whenrelevant,chira lity)onlyitcouldhavenonminimalte rmsnotresultingfromdirectbackgroundcovariantization(i.e., Wterms). The“nalresultofthequantizationprocedureisthefollowing:Wewritethewhole eectiveLagrangianasasumofaquadraticpartandtherest,withthequadraticpart being S = d4xd4 E EŠ 1[ Š1 2 H€ H€Š9 5 +( €i €+ i € €+ 3€i €3 ) +(3 V 1 V1+ V 2 V2)+4 i =11 2 V i V i+2 i =1 (1 2 i 2i + h c .)], (7.4.14a) where = + W M + W€€€ € M€€.(7. 4.14b) Intheseformulae , 3 Viand V ihaveabnormalstatistics.Thisexpressionis sucientforone-loopcalculations.Notethatatone-loopthecontributionsfromthe various V scan celduetostatistics. Thehigher-loopcontributionscomefromquantuminteractiontermsoriginatingin threeplaces:(a)thehighero rder(cubic,quartic,etc.)termsintheexpansionofthe classicalaction(7.2.24);(b)thegauge-“xingterm;(c)thehigherordertermsintheFaddeev-PopovL agrangian.Thelatterhasthesymbolicform ( antighost ) g host( gaugefixingfunction ),wherethevariationi sthefu llnonlinear

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4367.QUANTUMN=1SUPERGRAVITYvariation( 7.2.19,22,23),withthegaugeparameter Lreplacedbytheghost .Since wehavema detheshifts(7. 4.10)fortheghosts,the“elds Vi, V iwillalsoappear.Thus, thequant umverticesareobtainedfromthehigherorderterms(beyondquadratic)in theexpansionof(see(7.2.24,7.4.6)) SC+ SGFŠ{ [ €+ €( V 1Š iV 2)] Š €[ + ( V 1+ iV 2)] } H€(+ ( V1+ iV2)),(7.4 .15) where H€( L)istheex pression obtainedbysubstituting(7.2.22,23)into(7.2.19).We haveperformedanintegrationbypartsinthesecondterm.Wenotethatwhileboth termsinthegauge“xingfunction(7.4.3)leadtointeractionsoftheghostswiththe background“elds,onlythe“rstterm €H€leadsto(local)interactionsbetweenthe ghostsandthequantum“elds.Thisistheendofthequantizationprocess. Wehave discussedthequantizationprocedureintheformulationwiththechiral compensator .Asdiscu ssedinsec.5.2.d,anotherpossi blechoiceforcompensatorisa realscalarsuper“eld V introducedthrougha variantrep resentation.Thetreatmentof thecorrespondingformulationcanbeobtainedbymakingthesubstitution(eveno she ll) 3 1+( 2+ R R ) V V = V .(7. 4.16) V hasthetransf ormationlaws Ba ck gr o und: V= eiKV Quantum: V= V +( L+ € L€).(7.4 .17) Weusenowth e gauge“xingfunction F= €H€Š1 5 V .(7. 4.18) Theshifts(7.4.10)areagainmade,andbyaproceduresimilartotheonedescribed aboveweobtainthefollowingresults:Thehigher-ordertermsintheactionareagain givenby(7.4.15),withthesubstitution(7.4.16)(butnow SGFdoesnotcontribute).

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7.4.Quantization437However,thequadratictermsarenow S = d4xd4 E EŠ 1[ Š1 2 H€ H€Š1 10 V V +( €i €+ i € €+ 3€i €3 ) +(3 V 1 V1+ V 2 V2)+7 i =11 2 V i V i+7 i =1 i i].(7.4 .19) Here , 3 Vi, Vi ,and ihaveabnormalstatistics,and iarechiral. Ingeneral,thetotal“eldcontentisthefollowing:(a)physical“elds H and (or V ),whichcontributeatallloops;(b)the“rst-generationFaddeev-Popovghosts and ,whichcontri buteatallloops;(c)the“rst-gen erationN ielsen-Kalloshghost 3 and allhigher-generationghosts,whichcontri buteonlyatoneloop;(d)thecatalystghosts Viand Vi ,whichcontri buteatonlymorethanoneloop(beingcanceledattheone-loop levelbythecont ributionfromthe Vs).Wewilldiscussinsec.7.10someofthedierencesbetweentheformulations(7.4.14)and(7.4.19).

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4387.QUANTUMN=1SUPERGRAVITY7.5.Supergravitysupergraphs a.Feynmanrules Inthenextsectionweshallconsiderfurtherthebackground“eldquantization anddiscussitsapplications.InthissectionweconsiderordinaryquantizationanddiscusstheFeynmanrulesforsupergravity-mattersystems.Thereisnoneedtogoagain throughthegauge-“xi ngprocedure.Wesimplytaketheresultsoftheprevioussection andsetthebackground“eldstozero.Thereforethesupergravityquantumactionis givenby(7.4.14,15),where E E =1,allthed erivativesare”atspacederivatives,andall chiral“e ldsordinarychiral.Furthermore,the“elds 3 Vi and i canbedropped sincetheyhavenointeractions(buttheFaddeev-Popov“elds ,andthec atalysts Vi, V ido).Equivalently,wecanworkwith(7.4.15,19),dropping 3 Vi ,and i. Matteractions,covariantizedwithrespectto H€and ,canbe a dded. Fr omthe”atspaceformof(7.4.14a)weobtainordinarypropagators.Inparticularwehave HHpropagator : Š €€p2 4( Š )(7. 5.1) propagator : p€p2 4( Š ),(7.5 .2) andtheusualpropagatorsfor and V .Verti cesareobtainedfromtheexpansionof (7.4.15),aswellasfrommatteractions.For example,considerthekineticactionofa scalarmu ltiplet cov: S = d4xd4 EŠ 1cov cov= d4xd4 ( E )Š1 3 (1 eŠ H)1 3 eŠ H .(7. 5.3) Wehaveex pressed covintermsofa”atspacechiralsuper“eld andweareworkingin thechiralrepresentatio n.Wemustexpandnowthevariousfactorsinpowersof H€and = Š 1.However,theexpansionswerecarriedoutin(7.2.30-32).Replacing backgroundcovariantderivativeswith”atspacederivatives,we“ndthecubic

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7.5.Supergravitysupergraphs439interactions S(3)= d4xd4 { ( + ) + [ Š H ai aŠ1 3 ( D€DH a) Š1 3 ( i aH a)] } = d4xd4 [( + ) + H a(1 2 i a Š1 6 [ D€, D] )].(7.5.4) Thesameexpansionscanbeusedto“ndthesupergravityvertices,butwithbackgr o undssettozerothealgebraismuchsimpler.Thus,from = eŠ HDeH, €= D€, weobtain € B= =€= a = a€=0, a b= Š i D€ b,(7. 5.5) where bisobtainedasthecoecientof bin(7.2.27b): b= i ( DH b) Š1 2 [( DH c) cH bŠ H c( cDH b)]+ ... .(7. 5.6) (To“ndthec ubicinteracti onswedonotneedthethirdordertermin bsinceitonly contributesatotal( D€)derivat ive.)Therefore EŠ1 3 =[ det ( a b+ a b)]Š1 3 =1 Š1 3 a a+1 6 a b b a+1 18 ( a a)2Š1 18 a a b c c bŠ1 9 a b b c c aŠ1 162 ( a a)3.(7. 5.7) We alsoexpand(7.2.31)oneorderhigherwhichgives,againdroppingatermwhichonly contributesatotalderivative, (1. eŠ H)1 3 =1 Š1 3 i ( H ) Š1 18 ( H )2Š1 6 H ( H )+1 162 i ( H )3,(7. 5.8a) eŠ H =1+ + + Š iH Š iH Š1 2 H ( H ).(7.5 .8b) Thecubicsupergravityactionisobtainedfromtheproductof(7.5.7)and(7.5.8).

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4407.QUANTUMN=1SUPERGRAVITYWeobtain thec ubicghost-antighost-quantum“eldverticesfrom(7.4.15)and (7.2.19,22,23).Weneed H€to“rstor derin H€and : H€=( D L€Š D€L)+3( D€LŠ D L€) Š1 2 [ Š i ( D L€+ D€L) €H a+( D2L) DH a+( D2 L€) D€H a+ iH ( D L€+ D€L)].(7.5.9) Thecubicghostactionisobtainedbysubstituting L= + ( V1+ iV2)(cf. (7.4.15)).Theseverticesaresucientfordoingsomeone-loopcalculations.However,as wehavealre adymentioned,atleastforon-shell“ elds,thebackground“eldmethodis muchsimple r.Inthismethod,theabove(covariantized)verticeswouldbeneededonly fortwo-loopc alculations. b.Thetransversegauge TheFeynmanruleswehavediscussedaboveusetheparticular(weighted)gauge of(7.4.3),whichisthemostconvenientforinternallines.However,whencomputing gaugeinvariantquantities,wecanuseanygaugefortheexternallines(thisistruein bothordi naryandbackground“eldmethods).Wediscussherethechoiceofaglobally supersymmetricgaugethatisconvenientformostcalculations. Thesuper“elds H€, containseveralirreduciblerepresentationsofsupersymmetry.Accordingto(3.9.40) thesuperspin contentof H€is(3 2 +1+1 2 +1 2 +0),while has superspin0.Accordingto(3.9.36,37),thespinorgaugeparameters L+L€contain superspins(1+1 2 +1 2 +1 2 +1 2 +0).(Theextrasuperspin1 2 representationsinthegauge parametercorrespondtosecond-generation ghosts.)Therefore,weshouldbeableto“nd a gaugewherewehaveeliminatedallsuperspinsbut3 2 and0.Wehavetwochoices,correspondingtoeliminatingthesuperspin0in (gauging to1),orin H€(inwhich case mustbekep t).Thesecondchoi ceismuchmoreuseful,andcanbeachievedby imposingthe transversegaugeconditionDH€=0.Notethatt hiscondition(andits complexconju gate)implies H =[ D, D€] H€= D2H€=0.

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7.5.Supergravitysupergraphs441c.Linearizedexpressions Forsomec omputations,weneedtheexplicitexpressionsforthegeometrical quantitiesintermsoftheprepotentials H€, .Hereweout linethe procedurefor obtainingthelinearizedexpressions;higher orderscanb eobtai nedinsim ilarfashion.We workinthec hiralrepresentation, andintheLorentzgauge N = .Afterweobtain theresultswewillconsiderotherLorentzgauges. We be ginwiththelinearizedexpressionsof(5.2.78a)(cf.also(7.5.5)) E€= D€ E= D+[ D, H ]= D+ i ( DH m) m(7.5.10) Weset =1+ and,usingforexample(7.5.5-7),wehaveatthelinearizedlevel E =1+ D€DH€(7.5.11) Fromthef ormofin(5.2.78c)weobtain =1+ X (7.5.12) where X 1 2 Š Š1 6 (2 D€D+ D D€) H€.(7. 5.13) IntheparticularLorentzgaugeweareusingweneednotdistinguishbetween”atand curvedindices.Weobtainthen E= E= D+ XD+ i ( DH b) bE€= E€= D€+ X D€.(7. 5.14) To “nd E awe write(againusing N = ) E a= E a+ i1 2 C (€ € ) E€+ i1 2 C€ ,( € )€ E(7.5.15) where E aŠ i { E, E€} .There fore E a= aŠ i ( DX ) D€Š i ( D€ X ) D

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4427.QUANTUMN=1SUPERGRAVITY+[ D€DH b+(1 6 [ D, D€] H dŠ1 2 ( + )) a b] b(7.5.16) Thelinearizedexpressionsforthe C scanbewor kedoutfromtheird e“nition.We“nd “nally E a= a+ i [1 2 D2D( H )€Š ( D€ X ) ] D+ i [ Š1 2 D2 D(€H€ )Š ( DX ) €€] D€+[( D€DH b)+( X + X ) a b] b.(7. 5.17) To “ndtheconnectionsweevaluate“rstthe CAB C,and usethetorsionconstraints. We “nd = Š C ( D ) X €€=1 2 D2 D(€H€ ),€= Š1 2 D2D( H )€, a = i1 2 D D2D( H )€+ iC ( D€D ) X .(7. 5.18) Theindependent“eldstrengthsare R = D2( Š i1 3 aH a), G a= Š2 3 D D2DH aŠ1 6 a b c d b[ D, D€] H dŠ1 3 a bH b+ i a( Š ), W=1 6 D2D( i €H )€.(7. 5.19) Theremaining“eldstrengthscanbereadfromthesolutionoftheBianchiidentities (5.4.16). Aswehavementionedseveraltimes,itis sometimesusefultochooseaLorentz gauge N = inwhich€ =0sothat,inthech iralrepr esentation,whenactingon a“ eldwithundottedindices, € ...= N€€ D€ ....(Thatsuch a gaugeispossible followsfrom R€€ =0,whichim pliesthattheaboveconnectionispuregauge.)We reachthisgaugebytheLorentztransformation A=[ L A], L = M + h c .(7. 5.20)

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7.5.Supergravitysupergraphs443sothat,inparticular, €€= €€Š E€€.(7. 5.21) Atthe linearizedlevel,setting €€=€€+ €€=0,w e “nd €€=1 2 D D(€H€ )(7.5.22) and = ( €€).Inthisgauge N€€= €€+ €€. Inthisgauge,thevariousquantitiesof(7.5.19)areshiftedaccordingtotheirindex structure.Inparticular,we“ndthatnow = Š C ( D ) X +1 2 D D€D( H )€.(7. 5.23) d.Examples Inthissubsectionweassumethataregularizationschemeexiststhatpreserves localsup ersymmetry.Suchaschemewillbediscussedinsec.7.9.We“rstcomputea masslesschiralloopcontributiontothesuperg ravityself-energy.Therelevantinteractionisgivenby(7.5.4),andthesupergraphisgiveninFig.7.5.1. k + p Hb( k ) Ha( Š k ) p Fig.7.5.1 Wenote thefollowingsimpli“cations:(a)The( + ) vertexleadstoo nlyatadpole contributiontothe or H self-energydiagram,andwesetthistozeroindimensional regularizationformassless s.Equivalently,weobservethatintheoriginalaction (7.5.3)thecompensator canbeabsor bedinto bya“eldr ede“nition.(b)Withsuitableregularizationtheresultshouldbegaugeinvariant,andwecanworkinthe

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4447.QUANTUMN=1SUPERGRAVITYtransversegaugewheretwoofthethreetermsinthe H vertexdo notcontribute.The Š H a i a ve rtexisthesameasintheYang-Millscase,ifwereplace VATAby H ai a. Theresultcanthenbereadfrom(6.3.31)(withtheadditionalmomentumfactorsfrom i a)1 2 d4k (2 )4 d4 H a( Š k ) d4p (2 )4 Š p2Š p c D€D+ D2D2p2( k + p )2 p a( k + p ) bH b( k ).(7.5 .24) Usingthegaugecondition thiscanber educedto Š1 8(D Š 1) d4k (2 )4 d4 H a( Š k ) k2k c D€DH a( k ) I ( k2),(7.5 .25) whereindimensionalregularization I ( k2)= dDp (2 )D 1 p2( k + p )2 =1 (4 )1 2 D (2 Š1 2 D)[(1 2 DŠ 1)]2(DŠ 2) ( k2)1 2 D Š 2=1 (4 )2 ( 1 Š lnk2+ const .).(7. 5.26) Whenac tingon H a( k ),againusingthegaugecondition,wecanrewrite k c D€D= k c1 2 { D€, D} = k2.Thefu llycovarian tresultcanbewri ttenasacontributiontotheeectiveactionoftheform[ c1 d2 ( W)2+ h c .+ c2 d4 ( G2+2 RR )] I However,thecoecients c1, c2cannotbedeterminedfromjustatwo-pointcalculation (ex ceptinthebackground“eldmethod:seesec.7.8).Onshellonlythe“rsttermsurvives.Althoughther esultisindependentof ,thecomp ensatorreappearsinthecourse ofseparatingoutthedivergentpart(seesec.7.10). Asas econdexamplewecomputesupergravitycorrectionstothechiralself-energy. ThegraphsarethoseofFig.7.5.2:

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7.5.Supergravitysupergraphs445 ( k ) ( Š k ) HH Fig.7.5.2 The“rstgraphgivesnocontribution(after D -algebra itisatadpole)whiletheothers addu pto d4k (2 )4 d4 ( Š k ) d4p (2 )4 1 p2( k + p )2 [ Š5 9 D2D2+1 9 p k Š2 9 ( p Š k )2] ( k ). (7.5.27) (The Š5 9 forthe pr opagatorfollowsfromitsnormalizationin(7.4.14).)Intheintegralwecanreplace p k = Š k2and p2=0.Thusthe totalresultvanishes.

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4467.QUANTUMN=1SUPERGRAVITY7.6.CovariantFeynmanrules Tothequanti zedsupergravityactionofsec.7.4wecanaddcovariantizedsupersymmetricmatteractionsandconsidergeneral matter-supergravitysystems.Ifthematteractionscontaincovariantderivatives,thesemustbesplitasinsec.7.2.Forconstrainedsuper“eldswemust“rstextractexplicitquantum“elddependence(e.g., eŠ H ,where ,arebackgr oundcovariantlychiral).Inprinciplewecanalsosplit mattersuper“eldsintoquantumandbackgroundpartsandconsiderageneralquantum systeminabackgroundofmatterandsupergravity.However,ingeneraltheprocedure ofsec.7.4isnotapplicable.Wecannotimposetheon-shellconditions R R = G G€=0. Theseconditionsmustbereplacedbytheequations R R = J ( matter ), G G€= J€( matter ) andthequantizationmustbecarriedoutwiththesupergravity“eldso-shell.Thisisa straightforwardbutalgebraicallycumberso meprocedure.Thereforeinthissectionwe willconsideronlypureon-shellsupergravitybackgrounds(noexternalmatter).General systemscanbehandledbyanextensionofourquantizationmethodsorbytheordinary (n on background)quantizationoftheprecedingsection. Giventhebackground“eldLagrangianwithquantummatterorsupergravity “elds,theFeynmanrulescanbederivedinexactlythesamewayasforglobalsupersymmetry.Ingeneral,forunconstrainedquantumsuper“elds,wecanreadtherulesdirectly fromtheL agrangian.Wehavetwotypesofvertices:thosearisingfromquantumselfinteractions,andthosecontainingalso(oronly)interactionswiththebackground“elds. Thebackground“eldsappearonlythrough“eldstrengthsandbackgroundcovariant deriva tives A= E EA MDM+ A,withthe ”atsuperspace DM.Therefo re,wewill encounterverticeswithallquantumlines,orwithamixtureof(atleasttwo)quantum linesandbackgroundlines.Forco nstrained,i.e.,backgroundcovariantlychiralsuper“elds,wemust“rstofallsolvethec hira lityconstraints,i.e.,write = e 0intermsof anordinarychiralsuper“eld.Thiswillintroduceinteractionsinvolvingexplicitlythe backgroundpotentials.Weshalldiscussbelowhowtoavoidthis,butatanyrateweend upwithanactiontowhichthemethodsofchapter6canbeapplied,withordinarypropagatorsandrulesforcalculation. Weobservethat attheone-looplevel, thecontributionfro mthege neralspinors canalsobeobtainedbysquaringtheirkineticoperatorandtakinghalfoftheresulting contributiontotheeectiveaction.Wehave

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7.6.CovariantFeynmanrules447( i €)( i €)=1 2 ( i €)( i €)+1 2 [ i €, i €] = + { W W W W M } ,(7. 6.1) wherewehaveused(5.4.16).Thespinoractionin(7.4.14a)thusbecomes d4xd4 E EŠ 1 3 i =11 2 i ( + { W W W W M } ) i + h c = d4xd4 E EŠ 1 i1 2 i i + h c .,(7.6 .2) andweobservethat allthe unconstrainedsuper“elds( H V )aredesc ribedbysimilar actions,withthesameoperator givenby(7.4.14b).Weshalldiscusslaterapplicationsofthi sresult. Wenowdes cribeamodi“cationoftheFeynmanr ulesforcovariant lychiralsuper“e lds,analogoustothemodi“cationfortheYang-Millscaseinsec.6.5.Theconsequencesofthemodi“cationare:ItguaranteesthattheFeynmanrulesforchiralsuper“eldswillnotintroduceexplicitbackground gaugepotentials,butonlythevielbeinand connections,anditactua llysimpli“essomeofthe D -algebra.Wefo llowapro cedure thatisidenticaltot hatofsec.6. 5.We“rst de“ne covariantfunctionaldierentiation forageneralsuper“eldby ( z ) ( z) E 8( z Š z),(7.6 .3) whichgives( ) d8zEŠ 1L = L .Wethende “necovariantfunctionaldierentiation foracovariantlychiralsuper“eld (w hichcouldcarryadditionalundottedspinor i ndices,butwedonotindicatetheseexplicitly)by ( z ) ( z) ( 2+ R ) E 8( z Š z)= Š 3 D28( z Š z),(7.6 .4) wherethesecondformisobtainedbyusingtheidentity(5.3.66b)andthechiralrepresentation( E€= N€€ D€)withthe particularLore ntz gaugewhere N = suchthat E€ = E€ €=€ =0.(7. 6.5)

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4487.QUANTUMN=1SUPERGRAVITYInthisrepresentationcovariantlychiralsuper“eldsarechiralintheusualsense: €...=0imp lies D€...=0 .J us ta si nt he Ya ng -M illscase,atthispointweneednot beexp licitastowhetherthechiralcovarianceiswithrespecttofullderivatives(containingbothbackgroundandquant um“elds)orjustbackground“elds,andtheobjects appearingin(7.6.4)canbefunctionsofboth,orjustbackground“elds(exceptwhenthe chiralsuper“elds aresupergravitysuper“elds,inwhichcasethecovariantderivativescan on ly be background).Wecanstayo-shell. Thecovariantizationoftheusualexpression D2D2 = b ecomesnow ( 2+ R )( 2+ R ) ... = + ... += + W M +1 2 i ( €G€) M Š1 2 iG€€Š R 2Š1 2 ( R ) + R R +( 2 R ),(7.6 .6) generalizing(7.4.4)oshell.Weobservethatonshell(recallingthatchiralsuper“elds canonlyhaveundottedindices) += ,wherethela tterquantityw asde “nedin (7.4.14b),aresultwhichweshalluselater. Asinsec.6.5cwestartwiththeaction S = S0+ Sint( ), S0= d4xd4 EŠ 1 .(7. 6.7) Sintalsocontainstheotherqu antum“elds butwehaveindicatede xplicitlyonlythe dependenceon .Wecon centrateonthefunctionalintegralover whichgives,using (7.6.3,4), Z ( J J )= ID ID exp [ S +( d4xd23J + h c .)] = [ expSint( J J )][ exp ( Š d4xd4 EŠ 1 J + Š 1J )],(7.6.8) whereisthefunctionaldeterminant = ID ID eS0.(7. 6.9) Ingeneraltheaboveexpressionfor Z dependson,andistobeintegratedover,theother

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7.6.CovariantFeynmanrules449quantum“elds.Weare consideringthemasslesscase,buttheresultsforthemassive casecanbeobtainedeasily.Exceptfor,theotherfactors,whichcontainquantum “eldself-interactions,contributeonlybeyon doneloop,ortod iagramscontainingexternalchirallines. Thedeterminantgivesthecompleteone-loopcontributionfromchiralsuper“eldsofdiagramswithonlyexternalsupergravitylines,andcouldbeevaluatedbyusing standardsuper“eldFeynmanrules,butwewishtoavoidthis.Instead,weshallusethe do ub lingtrickasinsec.6.5c.(Insupergravitywearealwaysdealingwithrealrepresentations).Wenowhave O O + J J =0, O O O O = 0 2+ R 2+ R 0 .(7. 6.10) Itssquare, O O2 Š J J =0, O O O O = ( 2+ R )( 2+ R ) 0 0 ( 2+ R )( 2+ R ) ,(7. 6.11) correspondstoanaction S 0= d4xd231 2 + = d4xd4 EŠ 11 2 ( 2+ R ) ,(7. 6.12) andintermsofitwecanwritethefunctionalintegral 2= ID ID exp [ S 0( )+ h c .]=( ID eS 0)2.(7. 6.13) Weinte grate S 0bysepara tingout 3D2from EŠ 1( 2+ R ),treating1 2 [ EŠ 1( 2+ R ) Š 3D2] asaninteractionterm.Theresultis = ID eS 0= { exp d4xd231 2 J [( 2+ R )( 2+ R ) Š D2D2] J } [ exp Š d4xd231 2 J 0 Š 1J ] |J =0.(7. 6.14) (Notethatwritinginstead( 2+ R )( 2+ R ) D2EŠ 1Š 3eŠ HD2EŠ 1 Š 3eHwouldgive

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4507.QUANTUMN=1SUPERGRAVITYtheusualru les,ex ceptfortheextra 3sfromthede“nition(7.6.4)).Therefore,acalculationoftheone-loopcontributionof totheeectiveaction(i.e., ln )consistsinevalua tinggraphswithpropagators pŠ 24( Š )andvert i ces[( 2... ]giv ingrisetoastring ... [( 2+ R )( 2+ R ) Š D2D2]i4( iŠ i +1)[( 2+ R )( 2+ R ) Š D2D2]i +1... (7.6.15) withd4iintegralsateachvertex.Weconcentrateonagivenvertexandatthenext onew erewrite( 2+ R )= D2Š 3EŠ 1.Wetempora rilytransferthe D2factor across the -functionandusetheidentity [( 2+ R )( 2+ R ) Š D2D2] D2=( +Š 0) D2.(7. 6.16) Wefurthersim plifytheexpressionbyusingtheanticommutationrelationstomovethe D sin +totherightuntiltheya reannihilatedbythe D2.Theresu ltingexpression, whichwecall +,containsno D s.Wenowreturnthe D2factortoitsoriginalplace, reexpressthevertexinitsoriginalform,andproceedtomanipulateitinthesameway. Wecancontinuea roundtheloopandtreatinthiswayallverticesbutthelast,andwe areledtothefollowingrules: onevertex : D2[ Š 3EŠ 1( 2+ R ) Š D2], othervertices : +Š 0.(7. 6.17) Themassivecaseisobtainedsimplybyaddingamassterminthedenominatorofthe pr opagator. Theserulesleadtoasimplerevaluationoftheone-loopcontribution,sincethere areno D sintheloopexcepttheone D2, butmoreimportantlythecontributionis manifestlyexpressibleonlyintermsofobjectswhichappearinthecovariantderivatives, andnotthegaugeprepotentials.Thisisevidentlytrueofthehigher-loopcontributions aswell.From Sintandthede“nitionofthecovariantfunctionalderivativein(7.6.4),the expression(7.6.8)leadstohigher-loopFeynmanruleswhichdonotexplicitlydependon thebackgroundprepotentials.Weobtainpropagators + Š 1forchirallines,wherethe full +canbeexpressedintermsofthequantum H€, ,andthebackgro undcovariant derivati ves.From Sint( J J )weobtainverti ceswithfactors( 2+ R ) E or( 2+ R ) E

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7.6.CovariantFeynmanrules451operatingoneachchiralorantichirallineleavingthevertices.(Thesegeneralizethe ordi nary”atspacerules.)Againwecanexpressthesequantitiesintermsofthequantum H€and ,andthebackgro undcovariantderivatives .For anactualmomentum spacecalculationthesehavetobefurtherexpressedintermsofordinaryderivativesand backgroundvielbeinandconnections.Wenowhavetheresultthatforallsuper“elds, whencalculationsarecarriedoutinthebackground“eldmethod,thecontributionsto thee ectiveactionfromindividualgraphsdonotinvolvethebackgroundsupergravity prepotentialsthemselves,butonlyvielbeinandconnections,whichdependon (multi)derivatives oftheprepotentials.Consequentlythereissomeimprovementinthe powercountingrul esforpotentiallydivergentgraphs.

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4527.QUANTUMN=1SUPERGRAVITY7.7.Generalpropertiesoftheeectiveaction Wean alyzeinthissectio nthege neralformoftheeectiveaction(background “eldfunctional),asconstrainedbytherequirementofbackground“eldinvariance. Thisanalysisisparticularlyimportantfordeterminingthedivergencestructureofsupergravity.Divergences,whichcouldbecanceledbycountertermsintheLagrangian,correspondto local termsintheeectiveaction,andtheirformislimitedbygaugeinvariance anddimensionality.Insomecasesweobtainstrongerresultsbyrestrictingourselvesto on-shellbackgro und“elds.Theon-shellrestrictionisnotserious:Thetheoryisnotperturbativelyrenormalizable,Greensfunctionsaregauge-dependentanddivergent,andat bestwecanhopethat gauge-indepe ndent,on-shellquantities(e.g.,theS-matrix)are “nite.Therefore,onlythedivergenceswhichdonotvanishon-shellaresigni“cant (divergenceswhichareproportionaltoth e“eldequatio nscanberemovedbya“eld rede“nitionwhichdoesnotaecttheS-matrix). Wew illdisc uss“rstthesituationin N =1superg ravity.Thediscussionisapplicablethentoextendedsupergravityexpressedintermsof N =1super“elds .Howev er, strongerstatementscanbemadeiftheextendedtheoriescanbeexpressedintermsof extendedsuper“elds.Sincethediscussiondoesnotdependondetailsoftheextended super“eldconstructions,butonlyonpropertiesthatgeneralizeour N =1back ground quantizationmethods,wedevoteasubsectiontothiscase. a.N=1 Ourbackground“eldsaresupergravity“elds,whilethequantum“eldscanbe supergravityormattersuper“eldsorboth.Sinceinthebackground“eldformalismthe eectiveactionisgaugeinvariant,itcanbeconstructedfromthe“eldstrengths R R G G€, W,andcovarian tderivat ives(withanoverallfactorof E EŠ 1,or 3forchiralintegrands).Furthermore,onshell R R = G G€=0.(Wecons ideronlyvanishingcosmological term:Otherwise, R R isanonvanishingdimensionalconstant,andthedimensionalanalysisischanged.)Thus,onlythechiral“eldstrength W(anditsc omplexconjugate) andcovariantderivativescanappear.Wealsohavetheon-shellconditions W= €W=0aswell asthecorrespondingequationsfor W .Indet ermining theformofthee ectiveactionwecanalsousethefollowingfacts:(a)Theeective actionisdimensio nless.Thedimensionsofthevariousquantitieswhichcanappearare

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7.7.Generalpropertiesoftheeectiveaction453[ d4x ]= Š 4,[ d2 ]=1,[ W ]=3 2 ,[ ]=1 2 .Ina ddition,withonlysupergravityinteractions,foran L -loopcontributio nwehaveafactor 2( L Š 1)withdimension Š 2( L Š 1).(b) F unctions G ( x1,....)arisefromloopintegralsafterallthe D -algebrahasbeencarried outa nd,iftheyhaveodddimension,mustcontainanoddnumberofspace-timederivatives(momentumfactors)wh ichhavea ni ndexstructure €.(c)Inte gralswiththechiralmeasure d2 musthavechiral integrands,i.e.,factorsof W or 2( W2)( 2 W =0 onshell),etc.(butinthelattercasetheycanberewrittenasfullintegralsanyway).(d) Dottedandundottedindicesmustbeseparatelysaturated. Anotherimportantfeatureisthefactthatallthe d4 termsinhaveanequal nu mb erof(spinor-)undierentiated W sand W s.Thisi sacons equenceoftheglobal chiralRinvarianceofthetheory(cf.(5.3.10);the Y transformationsareglobalinvariances;intermsofprepotentials,theyaresimplyphasetransformationsof ,leaving H invariant).Weshouldalsoremarkthat,apriori,asdiscussedintheprevioussection, pertur bationtheorydoesnotleadtoaforminvolvingonlythe W sandtheircovariant derivatives,butratherthequantitieswhichappearinthebackgroundcovariantderivatives,i.e.,backgroundvielbeinandconnectioncoecients,witha d4 integral.However, b ecauseofbackgroundinvariance,thesequantitiesmustarrangethemselvesintoaform thatismanifestlycovariantorcontainsonenoncovariantfactortimesacovariantobject thatsatis“esaBianchiidentity.(Thenon covariantterm,whenvaried,producesa derivativewhich,whenintegratedbyparts,giveszerouponuseoftheBianchiidentity.) Thus,thetermd4xd2 3W2reallyarisesfromanexpression(inthegauge(7.6.5))d4xd4 E EŠ 1 W,whichcant henberew rittenasachiralintegral. We“rst discussall local termsin,i.e.,allpossibleon-shelllocaldivergencesof thetheory.Ag enericlocaltermwillhavethestructure ( a)l( W W )m( W )n( € W )r,(7. 7.1) with W ( W )andtheindicescontractedinvariousways,andthespace-time derivativesdistributedinvariousways.WehaveusedtheinvarianceunderR-transformationstowriteonlytermswithequalpowersof W and W .Wenotethat unlesssome space-timederivativesactonthem, W and W cannotberaisedtoapowerhigherthan 4,becausetheyaresymmetricintheirthreespinorindicesandhencecontainonlyfour i ndependentLorentzcomponents.Thedimensionalityoftheabovetermis

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4547.QUANTUMN=1SUPERGRAVITYl +3 m +2 n +2 r .Ifitappearsat L loopsitismultipliedby( 2)L Š 1withdimension Š 2( L Š 1).Theoveralldimensionoftheactionmustbezero: d = l +3 m +2 n +2 r Š 2( L Š 1) Š 2=0.Theonly purelychiralterm,involvingjust W ,isthequadra ticexp ression d4xd2 3WW+ h c .= Š d4xeŠ 1ww+ h c .(7. 7.2) where w is th eW eyltensor.Ondimensionalgroundsitcanonlyappearattheone-loop level(withno factor,asf ollowsfromourdiscussionabove),andtheintegrandisa to ta ld erivativeonshell.Itis,infact,ondimensionalgrounds,theonlylocalterm(i.e., po ssibledivergence)whichcanoccuratthe one-looplevel,onshell.However,dueto theGauss-Bonnettheorem,itisjustatopologicalconstant,andvanishesintopologicallytrivialspaces.(Forafurtherdiscussion,seesec.7.10.) Atthetwo-loopl evelnolocaltermsarepossible.Wehaveafactor 2ofdi mension Š 2,anditiseasytocheckthatthereisnoway,fromamongthegenericexpression above,to“ndeitherachiralexpression(tobeintegratedwith d4xd2 ),orag eneral expression(tobe integratedwith d4xd4 ),whichcanleadtoatermwithdimensionzero. Thus,atthetwo-looplevelnoon-shelldivergencescanariseinsupergravity. Athigherlo opsthenumberandvarietyoflocaltermsincreases.Forexample,at threeloops ,thecomb ination W2 W2isapossiblelocaltermandthereforeapotentially divergentone(theonlyon-shellone,infact).Atanygivenloopthereareofcourselimitationsduetodimensionality,chirality,andindexcontraction.Inparticularitiseasyto ve rifythat,ondimensionalgroundsandinordertosaturateindices,allhigherloop termsmusthavefactorsofboth W and W andthereforevanishwheneither W =0or W =0.Thissit uationdescribesbackground“eldcon“gurationswhichareself-dualor antiself-dual.Weconcludethatsuchcon“gur ationsreceivenoradiativecorrections. Thenonlocalpartoftheeectiveactionhasastructureasin(7.7.1),including howeveranonlocalfunction G ( x1, x2,....)andwith“eldsevaluatedatdierentpoints ( x1, ),( x2, ), ... .W e “nd,usingsuperspaceperturbationtheoryanddimensionalanalysisthat,ifnomassive“eldsarepresent, theon-shelle ectiveactionhastheform d4x1d4x2d2 3W( x1, ) G ( x1, x2) W( x2, )+ h c .

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7.7.Generalpropertiesoftheeectiveaction455+ d4x1... d4x4d4 W( x1, ) W( x2, ) W€€€( x3, ) W€€€( x4, ) G ( x1... x4)+ ... ,(7. 7.3) wherethetermsnotexplicitlywrittencontainmorethanfour W sandtheircovariant derivatives.Thenonlocalfunctions G ( x1, x2,...)canbethoughtof aspolynomialsin thespace-time covariantderivativesactingonthe“elds,andfunctionsofthecovariant dAlembertian(e.g.,itsinverse),correspondingtotheresultofdoingvariousloopintegralsinmomentumspace.Theimportantpointisthatother,aprioripossibleterms withtwo,three,orfour W sarenotpresent.(Forexample, d4 WW W cannothaveits indicessaturatede venifderivat ivesareinc l udedwhile d2 ( W )4hasthewrongdimension,etc.)Insecs .7.8 ,10weshalldisc ussinmored etailtheformofthe G functions. b.GeneralN Weshalla ssumeinth issectionthatunconstrainedsuper“eldformalismsexistfor allsupersymmetricsystemsofinterest.Suchformalismshavenotyetbeendeveloped exceptfor N =2 ,a nd thereareindicationsthatiftheyexisttheyhaveanunfamiliar form.Weshallonlyassumethatthereexistco nstraintsonthecova riantderivatives thatallowthem,andtheaction,tobeexpressedintermsofordinaryderivativesand unconstrainedprepotentials.Wecanthenmimicthe N =1 background-quantumsplittingforg eneral N .Werepla cetheunconstrainedprepotentialsbyquantumprepotentials,andtheordinary derivativesbybackgroundcovariantderivatives.Furthermore,if covariantlyconstrained (e.g.,chiral)super“eldsarepresent,wecanderivecovariantrules forthe maswed idin N =1;the procedureisgeneral.Wewillnotrestrictourselvesto on -shellbackgrounds. Byanextensionofourfullycovariantbackground“eldmethodofsec.7.6,we obtainimprovedpower-counti ngrulesfordiscussinglocaldivergences.Theserulessimplyfollowfromthefactthat allquantumtermsintheeectiveactionareautomatically expresseddirectlyintermsoftheconstrainedbackgroundcovariantderivatives (andtheir “eldstrengths)andanexplicitexpansionintermsofunconstrainedbackgroundprepotentialsisunnecessary.(Onemightalsoexpect toneedthesup erspacegeneralizationof antisymmetrictensorgauge“elds,e.g.,thethree-formofD=11supergravity,butthe background-quantumsplitactioncanalwaysbewritteninaformwheresuch

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4567.QUANTUMN=1SUPERGRAVITYbackground“eldsappearonlyastheir“eldstrengths,andthese“eldstrengthsalready appearamongthe“eldstrengthsofthebackgroundcovariantderivatives.)Thisimplies thatalldivergenttermsmustbeexpressibleas localfunctionsofth ecov ariantderivatives. Thus,forsupersymmetricYang-Mills,wherethesameideasapply,allcountertermsmustbelocalfunctionsof ,andfors upergravityof andalso E M.(Conventionalconstraintsdetermineallof Afrom and E M.)Furthermore,becauseinthe derivation oftheFeynmanrulesverticesarealways integratedoverfullsuperspace, they willcarryafulld4xd4 N for N -extendedsupersymmetry. Forthecaseof extendedsupersymmetry,treatedwithextendedsuper“elds,there isatec hnicaldicultyinthebackground“eldmethodbecauseoftheappearanceofan i n“nitenumberofgenerationsofghostsuper“eldswithprogressivelyincreasingsuperspin.Forexample, N =2 Ya ng -M illstheoryisdescribedbyarealisovectorsuper“eld Va bwithgaugeinvariance Va b= Dc ( a bc ) + Dc€ ( ac b )€.Thistra nsformation implies thatthecorrespondingghosthasagaugeinvariance ( abc ) = Dd ( a bcd )( ),whichin turnimpliesaghostwithinvariance ( a bcd )( )= De ( a bcde )( ),etc.T he gaugesuper“e ldsunavoidablycontain“eldsofspinhigherthanthose(physicalandauxiliary)occurringinthegauge-invariantaction.Togaugetheseawaythegaugesuperparameters(and ther eforethecorrespondingghosts)mustcontainhigherspinsthanthegaugesuper“elds. However,onlya“nitenumberofghosts(i.e.,theusualFaddeev-Popovghosts,plusperhapscertaincatalystghosts)contributeatmorethanoneloop.Therefore,thehigherloopcontributionstotheeectiveactioncanbecalculatedinamanifestlybackground covariantformandwillobeythepower-countingrulesthatwederivebelow,whereasthe one-loopcontributionmayhavetobetreatedseparately.(Forexample,wecouldchoose backgroundnoncovariantgaugesforsomeoftheghostswhichcontributeonlyatone loopinsuchawaythatallbuta“nitenumb eroftheseghostsdecouple.Theeectof suchachoicewouldbetoproduceaone-loopeectiveactionwhichisnoncovariant,but thiswouldhavenoeectonphysicalquantities).Wediscussnowtheimplicationsof theser emarksandtheimprovedpower-countingrulestowhichtheylead.Forcompletenesswediscuss“rstthesituationinglobaltheories. The“rstexampleoftheimprovedpowercountingwasalreadygiveninsec.6.5for theFayet-IliopoulosD-termin N =1 Ya ng -M illstheory.Backgroundcovarianceimmediatelyimpliesthevanishingofsuchatermbeyondoneloop.For N > 1weobtain

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7.7.Generalpropertiesoftheeectiveaction457strongerresults: Si nc eY ang-Millstheoryisrenormalizable,theonlyalloweddivergenceinthebackground“eldmethodisproportionaltotheclassicalaction.However,beyondoneloopit musthavetheformd4xd4 N atthelinearedlevelbecauseofthecovariantFeynmanrules.Here hasdimension1 2 andisalocaloperator(nonnegativedimension). Sincetheactionisdimensionless,weobtaintheinequality Š 4+2 N +1 2 +1 2 0,which impliesthatonly N =0or1c anhavedivergencesbe yondonel oop. Thus N =2and N =4 su pe rsymmetricYang-Millstheorymustbe“nitebeyondoneloop.Furthermore, weknowfromex p licitone-loopcalculationsusing N =1super“elds(s eesec.6.4)that N =4isoneloop“niteaswell. Ontheotherhand, N =2does haveone-lo opdivergences.(Also,asfor N =1,loopcorr ections tothe N =2Fayet-I liopoulos termvanish.)Weem phasizethatwehadtomakeaseparate one-loopargumentbecauseofthe problemwithin“nitenumbersofghosts. Wecana pplysim ilarargumentsto N -extendedsupergravity.Thelocal(divergent)partoftheeectiveactionconsistsoftheintegralof EŠ 1timesa(covariant)productoffacto rsofvielbeinandconnections.At L loopsthelowestdime nsionalsuchterm is loc 2( L Š 1) d4xd4 N EŠ 1,(7. 7.4) mult ipliedperhapsbysomefunctionofadimensionlessscalar“eldstrengthfor N 4; suchafunctionmayhoweverbeforbiddenbyglobalon-shellinvariance(otheradditional factorswouldhavepositivedimension).Requiringthisexpression,possiblymultiplied byapolyn omialinthe “elds(withnonnegativedimension),tobedimensionless,we obtaintheinequality Š 2( L Š 1) Š 4+2 N 0,whichimplies L N Š 1.(SimilarargumentsforYang-Millsgivetheimproved Š 4+2 N +2 0inste adoftheabove Š 4+2 N +1 0.)Thus,fromtheseargumentsalone,we“ndthatin N -extended supergravitytheeectiveactioncanhavelocalterms,andthereforepossibledivergences, onlyat N Š 1loopsandbey o nd.(Thisissoeventhoughpossiblelower-loopinvariants canbeconstructed.TheimportantpointisthatourFeynmanrulesimplyintegration overfullsuperspa cewithintegrandsthatinvolvecovariantobjects.)Notethatthe divergencesexcludedbytheserulesareabsent bothon andoshell.

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4587.QUANTUMN=1SUPERGRAVITYAsim ilaranalysisinhigherdimensionsgivestheresultthathigher-loopdivergencesareabsentissupersymmetricYang-Millstheoryfor L < 2N Š 1 D Š 4 ,andinsu perg ravityfor L < 2N Š 1 D Š 2 (for L loopsinD-dimensions,where N referstothefour-dimensional value, i.e.thenumberofanticommutingcoordinatesis4 N ).Forlower dimensions (super-)Yang-Millsisrenormalizableanyway;forsupergravitytheaboveinequalityholds forD=3whileforD=2we“ndhigher-loop“nitenessfor N > 1. Ourbackground“eldapproachleadstoafurtherresultwhichisnotapparentin ordinaryquantizationornonsupersymmetricgauge“xing:Attheone-looplevel,in N =1lang uageandusingthebackground“eldformalism,theonlycontributionstothe (on-shell,topological)divergencesareproportionalto( W)2andcomefromchiral super“elds.Tounderstandthisweobservethatthedivergenceisjustacovariantization ofthedivergenceinthetwo-pointfunction,a nditscoecientcanbedeterminedbycalculatingaself-energydiagram.However,inourgauge,examinationofthequadratic actionin(7.4.14)(whichgivesthegeneralformforanytypeofsuper“eldinanon-shell supergravitybackground),revealsthatonlychiralsuper“eldverticeshaveenough D s and D stogivenonzerocontr i butions.Therefore inatheorywithanetzeronumber (physicalminusghost)ofchiralsuper“elds (any N 3theorywit ha ppropriatechoiceof auxiliary“elds(compensatingmultiplets)) thereareno(topological)one-loopdivergences. Atthetwo-l oopl evel no supergravitytheoryhason-shelldivergences. Wesu mmarizeourresultsinTable7.7.1,whichlistsallcaseswheredivergences mustbeab sentinpuresupersymmetricgaugetheories.Theresultscanbeclassi“ed intothreetypes:(A)absenceofdiverge ncesduetoone-loopcancellationsin N 3 supersymmetryofcontributionsof N =1chiralsuper“elds;( B)absenceoftwo-loop supergravitycountertermsbecauseinvaria ntsofa ppropriatedimensiondonotexist;(C) absenceofdivergencesathigherloopswhichisestablishedbyourargumentsabove. Theabsenceofhigher-loopdivergencescannotbeestablishedrigorouslyuntilthe correspondingsupergraphrule sareexp licitlyconstructed.Possibledicultieswithcarryingouttheprogramareinfraredproblemsduetolargenegativepowersofmomentain thesuper“eldpropagators,andtheexplicitconstructionoftheclassicalaction(whose formmaysurpriseus,ifthepropertiesofextendedsuperspacearenotasimpleextension ofthosefor N =1superspa ce).However,weemphasizethatoncetheactionhasbeen

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7.7.Generalpropertiesoftheeectiveaction459 loops N 123456 7 Ya ng -M ills0 1 2C CCCCC 4A C CCCCC supergravity0 1B 2B 3AB 4AB,C 5AB,CC 6AB,CCC 8A B, C CCCC Table7.7 .1.Absenceofdivergencesinsupersymmetrictheories written,thepowercountingrulesandourconclusionsimmediatelyfollow.Wenotethat inthe N =4 Ya ng -M illscasethe“nitenesscanalreadybeprovenwhenthetheoryis writteni nte rmsof N =2super“el ds, i.e., N =2 Ya ng -M illscoupledtoan N =2sc alar mult iplet;the N =2powercountingrul escanthenbeapplied.

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4607.QUANTUMN=1SUPERGRAVITY7.8.Examples Inthissectionweshallgivesomeexamplesandapplicationsofourcovariantformalismforcomputingsupergraphsinsupergravity.Werestrictourselvestoone-loop calculat ions.Higher-loopcalculationsarepossible,butthealgebraiscomplicatedif thereareinternalsupergravi tysuper“elds.W eshallco nsider“rstsomeone-loopcalculationswithmatter“eldsinsidetheloopand backgroundsupergravitysuper“elds.The algebrasimpli“esconsiderablyintheon-shellsituation. We be gi nb y “ndingone-loopchiral-“eldcontributionstothe(covariantized)onshelltwo-pointfunction,correspondingtothe“rsttermintheon-shelleectiveaction (7.7.3).Incontrasttothecalculationofsec.7.5.d,theseparatecoecientsofthe W2and G2+2 RR termsintheeectiveactioncanbede terminedfromthe two-poin tf unctionalonewhenthecovariantrulesareused(althoughherewe“ndonlytheformerterm sincethelattertermvanishesinouron-shellcalculation).However,wemustusedimensionalregularizationtokeeptrackoftermsthataretotalderivatives only infour dimensions,sinceinafour-dimensionalmomentum-spaceFeynman-graphcalculationtheyvanishbymomentumconservation.Weshallusetheon-shellconditionsonthebackground super“elds,butkeeptheexternalmomentum k oshell( k2 =0)intheloop integral. Also,weshallwriteourexpressionsinfourdimensions.However,thecalculationshould beca rriedoutinDdimensions,bothtoavoidultravioletdivergences,andtocircumvent thefactthat,whenD=4,thelinea rizedresultisatotaldivergence. Weconsid erachiralsuper“eld ...€€ ...with2 A undottedand2 B dottedindices, andaction1 2 d4xd2 3 + + h c ..(If B =0such“eldsc anexistonlyi non-shell backgrounds.)From(7.6.17),andintheLorentzgauge € =0,the lineari zedvertices are(onshell E EŠ 1= =1) One vertex : D2( 2Š D2) D2[ E E a aD+1 2 ( aE E a) D+ M D],(7.8 .1a) Other vertex : +Š 0 E E a aD+1 2 ( aE E a ) D+ W M D.(7. 8.1b) Thepropagatoris pŠ 2 ... €€... 4( Š ). The D -algebra istrivial.The M termsgiveacontributionproportionaltothe nu mb erofundottedindices,andwehaveafactorfromatraceoverallspinorindices.

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7.8.Examples461We “ndacontributiontotheeectiveaction 2=( Š 2)2( A + B ) d4k (2 )4 d4p (2 )4 1 p2( k + p )2 1 4 d2 [( p +1 2 k ) a( p +1 2 k ) bE E a ( Š k ) E E b ( k )+ A W( Š k ) W( k )]+ h c (7.8.2) Wehave usedthelinearized,on-shellrelations W= D2 E E a = D2E E a,and convertedthe d4 integraltoa d2 integral.Finally,doingthemomentumintegraland usingthe lineari zedrelation aE E b Š bE E a = C€€W,(7. 8.3) weobta intheresult 2=( Š 2)2( A + B )(1 12 Š A )1 2 d4xd21 2 W( x ) I ( Š ) W+ h c .(7. 8.4a) withthelogarithmicallydivergentintegral I of(7.5.26).Aftercovariantization2also containssomecontributionsfromgraphswith3,4,etc.externallines.Thechiralintegralabove,aftercovariantization,alsocontainsa 3factor. Separatingoutthedivergentpart,wehave 2= k11 (4 )2 1 2 d4xd2 31 2 W[ 1 Š ln 2 ] W+ h c .(7. 8.4b) where isarenormalizationmassand k1=( Š 2)2( A + B )(1 12 Š A )(7. 8.5) Forachir alscalar k1=1 24 .(Wehaveincl udedafa ctorof1 2 tocancelthe2dueto ourusingtheaction + h c .inste adof .)Forachiralspinor k1=20 24 .If thechirals pinorsuper“eldisthegauge“eldofthetensormultiplet,therewillbean additionalco ntri bution k1=5 24 fromthe“vesecond generationchiralscalarghosts discussedins ec.7.3.a.(The V ghostsdonotcontr i bute:seebelow.) Thenextcalculationwecouldimagineperformingisthatofatrianglediagram. However,sinceno WWW or WW W termispres entintheeectiveaction(cf.our

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4627.QUANTUMN=1SUPERGRAVITYdiscussionfollowing (7.7.3)),thecontributionfromsuchadiagrammustbecompletely containedinthethirdorder(in H€or )te rmsintheexpansionof2. Weobservethat, oncetheself-energycontributionfromachiralsuper“eldhas b eencomputed,thatfromavectormultiplet V istrivial,becausethewholecontribution comesfromthethreechiralghosts.Indeed,the V -backgroundinteractionsareextracted fromthec ovariant V V quadraticaction.However,ju stasinthebackgroundYangM illscalculation,each D or D containe dinthe operatorof(7.4.14b)bringswithit onefactoroftheexternal“eld,andforgra phswithlessthanfourexternallineswedo nothaveenough D s .T hisisanimportantfeatureofthebackground-“eldmethod:In o-shellYang-Millsoron-she llsupergravitybackground, general(nonchiral)super“elds donotcontributetoone-looptwo-andthree-pointfunctions;onlytheirchiralghostsdo. Thus,inthiscasethecontributiontothesupergravityself-energyfromavectormultipletis Š 3timesthatfr omaphysicalchiralscalarsuper“eld(3ghostswithwrongstatistics). Thecalculationoftheon-shellone-loopself-energycontributionsinself-interacting (q uantumandbackground)supergravityisnowtrivial.Nonewcalculationsneedtobe pe rformedbecause,fromtheactionin(7.4.14a)or(7.4.19),weseeagainthatonlychiral super“eldscontribute.Intheformwith V compensators(5.2.75a),theresultissimply Š 7timesthatfr omaphysicalchira l“eld(7chiralscalarghost swithwrongst atistics): k1= Š7 24 .Intheformwithac hiralcompensator,wehavethecontributionfromthe physical “eld,andcontributionsfromtwochiralspinorswiththeaction1 2 d4xd42+ h c ..The “nalansweris41timesthecontributionfromaphysical chiralsc alar: k1=41 24 .Fina lly,ifweusedthespinorcompensator(5.2.75b)wewould obtain k1= Š55 24 Thecalculationofachiral-orgeneral-“eldboxdiagramcontributingtothe WW W W termin(7.7.4)isinprinciplenomoredicult.Forachiral“eldwehave onevertex(7.8.1a)andthreevertices(7.8.1b)withone D2andfour D sintheloop,two ofwhichhavetobeintegratedbypartsontoexternallines.Forareal“eldwehaveone fact orof D or D ateachvertex(comingfromthelinearizationof ),sothe D -algebra istrivial.WhatisleftthenisaFeynmanintegralforaboxdiagram,withsome

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7.8.Examples463momentumfactorsinthenumerator. Ournextexample isthatofthecalculationofsomeone-loop,four-particleSmatrixelements,similartothecalculationwecarriedoutfor N =4 Ya ng -M ills.There wesawthatt herewerecancellationsandthewholecontributioncamefromthe V -loop. Thecorrespondingsituationthatwewill“ndhereisthatasimilarcancellationbetween ghosts,physical“elds,andcertain othercontr ibutionstakesplacein N =8superg ravity so thatinthatcasetheresultisessentiallyidenticaltotheYang-Millscase,andcanbe obtainedwithoutfurthercalculation.Wenowdiscussthesituationindetail. For N -extendedsupergravitydescribedby N =1super“elds,toc alculatecontributionstobackground N =1superg ravity,onesimplyaddscontributionsfromsuper“elds representingallthe N =1mult iplets.Thesuper“eldswhichenterinadditionto H are ofthesametypeasabove,namely V s, s,and s,andtheirLagrangianshavethe sameform(uptochoicesofcompensating“eldsanddualitytransformations,whichmay chan gethevaluesofcontributionstotopologica linvaria ntsandthec orrespondi ngsuperconformalanomalies(seesec.7.10)butdonotaecttheS-matrix).InTable7.8.1we givethenumberof“eldsofeachtype(themin ussignsindicateabnormalstatistics)for eachvalueof N N V H€ 1-74-31 2-21-21 30-1 -11 40-201 50-211 60021 80441 Table7.8 .1.Numberof“eldsofeachtypecontributingtotheone-loopeectiveaction Weuset heformof N =1supe rgravitywith V compensators.The(3 2 ,1)multi pletis describedbyageneralspinorsuper“eld(see sec.4.5.e)anditsquadraticLagrangian, incl udingghosts(whichisallthatisneeded)isgivenbythebackgroundcovariantization of(7.3.7).

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4647.QUANTUMN=1SUPERGRAVITYThefour-particleS-matrixisobtainedfromthediagramsinFig.7.8.1. Fig.7.8.1 Theexternalwavylinescorrespondto N =1superg ravity“elds,whilethesolidline loopcorrespondstovarious N =1mult iplets.Allbutthelastdiagramcorrespondto the“rstterm2of(7.7.3).However,bycovariancetheS-matrixmustcontainfourfactorsof W ,andby dimensionalitythecompletecontributionmustcomefromthesecond term4of(7.7.3),andthereforecanbeobtainedfromtheboxdiagram.(Onlythebox diagramhasenoughden ominatorfactorstobalancethedimensionsoffourfactorsof W .) Wewritethecontri butionfromtherelevantpartof4as =1 8 d4p1... d4p4(2 )16 d4 ( ( pi)) [ W( p1) W( p2) W€€€( p3) W€€€( p4)( C4G4+ C2G2+ C0G0)( pi) Š1 2 W( p1) W€€€( p2) W( p3) W€€€( p4)( C4G 4+ C2G 2+ C0G0)( pi)].(7.8.6) Here G0istheFeynmanintegralforascalarboxdiagram(6.5.68),while G2( G 2), G4( G 4)aresi milarcontributions extracted fromboxdi agramswithtwoandfourmomentumfactorsinthenumerator.Unlike N =4 Ya ng -M ills,theaboveexpressionisvalid onlyon-shell,whereaso-shelltheeectiveactiondiverges.(Becauseofcovariantization theexpressionabovecontainstermswithmo rethanfour“elds,butitdoesnotgivethe

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7.8.Examples465completecontributiontomore-th an-four-particleamplitudes.) Thecoecients Ciaredierentforeachvalueof N .Todet ermine Ciwemust computecontributionsfromeachofthesuper“eldslistedinTable7.8.1.Theonlydicultcalculationisofthecontributionfromthechiralsuper“elds i.Weshallth erefore restrictourselvesheretodiscussingresultsfor N 3where nochiralsuper“ eldsappear. Aswedisc ussedinsec.7.6,itisusefultosq uarethekin eticoperatorforthe spinors(andtakeonehalfofthecorrespondingcontributiontotheeectiveaction)in ordertomakeitsimilartotheotherkin eticterms.Thekine ticoperatorfor V ,and H€thentakesthe uni versal form = + W M + W€€€ € M€€,(7. 8.7) Therelativecoecientofthe and W termsis independent ofthechoic eof“eld. Therefore,inperformingone-loopcalculationsweneedonlykeeptrackoftheindex structure.Inparticular,thereisalwaysafactorfromatraceovertheLorentzindex:1 for V Š 1for Š 1for €and4for H€.( and €eachcountas Š 1 1 2 2= Š 1 duetoa Š 1forFe rmistatistics,a1 2 tocan celtheeectofhavingsquaredthekinetic operator ,and2forth etrace over .) Lookingatthekineticoperatorandagainrequiringthateachloopcontainatleast two Dsandtwo D€s,wediscovertwosourcesforsuchterms:Theexplicit W M (tothisorderwecanreplace by”ats uperspace D)andthos econtainedinthecovariantdAlembertian: =1 2 a a=1 2 ( E E a m m+ E E a D+ E E a€ D€+ a( M ))( E E a n n+ E E a D+ E E a€ D€+ a( M )), (7.8.8) where E E a mŠ a mandallotherquantitiescontainatleastonefactoroftheexternal “elds.(Theconnectiontermscanbedropped:Theycannotcontributetoanygraph withatmostfourexternallinesbecausetheydonotbringwiththemany D s.) Wenowmaket hefollowingobservations:

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4667.QUANTUMN=1SUPERGRAVITY(1)Since connectiontermscanbedropped, actsinthesamewayonall“elds. (2)Sincewe needtwo D sandtwo D s,andeachoneb ringswithitan E E ora W andan E E ora W ,ourresultw illcontainfoursuchfactors,twobarredandtwo unbarred.The E E verticeshavetheform E E a aDor E E a€ a D€. (3)Since [ aE E b ] W,andthev ectorindicesonthetwo E E sinatermwith onlytwosuchfactorsmustbecontracted(andthereforealsothespinorindices)inorder topro duceacovariantcontribution,thereare E E2 W2and E E2W2terms butno E E E EW W terms.(Similarlythereareno E E2 E E W termsor E EW W2terms.) (4)DuetothealgebraoftheLorentzgenerators,the W M W M factorsineither the E E2W2or W2 W2produceanextranumericalfactorof a (and b from W M W M ) relatedtothenumberofspinorindices. Therefore,therearethreetypesoftermstoconsider: (1)( E E )2( E E )2,(2)( E E )2( W M )2(and h c .),(3)( W M )2( W M )2.Eachterm takesthesameformforall N butwithacoecientdeterminedbysummingover V , ,and H thepro duct (numberofsuch“elds) (Lorentztracefactor) (M-factor). Thenumberof“eldsisgiveninTable7.8.1,thetracefactorwasdiscussedearlier,and the M -factoris,foreachofthethreetypesofterms,respectively:(1)1,(2) b ( a for theh.c.),(3) a b .Thevaluesoftheo verallnu mericalcoecientarepresentedinTable 7.8.2.Thesecoecientsarelabeled(1) C4,(2) C2,(3) C0,andappe arineq.(7.8.6). (Notetha tonly H contributestothelastcolumn,sinceonly H hasbothadottedand an undottedindex,andsohasboth W and W termsinitskineticoperator.)Our resultisthusthat:(1) N =1,2contain all typesofte rms,includingcontributionsfrom chiralsuper“elds ,whichwe havenotdiscussed;(2) N =3 ,4 receivenocontributions fromchir alsuper“elds;(3) N =5,6lack also the E E2 E E2typete rm;(4) N =8 receivesa contributionfrom only the W2 W2term,inanalogytothe N =4supers ymmetricYangM illscalculation.

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7.8.Examples467 NC4C2C0 3554 4244 5034 6024 8004 Table7.8 .2.Multiplicityofcontributionsofeachtypetotheone-loopeectiveaction Thecalculationoftheeectiveactionproceedsasfollows:For N =8superg ravity thereisnothi ngmoretodo.Onehasaboxgraph,withtwofactorsof W andtwofactorsof W andascalarloopintegraltoperform.Weobtainthe G0termsineq.(7.8.6) withafactor C0=4,w h ile C4= C2=0.For N =5,6oneh asaboxgraphwithtwo verti cesoftheform E E andtwowith W s,aswellasatrianglegraphwithonevertex containingtwo E E -factors.Theloopintegralfortheboxgraphcontainsnowtwoloopmomentumfactors,butgauge(localsupersymmetry)invariancecanbeusedtosplito apartwhi chgives [ aE E b ]soastoproduce W s,whiletherestmustcancelthetriangle graphcontributio n.(Theyactuallymaycontributeto2,whichis zerobymomentum conservation .)Finally,for N =3,4oneh asaboxgraphwithone E E factor ateach vertex,atria nglegraphwithonevertexcontainingtwo E E factors, andals oaselfenergytypegraphwithbothverticescontainingtwo E E factors.A gaingaugeinvariancecanbeusedtoextractthecompletecon tri butionfromtheboxgraph,theremainderaddinguptozero. TheS-matrixcanbeobtainedfromtheeectiveactionbydroppingthe piintegralsandta kingas umof G termsoverpermutationsofth eMa ndelstaminv ariants s =( p1+ p2)2, t =( p1+ p4)2, u =( p1+ p3)2.( IntheYang-Millscasethisalsoinvolves intercha ngeofinternalsymmetryindices).Thus,for N =8superg ravitywehave S ( s t u )=(2 )4 ( ( pi)) d4 W( p1, ) W( p2, ) W€€€( p3, ) W€€€( p4, ) [ G0( s t u )+ G0( s u t )+ G0( u t s )].(7. 8.9)

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4687.QUANTUMN=1SUPERGRAVITYThe -integrationsplitsuptheproductofthesuper“elds W intoasumoftermsinvolvingproductsofWeylte nsorsandgravitino“eldstren gthswhichcanbereplacedwith momentaandpolarizationvectorsforthevariousprocesses.Theactualvalueof G0is G0( s t u )= 2 Š (2 )4 (( Š ))2( ) ( Š 2 ) [sŠ 2 Š F (1,1,1 Š Š u s )+ tŠ 2 Š F (1,1,1 Š Š u t )],(7. 8.10) where =2 ŠD 2 and F isahypergeometricfunction.Itisultraviolet“nitebutinfrared di ve rgentforbothYang-Millsandsupergravity,butinthelattercasethedivergenceis milderbecauseofcancellationsinthe s t u permutatio ns.Theexpressionsfor G2, G 2, G4,and G 4aresomewhatmorecomplicatedandwillnotbegivenhere. Sofarourresultsarewithonly N =1superg ravityexternalparticles(gravitons andgravitini).However,theS-matrixcanbeextendedimmediatelytotheotherparticlesofan N > 1multi pleteitherbydirectglobalsupe rsymmetrytransformationsonthe S-matrixorbyre alizingthatthed4 ( W)2( W€€€)2canbeextende dt oasim ilar expressioninvolvingproductsoffouron-shell“eldstrengthsforextendedsupergravity. Wealsonote thatthe W2 W2formoftheresultimpliesthe he licityconservationpropertiesofthesupersy mmetricS-matrix. Theremarkablesimplicityofthecalculationsandresultsisdueinparttothe surprising decreaseinnumberofdiagramsonehastoconsiderasoneproceedsfrom N =1 to N =8.Inparticular, theabsen ceofchiralsuper“eldsfor N 3produ cesthecrucial simpli“cation,andtheensuingcancellationsbetweenvarious“eldsculminatesinthe absolutetrivialityofthecalculationfor N =8superg ravity.Attheotherextreme,the N =0theory (ordinaryEinsteingravity)wouldseemtorequireamajorcomputercalculation.

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7.9.Locallysupersymmetricdimensionalregularization4697.9.Locallysupersymmetricdi mensio nalregularization Iftheonlyinterestingsupergravitytheoriesarethosethatare“nite,theconstructionofaregularizationwhichmanifestlypreserveslocalsupersymmetryissomewhatofanacademicexercise.Nevertheless,weshalldiscusstheprocedurehereforcompleteness,andbecausethegeneralmethodisusefulfordiscussionofdimensionalreductiontointegraldimensions. Itispossibletoextendthesupersymmetricdimensionalregularizationmethodof sec.6.6forapplicationtosupergravity.Somemodi“cationsarerequiredbecause,unlike matter(scalarorvector)multiplets,supergravityisnolongeron-shellirreducibleafter dimensionalreduction.Thedimensionalreductionmustthereforebeperformedina mannerthatpicksouttheirreduciblepart.Insuper“eldlanguage,thedierenceoccurs b ecause( N =1)ma ttermultipletsare describedbyscalarsuper“elds,whereassupergravityisdescribedbyavectorsuper“eld.UponnaivedimensionalreductiontoDdimensions,thisvectorsuper“eldreducestoasuper“eldwhichisaD-dimensionalvector,describi ngpuresupergravity,plus4-Dscalarsuper“elds,describingvectormultiplets.Thedimensionalreductionmustthereforeberede“nedsothatonlytheD-dimensional-vectorsuper“eldappears.Wethereforeneedasuper“eldformulationthat describespuresupergravityforarbitraryD < 4.Forsimplicitywewilldescribetheconstructionfor N =1, butthemethodcanbeeasilygeneralizedtoanyfour-dimensional super“eldtheory.Forintegraldimensions,the N =1theory reducestopure N =2 supergravityforD=3or2,andpure N =4superg ravityforD=1. Webeginbyc onstructingcovariantderivatives.Sinceupondimensionalreduction theLorentzgroup SO (3,1)isbro kendow nto SO (D Š 1,1) SO (4 Š D),ourcovariant derivati vestaketheform A= EA+1 2 A b cM c b+1 2 Ab cMc b,(7. 9.1) wherethesupervectorindex A =( ,€ a ),andwehavere duceda4-dimensionalvector i ndex € intoaD-d imensionalvectorindex a pl usaninternalsymmetry ( SO (4 Š D))index a .The “eldstrengthsarede“nedasusual: [ A, B} = TAB CC+1 2 RAB c dM d c+1 2 RABc dMd c.(7. 9.2) Thederivativescontainedin EA= EA MDMrangeoverD(commuting)+4

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4707.QUANTUMN=1SUPERGRAVITY(anticommuting)coordinates.Thefact thatthespacetimecoo rdinateshavebeen reducedfrom4toDautomaticallytakescareofreducingthefundamentalsuper“eld H atoaD-componentvector ,asdis cussedabove( H = H ai a).However,ascomparedto theusual4-dimensionalcovariantderivatives,wehavelessgaugefreedomduetothe absenceofaLorentzgeneratorofthemixedtype Ma b,sothatana ddition alconstraintis neededtoaccountforthislostinvariance.Speci“cally,thismeansthattheobject N whichappearsuponsolvingtheconstraints,andwhichisgaugedawaybylocalLorentz transformationsinD=4,musthavethecomponentswhicharenotgaugedawayinD < 4 constrainedaway.Wethereforeimposethe followingsetofconstraintsforarbitrary D 4(forsim p licity,wechoosethecase n = Š1 3 ): Conventional : T = T [ b c ]= T€€= a €R€ c d= a €R€ c d=0, a €T€ c= i a c;(7. 9.3a) Chiral itypreserving : T c= T€=0;(7. 9.3b) Conformalbreaking : T b bŠ T€€=0;(7. 9.3c) Additionalconventional : a €T€ c=0;(7. 9.3d) wheretheadditionalconventionalconstraintistheonethatconstrainstheextracomponentsof N .( a €istheD-dimensionalPaulimatrix,whichprojectsouttheD-componentvectorindex a fromthe4-d imensional € ;sim ilarly, a €projectsoutthe4-D-component i ndex a .Our normaliz ationhereis a € b €= a b, a €b €= a b, a € a €+ a €a €= €€.)Theconventionalconstraintsaswrittenaresomewhat redundant,butitcanbeshownthat,inconjunctionwiththeremainingconstraints,they servetodetermine Aintermsof E,asusual.Thechir ality-preservingandconformalbreakingconstraintshaveasolutionsimilartothatofD=4,exceptthat HMin H = HMiDMisnowaD+4componentsupervector.Thesolutiontotheconstraintsis (cf.sec.5.3): E= N E,

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7.9.Locallysupersymmetricdimensionalregularization471E a= ( N a bŠ N a cAc b) E b+( f a E+ f a€ E€), E= eŠ De, E a= Š i a €{ E, E€} ,=MiDM, [ EA, EB} = CAB C EC, Aa b= i a € C€ b, N€ €= N N€€, detN =1, N = det ( N a bŠ N a cAc b), = 1 2 ŠD 2(D Š 2) [(1 eŠ )D(1 e )Š (D Š 2) E2 N2]Š1 4(D Š 1) E€ =0.(7. 9.4) Thematrix N€ €isdeterminedby(7.9.3d)totaketheform N€ €= N a bNa bN a bNa b = (1+ ATA )Š1 2 (1+ AAT)Š1 2 A Š (1+ ATA )Š1 2 AT(1+ AAT)Š1 2 (7.9.5) inanappropriateLorentz internalgauge,wheredoubles pinorindicesareconverted intovectorindicesandbackagainwith s(oftheappropriatetype),andthelastequationiswritteninmatrixnotationwith A = Aa b. N isdeterminedfromthisexpression for N€ €byusin gtherelation N€ €=( eX)€ €, X€ €= €€Y + Y€€ N =( eY) .(7. 9.6) Since N€ €isorthogonal, X isantisymmetric,andtheref orecanbeexpressedinterms ofthetraceless Y .Wehavenot giventheexplicitexpressionfor f a in(7.9.4),northe solutionsfortheconnections,buttheycanbeobtainedasinD=4withoutfurthercomplications,andwillnotbeneededhere. Thesupergravityactionis S = Š 2D Š 1 D Š 2 Š 2 dDxd4 EŠ 1

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4727.QUANTUMN=1SUPERGRAVITY= Š 2D Š 1 D Š 2 Š 2 dDxd4 EŠ1 (D Š 1) [ det ( a b+ Ac aAc b)]Š1 2(D Š 1) [(1 eŠ )(1 e )](D Š 2) 2(D Š 1) (7.9.7) Projectionoperatormethodscanbeusedtoshowthatthelinearizedactioncontains onlytheusualsuperspins3 2 +0.CouplingtomattercannowbeperformedasinD=4, withchiralLagrangiansintegratedbydDxd22(D Š 1) (D Š 2) Supergr aphcalculationscanbe perfo rmedwiththeusualfour-dimensional D -algebra.Wedomomentumintegrationas inconventionaldimensionalregularization,andminimallysubtractthedivergentpart using 1 timesalocal,covariant,D-dimensiona lcounterterm constructedfromtheDdimensionalcovariants. Thesameinconsistenciesthatoccurredi ngloba llysupersymmetricdimensional regularizationofcourseremaininthelocalcase.Nevertheless,asintheglobalcase,in actualcomputationstheinconsistenciesseemtodisappearaftertakingD 4.After minimalsubtraction,theremaining“nitequantitysatis“esthe4-dimensionallocal supersymmetryWard-Takahashiidentities(aftertakingD 4).Furthermore,the methodisperfectlyconsistentforreductiontointegraldimensions,andcanbeusedfor describingextendedsupergravityinlowerdimensions.However,weobservethatthe ab ov es uper“elddescriptioninnonintegraldimensionsde“esunderstandingintermsof components.(E.g.,sincetheD-beinin H ahasno ea bpart,what“eldgaugesthe internal Ma bsymmetry?) Sincethesubtractionprocedurepreserveslocalscaleinvariancewhenthecompensator isincluded,therenormalizedeectiveactionwillbesuperconformallyinvariant. However,D=4superconformalanomaliesare ingeneralpresentpr eciselybecausethe renormalizedeectiveactiondependson .Wediscuss thisinthenextsection.

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7.10Anomalies4737.10Anomalies a.Introduction Anomaliesinlocalconservationlawsareharmlessaslongasno“eldscoupleto thecorrespondingcurrent.Indivergentcomponenttheoriesthereisalwaysatleastone suchanomaly:thescaleanomaly.Thisanomaly,whichcanbeexpressedasanadditionalcontributiont othetraceoft heenergy-momentumtensor,occursbecauseanew massscaleisintroducedatthequantumlevel,therenormalizationmassparameter.For example,atheorythatisclassicallyconfo rmallyinvariant,andthushasaclassical energy-momentumtensorwithvanishingtra ce,getsquantumcontr i butionstothetrace. WhenEinsteingravityiscoupledtothequantumsystem,thisanomalyisharmless,as generalcoordinateinvariancemerelyrequire sconserv ationoftheenergy-momentumtensor( i.e.,thevanishingofitscovariantdivergence).However,itwouldbeharmfulin conformalgravity,sincelocal(Weyl)scaleinvariancedoesrequirevanishingofthetrace. Quantumc orrectionstotheenergy-moment umtensorar emostc onveniently de“nedbycouplingthequantumsystemtobackgroundgravityandde“ning T m n= R g m n (7.10.1) whereRistherenormalizedeectiveactionandisasuitablyde“nedvariation(see below).I tstraceisgivenby T m m= g m nR g m n (7.10.2) Alternatively,itcanbeobtainedby“rstint roducingacompensatingscalarintothetheory(5.1.34).Forconformaltheoriesthe compensatordecouplesfromtheclassical action,butingeneralitent ersintherenormalizedactionwhereitcouplestothetrace. Therefore,varyingRwithrespecttothecompensatordetermines T m m. Insupersymmetry,theenerg y-momentumtensoristhe componentofasuper“eld,the supercur rentJ€.(Morepr ecisely,1 8 [ D€, D] J€| + a b = T a bŠ1 3 a bT c c.) Thetraceoftheenergy-momentumtensorisacomponentofarelatedsuper“eld,the supertraceJ (1 2 ( D2J | + h c .)=1 3 T a a).Justastheenergy-momentumtensorcanbe de“nedfromthecouplingtogravity,thesupercurrent J€canbede“nedfromthe

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4747.QUANTUMN=1SUPERGRAVITYcouplingtothesupergravitysuper“eld H€: J€= H€ .(7. 10.3) Aswewilldisc ussbelow,thesupertracecanbede“nedbyfunctionaldierentiationwith respecttothecompensatingsuper“eld.Inclassicallocallysuperscaleinvarianttheories thecompensatordecouplesandthereforethesupertracevanishes.Ingeneral,itspresenceisameasureofthebreakingoflocalsuperscaleinvariance. Thesupercurrentalsocontainsthesupersymmetrycurrent(atlinearorderin ) andtheR-symmetryaxialcurrent(at =0); thesupertracealsocontainsthe -traceof thesupersymmetrycurrent(atlinearorderin )andthediver genceoft heaxialcurrent (theimaginarypartofthe 2component).Thus,inasupersymmetrictheorywhere scaleinvarianceisbroken,theaxialcurrenthasachiralanomalyandthesupersymmetry currenthasan S -supersymmetryanomaly,andthecoe cientsofallthreeanomaliesare equal.However,justastranslationalinvarianceisnotviolated(thetraceoftheenergymomentumtensorisanomalous,notitsdivergence),neitherisordinary Q -supersymmetry(the -traceofthesupersymmetrycurrentisanomalous,notitsdivergence). Inlocallysupersymmetrictheories,ina dditiontothesuperconformalanomalies describedbythesupertrace,theremayexistanomaliesintheWardidentitiesoflocal (Poincar e)supersymmetry.Forthecasesthathavebeenstudiedtheydonotoccurin N =1theoryfor n = Š1 3 (theminimalsetofauxiliary“elds)becausewecanregularize inama nnerconsistentwithlocalsupersymmetry;theydooccuringeneralfornonminimal(andnewminimal) N =1, n = Š1 3 theori es.Wewillusetheexistenceofsuperconfo rmalanomaliestoinfertheexistenceoftheseauxiliary-“eldanomaliesandconclude thatingeneralonly n = Š1 3 theoryisquantumconsistent. b.Conformalanomalies We“rstreviewon e-loopon-shellscaleanomaliesincomponenttheories.We areinterestedinquantumcorrectionstoma trixelementsoftheenergy-momentumtensorbetweenthevacuumandastatecontainingtwoormoregravitons.(Wecouldconsiderotherexternalparticle s,andalso o-shellanomalies,butwhengravityisquantizedonlytheonshellonesareunambiguous.)Equivalently,wecomputetheone-loop

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7.10Anomalies475eectiveactionforasysteminagravitationalbackground,functionallydierentiatewith respectto g m nand then setthebackground“eldonshell.AtthelevelofFeynmandiagrams,becauseofcovariance,weneedonlyconsiderthetwo-pointfunction(graviton pr opagatorcorrection),whichdeterminesalltheone-loopdivergences,andthethreepointf unction(trianglegraph),whichdeterminesthetraceoftheenergy-momentum tensor.Infact,iftheclassicaltheoryforthe“eldintheloopisconformallyinvariant, all therelevantinformationcanbeextractedfromthetwo-pointfunction. Inclassicaltheoriesconformalinvaria nceisbrokenintwoways:(a)bymassterms thatbreakitsoftlyandwhos ee ectcanbeseparatedout,astheycanbeforthedivergenceoft heaxialcurrent,and:(b)byhardterms,e.g.,derivativesof“elds,asfor Ya ng -M illsinD =4dime nsions,andforantisymmetrictensor“elds.Inthesubsequent discussion,whenwerefertononconformaltheorieswemeanclassicaltheorieswherethe breakingishard. Weconsid er“rstaclassicallyconformallyinvarianttheorysothatthetraceofthe classicalenergy-momentumtensoriszero.Wh encoupledtogravity,theclassicaltheory islocallyscaleinvariant.Byintroducinga ppropriateD-dependencethetheorycanbe dimensionallyregularizedsothatthisinvarianceispreservedintheregularizedeective actionnearD=4.(Theinvarianceisbrokenonlytoorder(D Š 4)2,andth ushasno eectevenin(D Š 4)Š 1divergentterms.)However,thecoecientofthe(D Š 4)Š 1factorisnotseparatelylocallyscaleinvariantexceptatD=4,andthereisnolocal“nite termthatcanbeaddedtoittomakeitso.Ther efore,therenormalizedeectiveaction, de“nedbysubtractingthisD-dimensional,local,covariantdivergenttermfromtheregulari zedeectiveaction(i.e.,byaddingacounterterm S)isnotloca llyscaleinvariant. Consequently,whenwecomputethetraceof T m nde“nedintermsoftherenormalized eectiveactionwe“ndanonzeroresult.Sincetheregularized,unrenormalizedeective actionUwasscale invariant,thescaleanomalyoft herenormalizedeectiveactionRisjustthetrace computedfromtheD-dimensionalcounterterm S. Wehave de“ned T m nin(7.10.1).Thevariationisde“nedintermsof by f g m n = gŠ1 2 f g m n (7.10.4) ( g = det ( g m n)),ordirectlyby

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4767.QUANTUMN=1SUPERGRAVITY g m n( x ) g p q( x) =1 2 ( m p n ) qgŠ1 2 4( x Š x).(7. 10.5) Thelocalscale(trace)anomalyisthen g m nT m n= g m nR g m n = g m n S g m n .(7. 10.6) Th el as te qu al it y hol ds on ly becauseweareconsideringclassicallyconformaltheories. Otherwise,theformertwoexpressions, thetotaltrace, donotequalthel astexpression, thetraceanomaly. (Ingen eral,the anomaly is understoodtobeacontributiontothe traceduetothedivergencesofthetheory.) SinceinclassicallyconformaltheoriesUislocallyscaleinvariantnearD=4, Stakestheformof(D Š 4)Š 1timesalocal(generalcoordinate)invariantthatisthe dimensionalcontinuationofa4-dimensiona lobj ectthatisloca llyscaleinvariant.From dimensionalandcovarianceconsiderationswe“ndtwoindependentfour-dimensional objectsofthisform:Intermsoftheirreduciblepartsofthecurvatureof(5.1.21),they aretheEule rnumber =1 (4 )2 1 2 d4xg1 2 [1 2 ( ww+ h c .) Š r€€r€€+3 r2],(7. 10.7) atop ologicalinvariantwhosefunctionalvariationvanishesandwhichitselfvanishesin topologicallytrivialspacetimeforD=4,andtheintegralofjusttheWeyltensor ( w2+ w2).Thedier encebetweenthetwovanishesonshell( r€€= r =0). Inquantumgravitythecoecientofanytermthatvanishesonshellisingeneral gauge-dependent,andinfactcanbemadetovanishbyanappropriategaugechoice,or canbeeliminatedbyalocal“eldrede“nitionofthemetric(sincesuchrede“nitionsof theactionareproportionaltothe“eldequat ions).Ther efore,weconsideronlytheonshellpartofthetraceanomaly,whichwewriteintermsof w2=1 2 ww.Wenote that w2hasthesimplescalingpropertyfor arbitrary D ( g m n g m n )( x )( g1 2 w2)( x)=1 2 (DŠ 4)( g1 2 w2) 4( x Š x).(7. 10.8) InD-dimensionstherelevantpartofUisgivenbyacovariantizationofagraviton self-energygraphandhastheform

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7.10Anomalies477U1 D Š 4 dDxg1 2 1 2 w ( `` 2 )D 2 Š 2w + h c .,(7. 10.9) where`` means +curvature termsnecessarytomakeUlocallyscaleinvariant inarbitraryD,and isarenormalizationmass.(Ualsocontains“nitelocallyscale invariantterms,anddivergenttermsthatvanishonshell.)Wethenhave SŠ1 D Š 4 dDxg1 2 w2+ h c .,(7. 10.10) R d4xg1 2 1 4 wln ( 2 ) w + h c ..(7. 10.11) Byintegrationbyparts(dropping“nitetermsproportionaltotheEulernumber,which canbeconsideredpartofthecorrespondingin“niteterm(7.10.10)),Rcanberewritten atD=4 R d4xg1 2 {1 2 r€€ln ( 2 ) r€€Š3 2 rln ( 2 ) r +( w2+ w2) ln [1 Š 1 + r r ] } (7.10.12) plusmore“nitetermsthatarelocallyscalei nvariant,andtermsof thirdorhigherorder in r and r€€.Sin ce[1 Š ( + r )Š 1r ]satis“est hescalec ovariant equation ( + r ) =0(withour conventionsofsec.5.1, + r isthekineticoperatorofalocally scale-covariantscalar),itcanbeshownthat ( g m n g m n )( x ) ln [1 Š 1 + r r ]( x)= Š1 2 4( x Š x),(7. 10.13) Therefore,using(7.10.8),weseethat(7.10.12)givesthesame(on-shell)trace(fromthe lastterm)asRin(7.10.11)(oras Sin(7.10.10)): g m nT m nŠ1 2 ( w2+ w2).(7. 10.14) We “nd(7.10. 12)amoreconvenientformofrepresentingR.The “rsttwotermsare covariantizedself-energycontributionsunambiguouslyexpressedintermsofthecurvaturescalarandRiccitensor,andareofnointerestforon-shelltracessincetheirvariationvanishesonshell.Thelastterm,whenthe ln isexpa ndedinpowersof r ,hasthe

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4787.QUANTUMN=1SUPERGRAVITYform d4xg1 2 w21 r + ... = d4xg1 2 r 1 w2+ ... ,(7. 10.15) sothatitreceivescontributionsonlyfromdiagramswithatleastthreeexternallines. Thistermisacovariantizedtrianglegraphcontributionandcouldbecomputedusing, forexample,the Adler-Rosenber gmethodwith r atthetopvertex(cf.also (6.7.10-13).Thetraceoperation g g actingon r in(7.10.15)isanalogoustothe 2in (6.7.13)). Inthemoregeneralcasewhenthequanti zedtheoryisnotcla ssicallyco nformally invariant,orgravityisalsoquantizedsothat g g U =0,localsc aleinvariancecannot beusedto determine g m nT m nfrom S.Itisthenn ecessarytocalculatethetotaltrace fromRdirectlyfromatrianglegraph.(Inthecaseofquantumgravityweuseabackground“eldgaugetomaintaincovarianceofR.)Thegen eralformoftheunrenormali zedeectiveaction near D=4is U= k11 D+1 (4 )2 d4xg1 2 [ k21 2 r ( 1 Š ln ( `` 2 )) r + k31 2 r€€( 1 Š ln ( `` 2 )) r€€Š k4( w2+ w2) ln (1 Š 1 + r r )](7.10.16) where =2 ŠD2 and DisthedimensionalcontinuationoftheEulernumberof(7.10.7): D=1 (4 )1 2 D 1 2 dDxg1 2 [1 2 ( ww+ h c .) Š r€€r€€+3 r2].(7. 10.17) Risobtainedbysubtractingoutthe Š 1terms. Therelevanttermfortheon-shelltraceisagainthelastonein(7.10.16),although nowitscoecientisnotrelatedtothoseoftheprecedingterms.Theon-shelltrace computedfromRisnotequaltothet racecomputedfrom S.I tr eceivesadditional contributionsfromtheclassicallynonconfo rmalpartofthetheory.Asmentionedabove,

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7.10Anomalies479wew illrefertot hepartattributableto Sasthetraceanomaly,whilecallingtheentire contributionfromRthetotaltrace. Thenumber k1hasbeencomputedinavarietyofways,anddeterminestheonshelltraceanomalyof“eldsintheloo p.Thisquan tityisusuallywritten ( T m m)anomalous= k11 (4 )2 1 4 [ ww+ h c .](7. 10.18) with360 k1=4,7, Š 52, Š 233,848,364,forascalar,Majoranaspinor,vector,RaritaSchwinger“eld,graviton,andsecond-rankan tisymmetrictensorgauge“eld,respectively, incl udingtheirghosts.Wenotethatalthoughthe“rstandlast“eldsbothdescribethe samespinzeroparticle(ifthescalarhasno improvementterm),theirtraceanomalies aredierent.Ontheotherhand,itcanbea rguedthattheyhavethesametotaltrace, whichisthephysicallyrelevantquantity,determinedbyR.(Forthe improved scalar “eld( T m m)tot=( T m m)anom, butfortheantisymmetrictensororunimprovedscalarthey aredierent:Thelattertheoriesarenotclassicallyconformallyinvariant.)Inlikefashion,third-andfou rth-rankantisymmetrictensor“elds,whichhavenophysicaldegrees offreedom,havezerototaltrace(infact,zer oreno rmalizedeectiveaction),although b ecauseofthequantizationprocedure,theyhaveanonzerodivergentcontributiontoUandthereforeanonzerotraceanomaly. Ausefulmethodforma kingscalebreakingpropertiesmanifestistointroducea compensatingscalarasin(5.1.34).Wethenhave ( g m n g m n f )( 2g p q)=1 2 Š 3 f 1 2 f ,(7. 10.19) sotheexistenceofanonzerotraceisequivalenttohavingdependenceon .Forex ample,(7. 10.10,11)becomes SŠ1 D Š 4 dDxg1 2 2D D Š 2 w2+ h c .,(7. 10.20) R d4xg1 2 1 4 wln ( `` 2 ) w + h c .;(7. 10.21) (notethatweh aveabsorbed into )andthelasttermi n(7. 10.12)becomes d4xg1 2 ( w2+ w2) { ln [1 Š 1 + r r ] Š ln } .(7. 10.22)

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4807.QUANTUMN=1SUPERGRAVITYIftheclassicaltheoryisconformallyinvariant, decouplesfromtheclassical action,andthus doesnotappearintheFeynmanrulesorinU:Itsonly appearancein Risthrough S.Itmus tbeintro ducedin StomakethistermscaleinvariantinDdimensions,andtocompensateforthisitmustalsoappearinR,sinceUisinde pendentof .Ontheot herhand,iftheclassicaltheoryisnotconformallyinvariant, is presentinU,andw illenterinRinama nnerwhichisnotrelatedtothewayitenters in S. c.Classicalsupercurrents Inthissubsectionwederivetheclassicalsupercurrentsforvariousmultiplets. Thesearethesuper“eldsthatcontainthesuperconformalcomponentcurrents.They canbeobtainedi nprincip lefromtheclassicalactionsbymeansofNoetherstheorem,or canbecalculatedasthevariationalderivativesofthecovariantizedactionswithrespect tothesupergravityprepotentials.Ingen eralwedonotimmediatelyobtainthesame results,unlessweperformsome“eldrede“nitions.Theserede“nitionshavenophysical eectsincetheyonlychangethecurrentsbytermsproportionaltothe“eldequations. Weconsid erminimalsupergravitywiththechiralcompensator. Theactionforascalarmultipletinthepresenceof(background)supergravityis (inthechiralrepresentation) S = d4xd4 EŠ 1 eŠ H +[ d4xd23(1 2 m 2+1 6 3)+ h c .]. (7.10.23) Ifwemakethe“eldrede“nition Š 1 andusethelinearizedequation(see(7.5.4)) EŠ 1Š 1( eŠ H )Š 1=1 Š1 3 D€DH€Š1 3 i aH a(7.10.24) weobta inthe superc urrent J€ J€= S H€ = Š1 6 [ D€, D] +1 2 i€ = Š1 3 ( D€ )( D )+1 3 i€ .(7. 10.25) The -independentcomponentof J€isthe(R-transformation)axialcurrent

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7.10Anomalies481J€| =1 3 Ai€A Š1 3 €,the linear -componentisthesupersymmetrycurrent,and atthe levelwe“ndthe(im proved)energy-momentumtensor. Wede “nethe supertrace J S 3 =1 6 m 2, D€J =0.(7. 10.26) Wecanv erify,usingtheequationsofmotion,theconservationequation D€J€= DJ .(7. 10.27) Quitegenerally,thisequationisadirectconsequenceoftheinvarianceofthe ac ti on under L-transformations(5.2.7,7.4.2b), H€= D L€Š D€L, 3= D2DL: LS = S H€ LH€+( S 3 L3+ h c .)=0.( 7.10.28) Iftheclassicaltheoryisconformallyinvariantthecovariantizedactionissuperscale invariant(i ndependentof ,po ssiblyafter“eldrede“nitions,e.g.,inthecaseaboveif m =0), thesupertracevanishes,and D€J€=0.(7. 10.29) Thisequationexpressestheconservationoftheaxialcurrent,andthevanishingofthe supersymmetrycurrent -trace,andoftheenergy-momentumtensortrace. Forthev ectormultipletthe”atsuperspacecomponentcurrentsarecontainedin thesuper current J€= W€W,where Wisthe”atsuper“eldstrength.However,to obtainthisexpressionfromcouplingtosupergravityrequiressome“eldrede“nitions whichwenowdescribe.Forsimplicityweconsidertheabeliancase. Inthesupergravitychiralrepresentationwehavetherealitycondition V= eHV Introducing V= e1 2 HV wehave now V = Vand W= i ( 2+ R ) ( eŠ1 2 HV )= i D22 N eŠ HDe1 2 HV = i Š3 2 D2 EŠ1 2 N eŠ HDe1 2 HV .(7. 10.30) Itisconvenienttocalculateinthegauge(7.6.5)where N = sothatspinorchiral “eldsarech iralintheusualsense.Inthisgauge,atthelinearizedlevel(seesec.7.5.c)

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4827.QUANTUMN=1SUPERGRAVITY EŠ1 2 N = Š D€DH€.(7. 10.31) However,ifwecalculatethesupercurrentby J€= H€ d4xd231 2 WW(7.10.32) itwillnotbe(Yang-Mills)gaugeinvariantbecausethegaugetransformationof V (or V)dep endson H€: V= i ( eŠ1 2 H Š e1 2 H), D€=0.(7 .10.33) Were medythisbymakingafurther“eldrede“nition V=( cosh1 2 H + sinh1 2 H1 2 H 1 2 H€[ D€, D]) V0.(7. 10.34) Thus V0hasthe H -independenttransformationlaw V0= i ( Š ),whichindicates thatthecomponentvector“eldin V0hasacurvedvectorindex,incontrasttothe”at i ndexonthatin V .Atthe linearizedlevel,we“nd 3 2 W= W0 Š D2H€ W0€,(7. 10.35) where W0 = i D2DV0(7.10.36) isthegauge-invariant“eldstrengthof V0(containingthecompon ent“eldstrengthwith curvedindices).Fromtheaction(7.10.32)we“ndthen J€= W0€W0 J =0; D€J€=0.(7. 10.37) Ifthe(covariantized)supersymmetricgauge-“xingterm(6.2.17)ispresent,we haveadditionalcontributions(for =1) J€ GF= Š1 6 [ D€, D][ V0{ D2, D2} V0+( D2V0)( D2V0)] +1 2 ( D2V0) i€( D2V0) Š V0i a[ D2, D2] V0

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7.10Anomalies483Š1 2 ([ D€, D] V0) { D2, D2} V0,( 7.10.38a) JGF=1 3 D2( V0[ D2, D2] V0).(7. 10.38b) Fo rt he tensormultiplet(chiralspinorsuper“eld)thereareanalogouscomplicationsduetothetransformationlaw = i ( 2+ R ) ( eŠ1 2 HK).(7. 10.39) Wehavein tro duced Kbyan alogyto V.Howev er,ifwerede“ne Kintermsof K0, and intermsof 0 ,byanal ogyto(7.10.34,35),we“ndthatthecovariant“eld strength G=1 2 e1 2 H+ h c .= Š1 2 e1 2 HE ( EŠ 1E )+ h c = Š1 2 e1 2 HE ( 3 2 EŠ1 2 N eHD eŠ H)+ h c .(7. 10.40) canbeexpressedas G= G0Š1 3 ([ D€, D] H€) G0Š1 2 H€[ D€, D] G0+[( DH€) D€Š ( D€H€) D] G0,(7. 10.41) where G0=1 2 D0 + h c .. Fromtheaction S = Š1 2 d4xd4 EŠ 1eŠ1 2 HG 2(7.10.42) weobtain J€= Š1 12 [ D€, D] G0 2+1 2 G0[ D€, D] G0= Š1 3 ( D€G0)( DG0)+1 3 G0[ D€, D] G0,( 7.10.43a) J =1 6 D2G0 2.(7. 10.43b) Wenote thatthesubstitution G0 + (cf.sec4.4.c.2)givesthesupercurrent J€for

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4847.QUANTUMN=1SUPERGRAVITYthenonconformalscalarmultipletwithLagrangian1 2 ( + )2.ThisL agrangianforthe scalarmultip let,identicalin”atspac etothe usualone,givesdisimprovementtermsto J€and J b ecausetheextraterms1 2 ( 2+ 2)leadtononmini malcouplingsd4xd23R1 2 2+ h c ..Ontheotherhand,theimprovedtensormultiplet(4.4.46) withaction Šd4xd4 GlnG doeshave J =0. Fromthe gauge“xingterm SGF= Š1 2 d4xd4 EŠ 1(1 2 Š h c .)2(7.10.44) weobtain additionalcontributions.Thecombinedcurrentfrom(7.10.42,44)canbewritten J€=1 6 ( D2) i €Š1 6 ( D) i €D( )Š1 2 i €D2+1 2 €+ h c ., (7.10.45a) J =1 12 D2[( D)2+ h c .](7. 10.45b) Asmentionedearlier,the“eldrede“niti onswehav eperfo rmedchangetheformof thesuper currents,butonlybyaddingtermsproportionaltothe“eldequations.Such termshave nophysicalconsequences. Thesupercurrentforthesupergravitymultipletitselfcanbeobtainedfromthe background-quantumsplittingofsec.7.2,byfunctionaldierentiationwithrespectto thebackgro und“eld.Wewillnotgiveithere. d.Superconformalanomalies Thediscussionofsec.7.10.bcanbetakenoverdirectlytothe N =1supersymmetriccase.Weconsiderquantumcorrectionstothesupercurrent J€,andin particular toitssupertrace J .Forcl assicallyconformallyinvariantsystemsthesupertracecanbe obtainedfromtheone-loopcounterterm,andwewillgenerallyrefertothiscontribution asthe superanomaly. Iftheclassicaltheoryisnotconf ormallyinvariantthesupertrace, computedfromtherenormalizedeectiveaction,doesnotequalthesuperanomaly.We

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7.10Anomalies485discussinthissectiontheminimal n = Š1 3 theorywithachiralcompensator. Wede “netherenormalizedcurrents J€= R H€ (7.10.46) J = R3 .(7. 10.47) (IntheversionofthetheorywithvariantmultipletcompensatorswehaveR V = J + J orR = J .)Wewillassumeforthetimebein gthatt heminimaltheoryhasno lo calsupersymmetryanomalies.Invarianceoftheeectiveactionunderlocalsupersymmetrytransformationsgivesthen €J€= J .(7. 10.48) ThesupertraceiszeroonlyifRisindepe ndentof .(Theoper ationisd e“nedin (5.5.44).) Thesuperanomalyisgivenby Jan= S3 (7.10.49) J = Janonlyiftheclassicaltheoryissuperconformal. Therelevantone-loopexpressionscorrespondingto(7.10.9,10,17)are U1 D Š 4 dDxd22(D Š 1) D Š 2 W( +2 )1 2 D Š 2W+ h c ., (7.10.50a) SŠ2 D Š 4 dDxd22(D Š 1) D Š 2 W2+ h c .= Š2 D Š 4 dDxg1 2 ( w2+ w2),(7. 10.50b) D=1 (4 )1 2 D [1 2 dDxd22(D Š 1) D Š 2 W2+ h c .+ dDxd4 EŠ 1( G2+2 RR )],(7.10.51) where W2=1 2 WW, isasuper covariantizeddAlembertian,and DisthesupersymmetricformoftheEuler numberof(7. 10.7).Theexpressi oncorresp o ndingto (7.10.16)is

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4867.QUANTUMN=1SUPERGRAVITYU= k11 D+1 (4 )2 d4xd4 EŠ 1[( k2Š k3)1 2 R ( 1 Š ln +2 ) R Š k31 2 G€( 1 Š ln 2 ) G€] Š k41 (4 )2 { d4xd23W2ln [1 Š ( 2+ R ) 1 Š R ]+ h c } ,(7. 10.52) andrepresent san unambiguouswayoforganizingtheo-shellcovariantizedcontributionsfromsupergraphswithtwoorthreeexternallines.Otherterms,withmorefactors of W,donotcontri butetoon-shellsupertraces.ThedAlembertian Šwasde “ned in(7.4.4). Iftheclassicaltheoryissuperconformal k1= k4.Otherwise,the yhavetobecomputedseparately,e.g.,fromaself-energyan dfromatriang lesupergraph,respectively. Forexample,t helastterm in(7.10.52)canbeexpandedas k41 (4 )2 d4xd4 EŠ 1W21 Š R + h c .+ ... ,(7. 10.53) andgives,atthelinearizedlevel, J€=1 3 k41 (4 )2 i €1 ( D2 W2Š D2W2).(7. 10.54) Thiscorrespondstothecontributionfromatrianglegraphwithtwolegsonshell.Its formisuniquelydeterminedbycovariancea ndpowercount ing,andtheactualvalueof k4canbedetermined,forexample,bytheAdler-Rosenbergmethod. Thesupertraceandsuperanomalyaregivenby J =1 3 k41 (4 )2 W2, Jan=1 3 k11 (4 )2 W2.(7. 10.55) Thesuperanomalycanbereadfromtheresultscontainedin(7.8.5)whichgivetheonshellvalueofthe“rsttermin(7.10.52).Forascalarmultiplet k1=1 24 ,wh ileforatensormultiplet,includingghosts,itis k1=25 24 .Inabackgr oundcovariantgaugeforthe v ectormult ipletthecontributionto Scomesentirelyfromtheth reechiralghostssince,

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7.10Anomalies487asdiscussedinsec.7.8,genera lsuper “eldsdonotcontributetothetwo-pointfunction. Thus,fortheYang-Millsmultipletwehave k1= Š3 24 .Forthegra vitinomattermultiplet,withtheeectiveLagrangianof(7.3.6)or(7.3.7)we“nd,byaddingcontributions fromthechiralghosts, k1= Š19 24 or5 24 ,resp ectively,forthetwodierentsetsofcompensat ors.Finally,forthesupergravitymultipletweobtainthevalues k1=41 24 Š7 24 Š55 24 dependingonwhetherweusea V ,or compensator.These numberscan alsobeobtainedfromacomponentanalysisofthetheories,usingthevalues k1of(7.10.18)forthecomponenttraceanomaly.(Changingfromonecompensator toanothercorrespondstoreplacingsomeofthespinzeroauxiliary“eldswithdivergencesofvectorauxiliary“elds.) Forthescal armultiplet,whichisclassicallysuperconformallyinvariant, k4= k1=1 24 ,andfort hesamereason ,forthev ectormult iplet k4= Š3 24 .Sincethe tensormultipletisphysicallyequivalenttothescalarmultiplet,ithasthesamevalue k4=1 24 ( = k1sincetheclassicaltheoryisnotsuperconformal).Thisresulthasbeen checkedbya nexp licitcalculation. Forthesuperg ravitymultiplettheexplicitcalculationshavenotbeencompletely carriedout.Ifweconjecturethatthecontributionstothesupertraceagaincomecompletelyfromthechiral“eldsinthequantumaction,wecandeterminethecoecients k4. Sincewearediscussingthesupertrace,chiralspinorsareequivalenttochiralscalarsor, whatamountstothesamething,theresultisindependentofthetypeofcompensator weuse.Thisgi vesthevalue k4= Š7 24 forthesupergravitymu ltipletand,byasimilar reasoning, k4=5 24 forthe(3 2 ,1)gravitino mattermultiplet.(Forexample,inthe(2,3 2 ) mult iplet,replacingthechiralcompensatorwitha V compensatorreplacestwo sand two €switheight swithoppositestatistics.Forthe(3 2 ,1)multi plettheequivalent ofone andone €in(7.3.6)isfourmore s,asin(7.3.7).) Forthescal arandvectormultiplets,thesupertraceresultsarealsoconsistentwith thecalculatedvaluesofthecomponentaxialcurrentanomalies(providedweassignthe correctR-weights1 3 Š 1forthefe rmionsofthescalarandvectormultiplet,respectively). However,theconventionallyquotedvalueforthegravitinoaxialanomaly ( mj5 m=21 24 (4 )Š 2r r )doesnotmatchthe energy-momentumtraceforeitherthe

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4887.QUANTUMN=1SUPERGRAVITY Nsmaxtotaltrace( k4) 00 8/360* 1/27/360 152/360 3/2127/360 2232/360 =( Š 1)2 smax+1(15 smax 2Š 2)/90 1 1/21/24 13/24 3/25/24 27/24 =( Š 1)2 smax+1(4 smax+1) /24 2 1/21/12 11/12 3/21/12 21/12 =( Š 1)2 smax+1/12 3all0 Table7. 10.1.Valuesofthetotaltracecoecients(*Complexconformalscalar) (2,3 2 )or(3 2 ,1)multi plet.Thisisaconsequenceofthefactthatthecomponentanomaly wascalculatedf oraclassically conser ved gravitinoaxialcurrent,whereasthecomponent currentcontainedin J aisnotclassicallycons erved:Itcontainsadditionaltermswhich givenonvan ishi ngcontributionsto mj5 m.(Itsenerg y-momentumpartnerisnottraceless:e.g.,thetraceofthe quadratic partoftheEinsteintensor,representingtheenergymomentumtensorofthegraviton“eld,isclassicallynonvanishingevenonshell.Thisis duetotheconformalnoninvari anceofEinsteingravity.) Thevaluesofthe k4coecientscalculatedonthebasisofourconjecturearepresentedinTable7.10.1,whic hgivesthesup ertracein N =0, N =1,andexte ndedsupersymmetry.That k4=0for N 3r e”ectsagaintheabsenceofanetnumberofchiral super“elds. Theveri“cationofourstatementsawaitsanexplicitcalculationoftherelevanttrian gl es upergravitysupergraph,andabetterunderstandingofsomeofthecomponent

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7.10Anomalies489calculations.Ifourconjectureiscorrect,itisrathercurious,andnotunderstood,that thesupergravit ytheorywiththe V compensatorbehavesasifitweresuperconformal ( k1= k4)or,equivalent ly,thatthesuperanomalyandsupertracedieronlyifchiral spinorsarepresent. e.Localsupersy mmetryanomalies Wecanusetheexi stenceofsuperconformalanomaliestoinfertheexistenceof anomaliesintheWardidentitiesoflocalsupersymmetryfor n = Š1 3 .Wedemonstrate thisexplicitlyforthecaseofquantummatte rmulti pletscoupledtobackgroundsupergravity,butexpectsimilarresultswhensupergravityitselfisquantized.We“rstconsider N =1superg ravity. Atthe linearizedlevel,theWardidentitiesr e”ecttheinvarianceoftheeective ac ti on underthe(linearized)localsupersymmetrytransformations( L= L) H€= D L€Š D€L,(7. 10.56) n = Š1 3 : 3= D2DL, n =0: = Š 2 D2L+ i D2DK n = Š1 3 ,0: H= i ( Š1 3 D2L+1 3 D€D L€+1 2 n +1 3 n +1 D D€ L€+ DL).(7. 10.57) Wehave usedth e gauge H=0for n = Š1 3 ;the gauge H= Š i D€H€( 1 H=0, 1 eŠ H=1;sees ec.5.2.b)for n =0,sothat EŠ 1canbelinearizedas 1+1 2 ( D+ D€ €);andthegauge=1forother n .For n = Š1 3 ,0weha vemade theshift H HŠ1 3 i D€H€sothat J€isthesuper conformalcurrent(couplingto co nf ormalsupergravitysaxialvector,andnottheotherauxiliaryaxialvector).Wehave

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4907.QUANTUMN=1SUPERGRAVITY0= R= d4xd4 ( H€) J€+ d4xd2 ( 3) J + h c d4xd2 ( ) + h c d4xd4 ( iH) + h c ., (7.10.58) where J€ R H€ J R3 R R ( iH) .(7. 10.59) Ifwerequirethat(7.10.58)besatis“ed,weobtainthe(linearized)conservationlaws n = Š1 3 : D€J€= DJ D€J =0; n =0: D€J€= Š 2 D€= DŠ D€€=0; othern : D€J€=1 3 D2+1 3 D€D €+1 2 n +1 3 n +1 D D€ €, D( )=0.(7. 10.60) TheinvariancesusedtoderivetheseconservationlawsarethoseofPoincar esupergravity,andtheirviolationwouldimplythatthemultipletcontributingtoRcannotbe coupledconsistentlytothecorrespondingformofsupergravity.Ontheotherhand,the violationofthesuperconformalconservationlaw D€J€=0imp liesonlythatthemultipletcannotbecoupledconsistentlytoconformalsupergravity. Weevaluatema trixelementsoftheconservationequations(7.10.60)betweenthe vac uumandanon-shellsupergravitystate.Inparticular,ifweconsiderone-looptrianglegraphsweknowthepreciseformofthel eft-handside.Asdiscussedintheprevious subsection,powercountingandcovariancedeterminesuniquelythematrixelementof thesuper current:

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7.10Anomalies491< ( H H ) | J€| 0 > i €1 [ D2( W)2Š D2( W€€€)2].(7. 10.61) Thenwehaveforthematrixelementof D€J€< D€J€> DW2.(7. 10.62) Itisnotzero(exceptwhenthesupertracevanishes),andisindependentoftheformof thecompensator. Wenowexami nethematrixelementoftheright handsideof(7. 10.60).Webegin byconsid eringcontributionstotheone-loopeect iveactionfromaclassicallysuperconformalmultiplet.Its(locallysupersymmetr ic,covariantized)actionisindependentof thecompensator.However,asdiscussedint heprevioussubsection,thecompensator entersthe(one-loop)renormalizedeectiveactionafterthedivergenceshavebeensubtractedout.Wecanasknowiftheform(7.10.62)iscompatiblewiththerighthandside of(7.10.60).SincethecompensatorentersRonlybecausewehavesubtractedoutthe covariant, local, counterterm S( H ,compensator),thecorrespondingcurrentmustalsobe local.For n = Š1 3 asolutiono f(7. 10.60)is J W2, but forn = Š1 3 thereexistsno local or thatsatis“estheconservationequation. Weconcl udethatanysuperconformal N =1mult ipletthathasanonzeroo ne-loopsupertracegivesacontributionto RthatviolatesthePoincar esupergravit yconserv ationlawsfor n = Š1 3 i.e.hasa local supersymmetryanomaly. Therefore,ingeneral,superconformalmultipletscanbecoupledconsiste ntlyon lyto n = Š1 3 supergravity.(Theanalysi saboveis inconclusive, however,ifthesupertracevanishes,e.g.fo rasy stemofonevectorandthreescalarmultiplets,whichhasnoone-loop divergenceorsupertrace.) Inthecasewheretheclassicaltheoryisnonsuperconformal,thecompensatorsmay coupletononlocaltermsintheeectiveaction.Thus,for n = Š1 3 ,0, <> D1 D2 W2(7.10.63) cansatisfy(7.10.60)andwecannotconclude,withoutfurtheranalysis,thatPoincar e supergravityanomaliesarepresent.However,wecanstillconcludethatananomalyis presentfor n =0since( 7.10.60)implies D2J€= €J€=0,whereas D€J€ DW2implies D2J€ i €W2and €J€ i ( D2W2Š D2 W2),neit herofwhichvanisheven

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4927.QUANTUMN=1SUPERGRAVITYonshell.Thisoccursbecausethesupertraceisanirreduciblemultipletofsuperspin0 ( W2isachiralscalar),whereasthecompensatormultipletfor n =0hass uperspin1 2 Fornonmini malsupergravity( n = Š1 3 ,0)wew illseebelowthatanomaliesare ab sentonlyunderveryspecialcircumstances.Ingeneral,theirpresenceisrelatedtothe nonexistenceofachiralmeasure.Aninterestingwaytounderstandtheoriginofthe anomalyistousethefactthat(inappropriatesupersymmetricgauges)only(physicalor ghost)chiralsuper“eldscont ributetothedivergencesandreq uireregularization.Inparticular,wecanaskifPauli-Villarsregularizationispossibleforchiralscalarsuper“elds withthevariousnonconformalcouplingsofsec.5.5.Sinceonly n = Š1 3 hasachiral measurethatallowsmasstermsforchiralsuper“eldswith conformalkin eticterms, itis theonly n thatallowsPauli-Villarsregularizationforthosesuper“elds.(Inotherregularizationschemes,thesamedicultywithchiralmeasuresshowsupinotherways:e.g., in dimensionalregularization,“ndinganalogstothechiralintegrandsinthelasttermof (7.10.52).)Infact,wewillshowbelowthattheonlyquantum-consistentcouplingsto su pe rgravityarethosewhich:(1)allowPauli-Villarsregularization,(2)havevanishing supertrace,or(3)havecouplingsthatco rrespondtoextendedsupersymmetry.For n = Š1 3 allchirals uper“eldscanhavemassterms,soallcouplingsarepossible.For other n couplingtothevector mult ipletaloneisimpossible(itisclassicallysuperconformalandhasclassicallysuperconformalchiralghosts),couplingtoascalarmultiplet aloneispossibleonlyforthenonconformalc ouplingthatallowsmass ( butnotself-interaction)terms,andcouplingt othecombinat ionofthetwor equiresacancellationthat o ccursinextendedmultiplets(andprobablynowhereelse,ifthecancellationistobe exactlymaintainedathigherloops). Todisc ussthesituationquantitatively,we performanexp licitveri“cationofthe conservationlaw(7.10.60)forcontributionsfromchiralscalarswithnonsuperconformal couplings.Fromtheactionsofsec.5.5(withthede“nitionsin(7.10.59))we“ndthe classicalcurrents J€= Š1 6 [ D€, D] +1 2 i€ forn =0, Š2 n+1 2 [ D€, D] +1 2 i€ forn =0,

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7.10Anomalies493 J =1 3 D2 n = Š1 3 =3 n+1 2 D2D n =0 =3 n+1 2 D othern (7.10.64) Wenowim aginecom puting reno rmalized one-loopmatrixelementsofthesecurrents betw eenthevacuumandanon-shellbackgroundsupergravitystate.Thematrixelementof J€musthavethefor m(7. 10.61)and,inparticular,its =0compon enthas theform i € Š 1( w2Š w2).Weobservethat i€ givesnocontributiontothiscomponentandt hereforenocontributionatall,sinceanycovariantsuper“eldthatvanishes at =0vanishesid entically.(Thetopvertexofthegraphcontainsonlycrossterms A Bof | =A+ i B,whereasthegravitationalco uplingsareproportionaltoAAand BB.)Therefore,tocomputematrixelementsof any ofthecurrentsin(7.10.63)itissuf“cienttocomputematrixelements < ( H H ) | | 0 > fortwo-particleon-shellgraviton states( H H ),andthe na pplyappropriateoperators(e.g., < J€> < [ D€, D] > =[ D€, D] < > ,etc.). Bypowercountingandcovarianceargume nts,thereno rmalizedmatrixelementhas the uniqueform < > = c 1 ( D2W2+ D2 W2)(7. 10.65) wherecisanumericalfactor.Wenowsubstitutethecorrespondingexpressionsof (7.10.64)intotheconservationlaws(7.3.59).Since <> =1 2 (3 n+1) D2D< > =0(7. 10.66) always,we“ndthattheconservationlawsare never satis“edfor n =0( unless c =0). For n =0s ubsti tuting(7.10.64)into(7.10.60)gives Š1 2 D D2< > =3 n+1 3 n +1 D D2< > ,( 7.10.67a) whichissatis“edonlyfor n= Š1 2 ( n +1).(7. 10.67b) For n = Š1 3 ,thisistheonlyvalueof nde“ned,evenclassically(seesec.5.5.f.2).For

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4947.QUANTUMN=1SUPERGRAVITYother n ,thisval ueisexactlytheonethatallowsamassterm. Toinvest igateanomalycancelingm echanisms,weconsidercontributionsfromone v ectormult ipletand l identicalscalarmultipletswitharbitraryweight n.Theco ntri butionofthevectormultiplettothenonlocalpartof <> and <> mustva nish b ecauseofclassicalsuperconformalinvariance(furthermore,theghostsmusthave ng host= Š1 3 ).Thecontributionto < D€J€> is Š 3timesth atofaphysicalscalarmultiplet .The l scalarmultipletscontributetoboththeleftandrighthandsidesof (7.10.60).Theconservationlawnowbecomes Š1 2 D D2< > ( l Š 3)=3 n+1 3 n +1 D D2< > l ,( 7.10.68a) whichgivesthecondition nl=1 2 (3 l Š 1) n +1 2 (1 l Š 1).(7.10.68b) Inparticular,for l =3,which correspo ndsto the N =4v ectormult iplet,wheredivergencescancel,thesuperconformalcoupling n= Š1 3 isrequired.For l =1,the N =2 v ectormu ltiplet,we“nd nl= n .R ecallthatfor n =0theconserv ationlawsrequirethe supertrace D€J€tovanish identically eventhoughthetheorymaystillhavedivergences (i.e.,thesuperanomalymaybenonzero).Wethushave Š2 n+1 2 [ D€, D] < > l Š1 6 [ D€, D] < > ( Š 3)=0,agreeingwith(7.10.68)for n =0. Wehaveth usfoundthatfor N =1only n = Š1 3 isgenerallyquantumconsistent, whileforother n onlyveryspecial nonsuperconformalcouplingsareallowed.These argumentscanbeappliedtoextendedsup ergravity.Inparticular,thestandard N =2 theory,which(intermsof N =2super“elds)h asanisovectorcompensator Va b,isq uantumconsistent,basicallybecauseithaschiralmeasure.Thusan N =2v ectormu ltipletwillgivecontributionstotheeectiveactionwhichareanomaly-free.Whenanalyzedintermsof N =1super“elds, N =2superg ravitydecomposesintoa(3 2 ,1)multipletcoupledto N =1, n = Š 1supergravity.The N =2v ectormultipletdecomposes intoa N =1v ectormultipletandanonconformalscalarmultiplet,butwith n= n = Š 1whichisco nsistentwiththenoanomalyconditionwederivedabove.Itis

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7.10Anomalies495likelythatt hisextendedsupersymmetryisnecessaryfor n = Š1 3 forthiscan cellationof theanomaliesinthesupergravitationalco nservationlawtooccurathigherloops. f.NottheAdler-Bardeentheorem Insec.6.7weconsideredtheanomalyinthe(axial)Yang-Millscurrentand,on thebasisofthecovariantrules,concludedthatit(anditscomponentaxialcurrent)satis“estheAdler-Bardeentheorem.Ontheoth erhand,thesupertrace(anomaly)ingeneralreceiveshigher-ordercorrections(the -functionisnotzero),andthereforethecomponentR-cu rrentdoesnotsatisfytheAdler-Bardeentheorem.(Weareconsideringhere matrixelementsofthecurrentbetweenthevacuumandon-shellYang-Millsstates, ratherthansupergravitystates.)Although thecurrentslookthesameclassically(fora scalarmultip letinanexternalvectormultipletorsupergravitybackgroundthe A a A termdoes notcontribute),thedierencearisesbecauseofdierentrenormalizationprescriptions. Inthe“rstcase,whentheaxial-vectorgaugesuper“eldisexternal(otherwise,in thepresenceofo ne-loopanomaliesthequantumtheorymakesnosense),itispossibleto renormalizethehigher-loopeectiveaction( V+, VŠ( ext ))andd e“ne Jrenormsothatit isnota nomalous.Ontheotherhand,if VŠ( ext )isrepla cedwith H€( ext ),thehigherloope ectiveaction( V+, H€( ext ))isusuallyrenormalized inama nnerwhichisconsistentwithPoincar esupergravity gaugeinvariance.Inthatcase,wedonothavethe freedomtorede“ne J€ renormsoastoremoveitshigher-loopsupertrace(anomaly).Ifwe giveupsuper-Poincar einvarian ce,wecanrenormalizesothat €J€ renorm=0athi gher loops.However, thisJ€ renormwillnotcontainaconserved (symmetric)energy-momentumtensoratthe level.Therefor e,therenormalizedchiralR-currentwhichisinthe samemultipletwiththerenormalizedconservedenergy-momentumtensordoesnotsatisfytheAdler-B ardeentheorem.

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Contents of8.BREAKDOWN 8.1.Introduction496 8.2.Explicitbreakingofglobalsupersymmetry500 8.3.Spontaneousbreakingofglobalsupersymmetry506 a.Renormalizabletheories506 a.1.Classicaleects506 a.2.Loopcorrections509 b.Nonrenormalizabletheories511 c.Globalgaugesystems513 8.4.Traceformulaefromsuperspace518 a.Explicitbreaking518 b.Spontaneou sbreak ing520 8.5.Nonlinearrealizations522 8.6.SuperHiggsmechanism527 8.7.Supergravityandsymmetrybreaking529 a.Massmatrices532 a.1.Vacuumconditions532 a.2.Gravitinomass533 a.3.Waveequations534 a.4.Bosemasses535 a.5.Fermimasses536 a.6.Supertrace538 b.Super“eldcomputationofthesupertrace539 c.Examples540

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8.BREAKDOWN 8.1.Introduction Themoststrikingfeatureoftherelationbetweensupersymmetryandthe observedworldistheabsenceofanyexperi mentalevidencefortheformerinthelatter. Theparticlesweseedonotfallintosupersymmetricmultiplets,nordotheyshoweven anapproximatemassequalitythatwouldindicatetheywereinmultipletsbeforesymmetr yb reaking.Thusifsupersymmetryisanunderlyingsymmetryofthephysicalworld, itmustbeba dlybroken,orotherwisehiddenfromdirectexperimentalveri“cation. At af undamentallevel,itisdiculttoaccepttheideaofaglobalsupersymmetry withoutbelievingthatthereexistsanunderlyinglocalsupersymmetry:Sincewebelieve thatgravitymustbequanti zed,andsinceevenglobalsupersymmetryimpliesthatthe gravitonrequiresaspin3 2 gravitinopartner,thenthegravitinomustbethegaugeparticleoflocalsupersymmetry,howeverbadlybrokenglobalsupersymmetrymaybe.Then, asinanygaugetheory,thesupersymmetrybreakingmustbespontaneous(i.e.,bythe vac uum)andnotexplicit( i.e.,intheactionitself).Ifwebelieveinlocalsupersymmetry withsymmetrybreaking,wemustunderstandmechanismsforthisbreaking.Itcanbe th roughtheHiggsmechanism,orduetocosmologicalfactorssuchasboundaryconditionsorhightemperaturee ectsintheearlyuniverse,ornonperturbativedynamical eects,orviadimensionalcompacti“cation.Itisalsoreasonabletobelievethatthe breakinghappensatalargeenergyscale.Ifthisisso,wemayhopethatthedynamical eectsofthesupergravity“eldscanbeignoredatalowerenergyscale,andthatthe eectivelowenergytheoryisabrokengloballysupersymmetrictheory.Wecanstart withanexactgloballysupersymmetrictheory,atsomescalewheresupergravity“elds havedecoupled,andinvestigateitsspontaneousbreakingabinitio,orwecanputthe breakinginbyhand,asanexplicitmanifestat ionoftheoriginallocalbreaking.(Ingeneral,ifwestartwithalocallysupersymmetr ictheorythatexhibitssymmetrybreaking andsetgravitational“eldsandcouplingstozero,softbreakingtermsareinduced). Unlikeothersymmetries,therearesomein terestingand unexpectedrestrictionson thepossiblebreakingofglobalsupersymmetry.Someofthesehavetheirorigininthe supersymmetryalgebraitself ,wh ileothersaremosteasilyobtainableinthecontextof super“eldperturbationtheory.The“rstres triction,whichfollowsfromthealgebra,is

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8.1.Introduction497thefollowingtheorem: Ifsupersymmetryisnotspontaneouslybroken,i.e.,ifthevacuumisinvariant undersupersymmetrytransformations,thenitsenergyiszero;conversely,ifthere existsastateforwhichtheexpectationvalueoftheHamiltonianiszero,supersymmetryisnotspontaneouslybroken.Furthermore,ifsupersymmetryisspontaneouslybrokenwithoutanattendantmodi“cationofthesupersymmetryalgebra, thenthevac uumenergyispositive. Asdiscussedinsec.3.2,thisresultfollowsdirectlyfromthecommutationrelations { Q Q } = P ,whichgivein particular Evac= Š 1 2 N €< 0 |{ Qa Qa€}| 0 > = 1 2 N a < 0 || Qa |2| 0 > (8.1.1) Thus: Ifallthecomponentsofthesupersymmetrycharge(generators)annihilatethevacuum,itsenergyiszero.Ifanyoneofthemdoesnotannihilatethevacuum,then itsenergyispositive. Weem phasizethat thistheoremassumest hatthesupersymmetryalgebraisnot chan ged.Withanappropriateinterpretationofthetotalenergyandcharge,thetheoremalsoholdsinsupergravity.Ontheotherhand,explicitbreakingdoeschangethe algebraandthennegativeo r zeroenergyispossible. Thesecondimportantresult,provedforafairlylargeclassofrenormalizablemodels,isthatinspontaneouslybrokenglobaltheories,therearemasssumrulesrelating fermion andbosonmasses,whichtaketheformstates mB 2Šstates mF 2=0.(8. 1.2) Thesesumrulesareextremelyrestrictive,andmaketheconstructionofrealisticmodels dicult;however,forlocallysupersymmetr icandexplicitly(softly)brokenglobally supersymmetrictheories,t hegeneralizationsofthisformulaarephenomenologically acceptable. Athird resultisthefollowingtheorem,whichcanbeproveninperturbationtheory forfour-dimensionaltheories:

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4988.BREAKDOWNIfsupersymmetryisnotspontaneouslybro kenatthetreelevel, thenitisnotbrokenbyradiativeco rrections.AColeman-Weinbergmechanismisnotpossible. Thistheoremisnotvalidintwo-dimensionalsupersymmetry. Ifsupersymmetryisspontaneouslybrok en,amasslessGoldstonefermionmustbe present.Therefore,ifonecanprovethatnomasslessfermionstatescanexist,supersymmetrycannotbebrokenspontaneously.Usingthisfact,Wittenhasgivencertaincriteria (indextheorems)thatallowonetoruleoutinasimplemanner,incertaincases,the po ssi b ilityofspontaneoussupersymmetrybreaking.Inalocallysupersymmetrictheory, theGoldstonefermionisabsorbedbythespin3 2 gravitinoviaaconventionalHiggs mechanism.Indextheoremshavenotbeeninvestigatedinsupergravity. Globalsupersymmetrybreakingismosteas ilydiscu ssedinsuper“eldlanguageasa breakingof Q -translationalinvarianceinsupers pace.Thiscanhappeneitherbecause thevac uumisnot Q -translationallyinvariant(spontaneousbreaking),orbecauseone hasexp licit -dependenceintheeectiveaction(eitheratthetreelevelornonperturbatively,forexampleviainstantoneects,whichcouldintroducesuchexplicitdependence).Ifthebreakingisspontaneous,it meansingeneralthatsomesuper“eldhasa nonzerovacuumexpectationvalue(ifLorentzandinternalsymmetryinvariancearenot tobebroken,ithastobeaneutralscalarsu per“eld).F urthermore,thenonzeroexpectationvaluemustresideinotherthanthe -independentcomponentofthe“eld,sothat someex p licit -dependenceisintroduced.For N =1ma ttersuper “elds, itmeansthat oneoftheauxiliary“eldsmusthaveanonzerovacuumexpectationvalue.(Unlessgauge invarianceisbroken,vacuumexpectationvaluesforthegaugecomponentscannotbe physica llyrelevant.) Supersymmetrybreakinginalocalconte xtandthesuperHiggsmechanismcan alsobedescribeddirectlyinsuperspace.Allthestandardmethods,suchasthetheory ofnonlinearrealizations,canbeappliedand allthestandardresults,suchastheconversionoftheGoldstinointohelicitymodesofamassivegravitinoandtheexistenceofUgauge,canbegeneralizedtothesuper“elddiscussionofspontaneouslybrokensupersymmetry;theresultingformalismisconsiderablysimplerthanacomponentapproach. However,some i ssues(atthepresenttime)canbesettledonlybyconsideringcomponentsdirect ly,e.g.,whatarecomponent“eldmasses,whataretheconditionsforspontaneousbre akingtooccur,whatistheWittenindex,etc.Therefore,althoughmostofthe

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8.1.Introduction499materialinthischapterisatthesuper“eldl evel,wecannotavoidsomecomponentcalculations,andwealsoomitsometopicsthathavenot,asyet,receivedanadequate superspacetreatment. We“rst discusssoftexplicitbreakingofglob alsupersymmetry(sec.8.2).OurcriterionforsoftnessistheanalogofSymanzikscriterioninordinary“eldtheory:In renormalizablegloballysupersymmetrictheo ries,theonlyrelevantdivergencesarelogarithmic.Weaskwhatnonsupersymmetric termscanbeadde dtothecl assicalaction withoutspoilingthedelicatecancellationsbetweenbosonandfermioncontributionsthat areresponsibleforthe absenceofq uadraticdivergences.Sincewecancasttheproblem insuper“eldlanguage,weareabletotakeadvantageofthesuper“eldpowercounting rulesofchapter6. Wenexttreatspontan eousbreakingofglobalsupersymmetryforbothrenormalizableandnonrenormalizabletheories(sec.8.3).(Nonrenormalizabletheoriesarerelevant toourdiscussionofbreakinginthecontextoflocalsupersymmetry.)Wedonotdiscuss Wittensindextheorem,orbreakingofsupersymmetrybyinstantons;asnotedabove, withourpresenttechniquestheseissuescanbehandledonlyatthecomponentlevel. Wedo,howev er,giveasuperspacederivationofthesupertracemassformulae(sec.8.4). Thisderivationismuchsimplerthanthecomponentcalculation(whichwealsogive, partlyforcomparison,butalsobecauseitprovidessomeextrainformation,e.g.,the massesoftheindividualcomponents). Finally,wediscussthesuperHiggseect.WeshowhowtheGoldstinocanbe describedbyanonlinear(super“eld)realizationofsupersymmetry,andhowstandard radialandanglevari ablescanbeintroducedinmodelswithspontaneouslybroken supersymmetry(sec.8.5).W eexhib itthesuperHiggsmechanism(sec.8.6)andgivea deta ileddiscussionofthecaseofarbitrarysupe rsymmetricmattersystemscoupledto supergravity(sec.8.7).

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5008.BREAKDOWN8.2.Explicitbreakingofglobalsupersymmetry Oneoftheimportantfeaturesofsupersy mmetrictheoriesistheperturbativenorenormalizationtheorem(sec.6.3.c):ThesuperspacepotentialP()forchiralsuper“eldsreceivesnoradiativecorrections,sotha ts calarmultipletmassesandcouplingconstantsarenotrenormalized(asidefromtheeectofwavefunctionrenormalizations). Fu rthermore,forrenormalizablemodels,onlyl ogarithmicdivergencesarepresent(asdiscussedinsec.6.5,quadraticallydivergentD-termsarenotgeneratedifgaugeinvariant regularizationisused).Whensupersymmetryisexplicitlybrokenthisisnolongerthe case,and,ingeneral,quadrati callydivergentcorrectionscanbeinduced.Equivalently, theparametersofane ectivelowenergytheorycandependquadraticallyonmasses associatedwiththetheoryde “nedathighenergies,andsomeofthenaturalnessof supersymmetrictheoriesisdestroyed.Ho wever,thereexistsa setofsupersymmetry breakingtermswhoseeectis soft :Whena ddedtoasupersymmetr icLagrangian,any ne wd ivergencesthatthesetermsgeneratearel ogarithmic.Moreprecisely,ifweintroducecountertermsintheclassicalLagrangiantocancelthenewdivergences,afterrenormalizationtheirdependenceontherenormalizationmass(orhighenergycuto)isonly logarithmic.Inthissection,wedescribethesetofsoftbreakingterms,andtheadditionaltermsthattheyinduce. Breakingsupersymmetryisbreaking Q -translationalinva riance.Thisisdoneby introducingexplicit -dependenceintotheLagrangian.Equivalently,wecanintroducea super“eld( x )witha“xed -dependentvalue.Thissuggeststhefollowingprocedure: Givenasupersymmetricaction,wegeneratenewtermsbycoupling,inamanifestly supersymmetricfashion,someexternal(spurion)super“eld(s)tothequantum“elds. Supersymmetrybreakingisachievedbygivingthese“eldssuitable( -dependent)“xed values.Atthe componentlevel,thisintroducessomenonsupersymmetricterms.Soft breakingisachievedbyonlyallowingnewco up lingsthatareconsistentwiththe(power co un ting)renormalizabilitycriteriaofsuper“eldperturbationtheory,sothatnodivergencesworsethanlogarithmicareintroduced .T heinducedin“nities correspondtoconventionald ivergenttermsintheeectiveactioni nvolving productsofthequantumand spurion“elds.Whenthespurion“eldsaregiventheir“xedvalueswecandeterminethe correspondingnewcomponentin“nities.Generally,wewill“ndthatinacomponent languagesymmetrybreakingtermsofdime nsiontwoaresoft,buttermsofdimension

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8.2.Explicitbreakingofglobalsupersymmetry501threearenot(withsomeexceptions).Thesetermscorrespondtosplittingthemassesof particlesinmultipletsbyhand,oraddingsomenew,nonsupersymmetricinteractionsof averysp ecialform.Wewill“ndthattherearee ssentia lly“vedistincttypesofsoft breakingtermsthatcanoccursingly,orincombinations.Ingeneral,onesuchterm inducestheothers,sothatwe shoulddiscussthemallatthesametime.However,since theirphysi calsigni“canceisdierent,wep refertotreatthemoneatatime. Weconsid erconventionalrenormalizableLagrangians(cf.sec.4.3)oftheform S = d4xd4 [ ieVi+ trV ]+ d4xd2 [1 2 WW+P(i)]+ h c .(8. 2.1) wherePisapolynomialofdegreethreeorless.Bypowercountingweknowthatthe onlydivergencesofthetheorycorrespondtotermsintheeectiveactionoftheform d4xd4 d4xd4 Vn d4xd4 V ( D )2( D )2Vn d4xd4 V (8.2.2) wherethe D -derivat ivesaresui tablydistributedandtermswith n > 1are relatedto termswith n =1by gaugeinvariance(weincludeghostsamongthechiral“eldsin (8.2.2)).Webreaksupersymmetrysoftlybycouplingadditionalexternalsuper“eldsina mannerconsistentwiththepowercountingcriteria(seesec.6.3):Nomorethanfour D sactingontheinternallinesshouldappearatanyvertexwheretheexternalspurion “eldisinserted.Inadditiontotheoriginald ivergen cesofthetheory,wemaygenerate newones,involvingthespurion“eldsaswell,andtheyaretheonesthatinterestus.In thissectionwedonotconsiderdivergencesinvolvingspurion“eldsonly,whichcorrespondtoinsertionsintovacuumdiagramsandcontributeonlytothevacuumenergy (cosmologicalconstant);see,however,sec.8.4. Sincethespurion“eldscanneverintroduceanyadditionalspinorderivativesintoa loop,ifinanysoftbr eakingtermthespurion“eldissetto1theresultingtermmustbe eitheraconventionalrenorma lizablesupersymmetrictermor atot al(spi nor)derivative. Thepossibleadditionalcouplingsthatintroduceexplicit -dependenceintotheaction correspondtomultiplyingaspurionfactorinto ,2, W2,3,or D( W).Indetail, wehave:

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5028.BREAKDOWN(a) Sbreak= d4xd4 U d4x 2A A (8.2.3) where U = 22 2isaneutraldime nsion-zero“xedgeneralscalarsuper“eld.Atthe classicallevel,whenaddedto(8.2.1),suchatermbreakstheequalityofthebosonand fermionmassesofascal armultipletbyadding Š 2tothemassesofA=2Š1 2 ReA and B=2Š1 2 ImA .Toinvesti gatethedivergencesitintroduces ,weconsiderloo pswithordinaryverticesandexternal U verti ces.Welookforlocaltermsintheeectiveaction, involvinga d4 integralandfactorsof U andthequantum“elds,ofdimensionnogreater than2.(Thisisourstandardpowercountingofsec.6.3.)Since U isdime nsionless, suchtermsare: U ,correspondingtoalogarithmicrenormalizationof(8.2.3),i.e.,of 2; U (+ )(butonlyifsomechiral“eldismassive);and UD D2DV (butonlyifthe gaugegrouphasa U (1)factor;the D -factorsarerequiredbygaugeinvariance).Therefore,theactionmayreceiveadditionall ogarithmicallydivergentcorrections: d4xd4 U + d4xd4 UDW+ h c d4x [ 2m A+ 2D](8. 2.4) (b) Sbreak= d4xd2 2+ h c d4x 2(A2Š B2)(8. 2.5) where = 22isaneutraldi mension-onechiralsuper“eld.Thisadditioncorresponds toanotherwayofsplittingthemassesofscalarsandpseudoscalarsawayfromthemass ofaspinorinachiralmultiplet.New,logar ithmicallydivergenttermsaregivenby d4xd4 + h c d4x F(8. 2.6) whereF= ReF .Since is neutralunderwhateverinternalsymmetrygroupsmaybe present,noin“nitiesinvolvinggauge“elds(asmightarisefroma eV term)canbe i nduced.

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8.2.Explicitbreakingofglobalsupersymmetry503(c) Sbreak=1 2 d4xd2 WW+ h c 1 2 d4x + h c .(8. 2.7) where = 2isaneutral,dimensionzerochiralsuper“eld.Againthisinvolvesvertices withonlyfour D sandisthereforesoft ,and providesamechanismforgivingmassesto fermionsin gaugemultiplets.Thefollowingdivergenttermsmaybegeneratedinadditiontocorrectionsto Sbreakitself: d4xd4 + h c d4x F d4xd4 ( + h c .) d4x A d4xd4 + h c d4x [FA Š GB] d4xd4 d4x [A2+B2](8. 2.8) Foragiventhe ory,notallofthesetermsneedappear;forexample,thethirdtermwill onlybegeneratedatthetwolooplevel,andonlyifamassivechiralsuper“eldispresent. (d) Sbreak= d4xd2 3+ h c d4x Re ( A3Š 3 AB2)(8. 2.9) with asin(c).Unlikethepreviouscases,thisintroducesanallowednonsupersymmetricinteractionterm.Ingeneralweinducethesamedivergencesasincase(c). Thebreakingtermd4 ( + ) canber educedbya“eldrede“nition (1+ )to thepreviouscases. Anotherpossibility,whichgivesagaugeinvariantmassmixingbetweenthe fermionsofa gaugemultipletandofascalarmultipletintheadjointrepresentation,is (e) Sbreak= d4xd4 DU W+ h c .

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5048.BREAKDOWN= d4xd4 D2DU DV + h c d4x Re [ + A D] (8.2.10) withadimension Š 1“eld U = 2 2,or,equivale ntly,w ithadime nsion1 2 chiral spinor super“eld = D2DU .L ogarithmiccorrectionsareinducedfor Sbreakitselfand for: d4xd4 + h c d4x 2F.(8.2 .11) Theabovepossibilitiesforsoftbreakingare”exibleenoughtocoverallinterestingphysicalsituationswithoutintroducingalargenumberofarbitraryparameters.(Withseveralmultiplets,becausecancellationsarepossible,othertypesoftermscanbesoft,e.g.,d4xd4 DU 1D2d4x ( F1A2Š F2A1).) Itisalsointerestingtoexaminesomeca sesofbreakingthatarenotsoft.Wementiontwo: (a) Sbreak= d4xd4 U ( D)( D)+ h c .(8. 2.12) with U asin(e),shiftsthemassofthespinorinascalarmultiplet.Butitleadstoverticeswithsix D s,asdoes (b) Sbreak= d4xd4 U (+ )3 d4x A3.(8. 2.13) Bothwillpro ducequadraticallydivergentterms,forexample d4xd4 U + h c d4x A(8. 2.14) We canunderstandthedierencebetweencases(a)and(b)ononehand,and(a) ontheotherasfollows:Theybothleadtofermion-bosonmasssplittingsforthescalar mult iplet.However,theformer,inadditiontosplittingmasses,alsoaectssomeofthe componentinteractionterms,anditis thedelicatebalanceofmasstermsand

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8.2.Explicitbreakingofglobalsupersymmetry505in te ractionsthatkeepsthedivergencesundercontrol.Ontheotherhand,thereisno dicultygivingmasstothefermionofavect ormultiplet,orintroducingmassmixing betw eenthefermionsofthetwomultiplets. Itisausefulandsimpleexercisetochecksomeoftheaboveconclusionsbyexaminingsupergraphsinvolvingthespurion“ elds.Wenotethatsomeoftheinducedterms wehave listedmaybemissing b ecauseofgrouptheoryrestri ctions,or,insomecases, b ecauseoftheabsenceofmasses,e.g.,thethirdtermincase(c).Incertaincasespossibletermsaremissingbecausethecorrespondinggraphsrequireor p ropagators andthese bringwiththemnumeratormassfactorsthatreducethedegreeofdivergence ofthediagrams.Forexample,incases(c)and(d)aterm 2cannotbeproduced b ecausethecorrespondingdiagramsmustcontaintwomassfactorsandhenceareconvergent.

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5068.BREAKDOWN8.3.Spontaneousbreakingofglobalsupersymmetry Ifglobalsupersymmetryisspontaneouslybroken,amassless(Goldstone)fermion mustbepres ent.ThiscanbeestablishedbytheusualreasoningthatprovestheGoldstonetheorem:Ifthesupersymmetrychargedoesnotannihilatethevacuum,thereexist operatorswhose(anti)commutatorwiththesupersymmetrychargehasnonzerovacuum expectationvalue,and,inparticular,wecanwrite < 0 |{ Q, S a }| 0 > = d4x x b < 0 | T ( S b ( x ) S a (0)) | 0 > (8.3.1) where S a isthesupersymmetrycurrent,satisfying aS a =0,and Q= d3xS 0 .The leftha ndsidenotbeingzero(itisactuallyproportionaltothevacuumenergydensity (8 .1 .1 )) im pliesthattherighthandsidereceivesacontributionfromasurfaceterm;this is th ec as eo nlyifthematrixelementvanishesatin“nitynotfasterthan | x |Š 3,whichis po ssibleonlyifamasslessfermion intermediatestateispresent. Thespontaneousbreakingofsupersymme tryingloballysupersymmetrictheories canbeinvestigatedbyexaminingtheeectivepotentialatitsminimum,whereitequals thevac uumenergy.Theeectivepotential U isobtainedfromminu sthee ectiveaction byse ttingallmomentaandallcomponent“eldsthatarenotscalarstozero.Wemust thenminimize U withrespecttoalltheremainingcomponent“eldsandaskifitvanishesa tthemi nimum. a.Renormalizabletheories a.1.Classicaleects We“rstco nsiderasystemwithonlychiralscalarsuper“elds,andarenormalizableclassicalactiongivenby(4.1.11): S = d4xd4 ii+ d4xd2 P(i)+ h c .(8. 3.2) wherePisapolynomialofdegreenohigherthanthree.Toinvestigatetheclassicalvacuumwesetallmomentaandfermion“eldstozero;wethenhaveeectively = A Š 2F ,andsi nceeach integrationrequiresa 2 2factor,weobtaintheclassical potential

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8.3.Spontaneousbreakingofglobalsupersymmetry507U = Š FiFiŠ [ FiPi( A )+ h c .](8.3 .3) wherePi= P Ai asin(4.1.13).Theclassicalvac uumisdescribedbytheconstant( x i ndependent)classical(expectation)valuesofthescalar“eldsobtainedbysolvingthe classical“eldequationsforconstant“elds,i.e.,byextremizingtheclassicalpotential. Extremizingwithrespecttothe Fi“rst,we“nd Fi= Š Pi( A );substitutinginto U we obtain U =i | Pi( A ) |2.(8. 3.4) Werequire U Ai = Pij( A ) Pi( A )=0.(8 .3.5) Thepotentialwillvanishattheextremuma ndsupersymmetrywillnotbebrokenonlyif thesimultaneou sequationsPi( A )=0haveasolution.Thisreq uirementisequivalentto thatofrequiringthatallthe F shave zerovacuumexpectationvalue.Wecanwork directlyinsuperspace,byde“ningP()asthe supers pacepotential. Theconditionfor supersymmetrynottobebrokenisformallythatthesuperspacepotentialhavean extremumwithrespecttothesuper“elds: P i =0. Weconsid ertwoexamples: (a)TheWe ss-Zuminomodel,withaction d4xd4 + d4xd2 [ a +1 2 m 2+1 6 3]+ h c .(8. 3.6) Thesuperspacepotential,whendierentiatedwithrespecttogives a + m +1 2 2andsettingthistozeroalwaysgivesusasol ution.Hencethevacuumenergyiszeroand thereisnosupersymmetrybreaking.Thisis thecaseevenifweconsideranarbitrary (nonrenormalizable)polynomialpotential. (b)Ontheotherhandwecanconsider theORaiferteaighmodel,givenby d4xd4 [ 00+ 11+ 22]+ d4xd2 [01 2+ m 12+ 0]+ h c .(8. 3.7) forwhichweobta intheequations

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5088.BREAKDOWN1 2+ =0 201+ m 2=0 m 1=0,(8. 3.8) whichhavenosolution.Inthiscasesupersymmetryisbrokenattheclassicallevel. Inthegeneralcase,thesituationdependsonthetopologicalstructureofP.In particularthepresenceofanextremum,andi tsstabilityundervariationsofparameters inP,canbestudiedinrigorousfashionandgivesrisetoindextheoremstodetermine whethersupersymmetrycanorcannotbespontaneouslybroken.Weremarkthatsupersymmetrybreakinginthesenseaboveimp liesthatthefermionmassmatrix,whichis givenbythematrixof secondderivativesPijevaluatedattheminimumof U (see (4.1.12)),hasazeroeigenvaluecorrespondingtothezeromassoftheGoldstonefermion. Indeed,if(8.3.5)issatis“edwithPj =0,Pijmustbe singular. Includinggaugeinvariantinteractionswitharealgaugesuper“elddoesnotfundamentallychangethediscussion.Gaugeinvarianttermsthatcanbeaddedto(8.3.2) havethegeneralformd4 [ eV+ V ] (thelasttermonlyif V isa U (1)gauge“eld), andtheonlycomponentof V thatcanhaveanonzeroexpectationvalueistheauxiliary “eldD;thislea dstoadditional termsintheclassicalpotentialoftheform Š1 2 (D)2Š DŠ AA D.Extre mizingwithrespecttoDandtheneliminatingitgives anadditionalcontribution1 2 | + AA |2totheclassicalpotenti al,whichmustbeseparatelyzeroforsupersymmetrynottobebroken.Theexpressionfortheclassicalpotentialcanbereadfrom(4.3.7).Wenotethatif =0,ifitcanbe arrangedfor + AA to equalzero,thensome(charged)scalar“eldmustacquirea(nonvanishing)vacuum expectationvalue,andthegaugegroupwillbespontaneouslybroken.Thus,tohave both gaugeinvarianceandsupersymmetry,itisnecessarythat =0. We observethatifsomeexpectationvalueofanauxiliary“eldisnonzero,i.e., f = < F > or d = < D> ,thesupers ymmetrytransformation ofthespi nor“eldofthe mult ipletbecomes(see(3.6.5,6)) = f + ... or

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8.3.Spontaneousbreakingofglobalsupersymmetry509= id + ... ,(8. 3.9) whichistypicalbehaviorforaspontaneouslybrokensymmetry.Thespinor“eld describestheGoldstino,and f or d sets thescaleofsupers ymmetrybreaking. a.2.Lo opcorrections Wenowestab lishthefollowingresult:Infourdimensions,ifsupersymmetryisnot spontaneouslybrokenattheclassicallevel,itisnotbrokenbyradiativecorrections. Thistheoremcanbeprovenmostreadilybyusingresultsofsuper“eldperturbationtheory,anditmightbeviolatedbynonperturbativeeects,althoughnoexampleisknown infour dimensions.We“rstconsiderthesituationwithonlychiralsuper“elds. Abasicfe atureofperturbationtheoryisthatt heeectiveactionisobtainedwith a d4 integral.Ifweconsiderclassicalconstant“eldsoftheform= A Š 2F Dactingonthemissimply ,sothatt hederivativesdonotintroduceany factors;consequently,inthe d4 integration,wemustget and factorsfromthe s,andtheseare accompaniedbyan F andan F factor.Therefore,addingth ecla ssicalpoten tialtothe quantumcorrections,wehaveatotalpotentialoftheform Ueff= Ši [ FiFi+ FiPi( A )+ Fi Pi( A )]+ij FiFjGi j( A A F F )(8. 3.10) Dierentiatingwithrespectto Aiand Fiweobtain Š U Ai = FjPij+ FjFk Ai Gk j(8.3.11a) Š U Fi = Fi+ Pi+ FjGj i+ FjFk Fi Gk j(8.3.11b) Now,ifattheclassi callevelthereexistvaluesofthe A ssuchthatPi=0,sothat Fi=0sati sfytheextremumequationsandmaketheclassical U vanish,iti sclear from theaboveformthatthisresultisnotcha ngedbythequantumcorrectionssincethe additionaltermsalsovanishesfor Fi=0.Theimport antingre dientisthatthe quantum correctionsarebilinear intheauxiliary“elds. Thereforetheclassicalminimumoftheclassicalpotentialisstillanextremumof thequantumcorrectedpotential,anditisstillsuchthatthetotalpotentialvanishes

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5108.BREAKDOWNthere.Furthermore,ifthesupersymmetrya lgebrastillholds,itmustbeanabsolute minimum(nonegativeenergy)andthereforesupersymmetrycannotbebroken.Conversely,ifsupe rsymmetryisbrokenattheclassicallevel(somePi =0forany A s),the aboveequationhasno Fi=0solution s,andhenceradiativecorrectionscannotrestore thesymmetry. Were markthatinlowerdimensionsthesituationisslightlydierent.Therethe superspaceintegrationsare d2 ,wh ilesuper“eldsstillhavetheform A Š 2F .Therefore,thequantumcorrectionstotheeectivepotentialcanhavetermsoftheform FG ( A F )witha single F .Whentakingd erivativeswithrespectto F ,thefactorin frontc andisappear,andwe“ndthattheclassicalextremumnolongerneedbean extremumofthequantumpotential. Inthepresenceofgaugesuper“eldswecanhaveadditionalcontributionstothe eectivepotential.TermsproportionaltoD 2orDF arequadraticinauxiliary“elds anddonotchangeourconclusions:If F =D=0ares olutionsoftheclassicalequations,theywillalsobesolutionsofthequantumcorrectedequations.However,itispossibletogeneratetermsoftheformDf ( A A ),andsuchterms,nolongerquadraticin theauxiliary“elds,couldchangeourconclusions(recallthatapureDtermisnotgenerated).Nevertheless,aslongasgaugeinvarianceandsupersymmetryareunbrokenat thetreelevel(whichimpliest hatthetheorydoesnothaveaFayet-Iliopoulosterm),even aterm linearinDisharmless.Thisisbecausesuchatermarisesonlyfromthecovariantizationoftermsintheeectiveactionoftheformd4xd4 g ( ,) d4xd4 g ( eV,) d4x D A g ( A A ) A ;t hu st hi st er mi sa tl ea st b ilinearintheD, A “elds,andhencewecanusethesameargumen tsasabovetoconcludethattheclassical solutionD= A =0(whichmustbe thecaseifgaugeinvarianceandsupersymmetryare unbrokenclassically)isstillasolutionatthequantumlevel.(Alinear A termwould spoilthisargument,butsuchatermcannotbewrittenasa d4 integral.) Ifclassicalgaugeinvarianceisbroken,andaDf ( A A )isgen erated,ithasbeen shownthatforaspeci“cclassofmodelsasupersymmetricsolution( F =D=0)ofthe quantumcorrect edequationsexistswiththe A sshiftedfromtheircl assicalvalues;thus, eveninthiscase,supersymmetryisnotbrokenbyradiativecorrections.However,the generalsituationisinneedoffurtherclari “cation.Alloftheseresultsholdforthe

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8.3.Spontaneousbreakingofglobalsupersymmetry511nonrenormalizab lesy stemsthatwediscussbelow. b.Nonrenormalizabletheories Wenowconsi dermoregeneralsitu ations.Forglobalmodelscoupledtosupergravity(seesec.5.5.h)therenormalizabilitycriterionistoorestrictive:Thecombined systemsarenotpower-countingrenormalizable,andsincewecanmake“elddependent Weylresca lings,thereisnoreasontoinsistonpolynomialityofthematteractions. However,no nderivativesuper“elddependentrescalingsdonotchangethenumberof derivativesintheactionandthereforewere strictourselvestoactionsthatleadtocomponentL agrangianswithnomorethantwospacetimederivativesinthepurelybosonic termsoftheaction,andnomorethanonespa cetimederivativeinthetermscontaining fermions;thisispreservedbysuper“elddependentrescalingsthatdonotinvolvespinor derivatives.Inthissubsectio nwediscuss interactingchiralscalarsuper“elds;weextend thediscussiontogaugesystemsinthefollowi ngsubsection.Thereadershouldreview ourdiscussionofK¨ ahlermanifoldsinsec.4.1.b. Weconsid erasystemof N chiralsuper“eldsidescribedbythesuperspaceaction S = d4xd4 IK (i, j)+ d4xd2 P(i)+ h c ., =( x , ), D€=0, =(), D =0.(8 .3.12) Asdiscussedinsec.4.1.b,the“rsttermoftheaction S canbegivenageometricalinterpretation:i, jcanbethoughtofascoordinatesofacomplexmanifoldwithK¨ ahler potential IK Wer ecallthatthe(complex)component“eldsofiarede“nedbyprojection Ai=i| i= Di| Fi= D2i| .(8. 3.13) Wede notevacuumexpectationvaluesofthecomponent“eldsby ai= < Ai> fi= < Fi> =0.(8. 3.14) Thevacuumexpectationvaluesareobtainedbysolvingtheclassical“eldequationsfor x -independent“elds.

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5128.BREAKDOWNTheaction S leadstothesuper“eldequations D2IKi+Pi=0(8.3 .15) andtheirhermitianconjugates(weusethenotationof(4.1.13,25a)).Takingvacuum expectationvaluesandevaluatingat =0usingt hede“nitions(8.3. 13,14),weobtainin particular IKi j( a ) fj+Pi( a )=0.(8 .3.16) Asintherenormalizablecase(sec.8.3.a),spo ntaneoussupe rsymmetrybreakingoccursif fi=0is not asolutiontothese equations.Furthercomponentequationsareobtained bydierentiati ng(8.3 .15)with D2andevaluatingat =0.W e “nd [ IKij k( a ) fk+Pij( a )] fj=0.(8. 3.17) Af te r “ndingthevacuumsolution(s),wecanchoosetoworkinnormalgauge(4.1.27)at thevacuumpoint.Inthatcasethevacuumequations(8.3.16,17)reduceto fi+Pi=0,Pijfj=0.(8. 3.18) IfPij( a )isnonsing ular all fj=0andsupersy mmetryisnotbroken.Conversely,if fj=0isnotaso lutionof(8.3.16,17)thensupersymmetryisbrokenandPij( a )mustbe singular. Returningtotheac tion(8.3.12),weshiftthe“eldsi i+ < i> andinvestigate”uctuationsaboutthevacuumstate. Inparticularwecanreadothemassesof thevariousparticlesfromtheresultingaction;alternatively,wecan“ndthemassmatricesofthecomponent“eldsbyexpandingthesuper“eldequations(8.3.15)tolinearized orderint he”uctuations: D2[ < IKi j> j+ < IKij> j]+ < Pij> j=0.(8. 3.19) A pplying Dand D2andevaluatingat =0asin(4 .1.21),we“nd Fi+PijAj=0 i € i€+Pijj =0 Ai+ IKik jl flfk Aj+PijkfkAj+PijFj=0(8.3 .20)

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8.3.Spontaneousbreakingofglobalsupersymmetry513wherew ehavedroppedthe <> onP...and IK... ...andassumedthatweareinnormal gauge.Eliminatingtheauxiliary“eldsweidentifythefermionandbosonmassmatrices: MF=Pij( a ) MB 2= Š IKik jl flfkŠ Pik Pkj Pijk fkPijkfkIKjl ik fkflŠ PikPkj .(8. 3.21) Again,asabove,ifsupersymmetryisbrokenPijhasatleastonezeroeigenvalueandone ofthecorrespondi ngmasslessfermionsistheGoldstino. Weevaluate thegradedtraceofthemassmatrixsquared.Thissupertracegives themassrelation strM2=J ( Š 1)2 J(2 J +1) MJ 2= trMB 2Š 2 trMFMF *= Š 2 IKik il flfk(8.3.22) Sinceweareinnormalgaugewecanrewritethisas strM2= Š 2 Rk l flfk(8.3.23) where Rk l= Ri l k iistheRiccitensorofthemanifoldevaluatedati= < i> (see (4.1.28)).Theresultismanifestlycovariant,andthus(8.3.23)holdsinanarbitrary gauge.Inparticular,formodelswithconventionalactions iitheK¨ ahlermanifoldis ”at andweobtainthesimplemassformulaJ ( Š 1)2 J(2 J +1) MJ 2=0.(8. 3.24) Wealso observethatincontrasttorenormalizab lemodels,spontaneoussupersymmetry breakingcanoccurinamodelwithasing lechiralmultiplet,forexamplewith IK = cos (+ ),P()=,where Š < A < c.Globalgaugesystems Inthissection,werepeatthepreviousanalysisbutincludegaugesuper“elds V = VATA.B ecausegaugesymmetriesareusua llydescribedbyexplicitmatrix

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5148.BREAKDOWNrepresentations( TA)i jofthegenerators,webeginwiththisformulation;wethenchange overt oamoregen eralformulationwherewedescribetheactionofthegeneratorsby K illingvectors.Asdiscussedinsec.4.1.b,thisallowsustochoosenormal coor dinates in whichthecomputationofthemassmatricessimpli“es(however,wecannotusenormal gauge ).Werestrictourselvestomodelswherethegaugedgroupisunbrokenor isotropic atoneormorepointsofthemanifoldofscalar“elds.(Theusualmatrixrepresentation assumesisotropyattheorigin,i.e.,theoriginiskept“xedbygaugetransformations.)A fo rmulationintermsofKillingvectorsshouldallowonetogaugegroupsthatarerealizednonlinearlyateverypointonthemanifoldofscalar“elds,i.e., hasac onstant termeverywhere(theconstan ttermca nnotbeelim inatedbyshiftingthescalar“elds); however,thesuper“elddescriptionofthemoregeneralcasehasnotbeenworkedout. Weconsid ertheaction S = d4xd4 [ IK (i,j)+ trV ] + d4xd2 [P(i)+1 4 QAB(i) WAWB]+ h c .(8. 3.25) withcovariantlychiral: j= k( eV)k j, WA= i D2( eŠ VDeV)A.(8. 3.26) Thechiralquantities QAB= AB+ O ()can generatemassesforthegaugefermionscontainedin V .Wehaveincl udedthegl obalFayet-Iliopoulosterm trV (4.3.3). WechoseaK¨ ahlergauge(see(4.1.26))where IK itselfisinvariant;wecanalways dothis ifthegaugegroupisunbrokensomewhereonthemanifoldasdiscussedabove. Thengaugeinvarianceof S requires IKj( TA)j iiŠ j( TA)j iIKi=0, Pj( TA)j ii=0, QDE, j( TC)j ii+( TC)D AQAE+( TC)E AQAD=0.(8. 3.27) Thematrices( TC)E Aformtheadjointrepres entationofthegenerators,andare,uptoan overallf actor,thestructureconstants;thustheyareindependentofanyspecialchoiceof

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8.3.Spontaneousbreakingofglobalsupersymmetry515coordinatesonthescalarmanifold. Webeginbyderiv ingtheeq uationsforthevacuumexpectationvalues.Wede“ne (Yang-Mills)covariantcomponent“eldsbycovariantprojection(see(4.3.4,5)) Ai=i| i= i| Fi= 2i| ,( 8.3.28a) = W| f=1 2 ( W )| ,(8. 3.28b) i € €=1 2 [ {, W} ] | ,D= Ši 2 {, W}| ,(8. 3.28c) where fBisthecomponentgauge“eldstrength(see(4.2.85)).Wealsoneedtheidentity( dA= < DA> ) < 2 2i> | = dA aj( TA)j i.(8. 3.29) Thesuper“eldequationsthatfollowfrom(8.3.25)are: 2IKi+Pi+1 4 QAB, iWAWB=0 j( TA)j iIKiŠi 2 ( QABWB)+1 2 i €( QAB W€B)+ trTA=0.(8. 3.30) Theequationsforthevacuumexpectationvaluesareobtainedbyevaluatingat =0 theaboveequations,andtheequationobtainedbydierentiatingthe“rstonewith 2. We “nd IKi j fj+Pi=0 IKij k fkfj+ ak( TA)k jdAIKi j+Pijfj+1 2 QAB, idAdB=0 aj( TA)j iIKi+( QAB+ QAB) dB+ trTA=0(8.3 .31) where IK QAB,Pareevalu atedwithi ai. Wenowgen eralizetoarbitrarycoordinatesbyrewritingtheaboveintermsof ho lomorphicKillingvectors kAi.W er ep lacethespeci“cformoftheYang-Millsgauge transformation(4.1.35) i= i A( TA)i jj, i= Š i jA( TA)j i(8.3.32)

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5168.BREAKDOWNwiththemoregeneralform(4.1.31): i=AkAi, i=AkAi(8.3.33) whereiisde “nedbyanalogywi th(4.1.34b): i exp ( iVAkAj j ) i.(8. 3.34) Theconditions(8.3.27)thatensuregaugeinvarianceoftheactionbecome IKikAi+ IKikAi=0 PikAi=0 QDE, ikCi+ i ( TC)D AQAE+ i ( TC)E AQAD=0.(8. 3.35) Asdiscussedabove,aformulationintermsofKillingvectorsenablesustousenormalcoordinatesandthustosimplifyourcomputations.Thus,forexample,wecancomputethemassmatricesofthevariouscomponent“eldsand“ndasupertracerelation thatgeneralizes(8.3.22).We“ndthelinearized“eldequationsforthecomponent“elds byexpa ndingthecovariantizedformofthesuper “eldequation s(8. 3.30)aroundthevacuumandapplyingtheoperators1, 2tothe“rstand1, ,[ ]tothes econdof theequationsandevaluatingat =0.Theresult,inno rmalcoordinates,is Fi+PijAj=0 i €€ i+ AkAi+Pij jŠi 2 QAB, idAA=0, Ai+[ idAkAi j+ IKik ljfk flŠ Pik Pkj] Aj+[Pijkfk+1 2 QAB, ijdAdB] Aj+[ QAB, idB+ ikAi]DA=0, ( QAB+ QAB)DB+( QAB, idB+ ikAi) Ai+( QAB, idBŠ ikAi) Ai=0,i 2 ( QAB+ QAB) €B€+( kAiŠi 2 QAB, idB) i+1 2 QAB, ifiB=0,

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8.3.Spontaneousbreakingofglobalsupersymmetry517( QAB+ QAB) €fBŠ ( kAikBi+ kBikAi) AB€=0.(8. 3.36) wherewehav edropped <> onP..., IK... ..., QAB....Inour normalizat ionthev ectorwave equationis €fŠ m2 VA€=0.(8. 3.37) Eliminatingtheauxiliary“eldswe“ndthemassmatricesfromwhichweobtainthe supertrace(innormalcoordinates) strM2= Š 2[ idAkAi i+ IKik lifk fl+ tr ( Qi1 Q + Q Qj1 Q + Q ) fi fj+ i ( Q + Q )Š 1AB( kAi QBC, idCŠ kAiQBC, idC)].(8.3.38) Acovariantform ula,va lidinanycoordinatesystem,isobtainedbyreplacing kAi iwith kAi ; iand IKik liwith Rk lstrM2= Š 2[ idAkAi ; i+ Rk lfk fl+ tr ( Qi1 Q + Q Qj1 Q + Q ) fi fjŠ itr ( Qi1 Q + Q ) kAidA](8. 3.39) wherewehaverewrittenthelasttermusingthegaugeinvariancerelations(8.3.35).In thecoordinatesystemwheretheYang-Millsgaugetransformationsaregivenby(8.3.32) wehave kAi ; i= Š i ( TA)i iŠ i ( TA)j i aji(cf.(4.1.29b,31,32d)).

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5188.BREAKDOWN8.4Traceformulaefromsuperspace Inthelasttwosections,wefoundthesupertraceusingessentiallyacomponent approach,andnottakingadvantageofthesuper“eldformalism.Thereisamucheasier waytoevaluatethesu pert raceexpressionwithoutevercomputingcomponentmass matrices:Iftheactionisexpandedincomponentsandtheone-loopeectivepotentialis evaluated,itsquadraticallydivergentpartisproportionaltothesupertrace strM2; moreover,wecaneasilyreadothisquadra ticallydiverg enttermfromtheclassical super“eldactionifweimagineperformingasuper“eldone-loopcalculation. a.Explicitbreaking Wecand evelopthemethod(andderivesomenewmassformulae)by“rstconsideringthecaseofexplicitsoftbreakingof supersymmetry.Forexample,weconsidera masslessscalarmultipletandaddtoittheexplicitsoftbreakingterm(8.2.3) Sbreak=d4xd4 U .Wenowcalculatet hequadraticallydivergentpartoftheoneloope ectivepotential;thecoecientisthecontributionoftheterm(8.2.3)tothe supertrace.Recallthatsoftbreakingtermsarede“nedbythepropertythattheygiveat mostlogarithmicallydivergentcontributionstotheeectiveaction,andyethereweare calculatingquadraticdivergences;however,insec.8.2weignoredvacuumdiagrams (which haveonlyspurion“eldsexternally),whereasherethatisallweareinterestedin. Inthecalculation,wehavetoconsiderthesumofone-loopdiagramswith n masslesschiralpropagatorsand nU -spurionvertices( U = 22 2;altho ughthecalculation simp li“esifweusetheexplicitformof U ,wew illkeep U general,si ncethentheresults canbeappliedtoothercases).Ateachvertexwehavefactors D2, D2actingonthe pr opagators;however,eachpropagatorisproportionalto pŠ 2,andth us,togeta quadraticdivergence,wemustcancelallbutonepropagatorwithanumeratorfactor. Thisrequires n Š 1facto rsof D2 D2Š p2;there mainingfactorisneededforthe loop (seesec.6.3,e.g.,(6.3.28)).Hencewe“nd =n d4 d4p (2 )4p2 Š 1 n ( Š U )n= d4 ln (1+ U ) d4p (2 )4p2 .(8. 4.1) Thereforethesupertraceis

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8.4Traceformulaefromsuperspace519strM2= Š 2 d4 ln (1+ U )= Š 2[ D2 D2ln (1+ U )] | = Š 2[ D2 D2U ] | = Š 2 2.(8. 4.2) Comparingtothecomponentexpressionin(8.2.3),weseethatthisisindeedthecorrect result:themassofthescalar A hasbeenloweredby 2(thefactorof2arisesbecause A iscomplex).Itisclearthatnodiagramcontai ningchiralself-int eractionscanchange theresult:Wen eededafactorof D2 D2ateachvertex,andachiralvertexcomeswith onlyafactor D2. Suc ha di agramcanbeonlylogarithmicallydivergent.Also,since supersymmetricmasstermscanbetreatedas interactions,ourresultsholdinthemassivecase.Thissameargumen talsoimp liesthatexplicitbreakingtermsofthetypes consideredin(8 .2.5,6,9)cannotcontributetothesupertrace. Nextweconsiderexplicitbreakingterms(8.2.7)foranabelianvector“eld: Sbreak=1 2 d4xd4 WW+ h c .=1 2 d4xd4 ( + ) VD D2DV .Thecalc ulation isalmostidenticaltotheabove:Eachpropagatorisstill pŠ 2,ex ceptthatavector pr opagatorhasanextra Š 1r elativetoachiralpropagator,andeach + vertex ( = 2)comeswith afactor D D2D,whichacts precisely inthesamewayasafactor D2 D2,ex ceptfora Š 1thatcan celstheextra Š 1f romthepropagator.Thuswe“nd strM2= Š 2 d4 [ Š ln (1+ + )]=2[ D2 D2ln (1+ + )] | = Š 2[ D2 D2 ] | = Š 2 2.(8. 4.3) Asbefore,thisagreeswiththecomponentexpression(8.2.7)(thefactor2comesfrom thetwohelicitycomponent softhefe rmion).Wecancombinetheexplicitbreaking terms(8.2.3-7)withthepreviousonesand“ndsimplythesumof(8.4.2,3). Finally,weconsider(8.2.10);sincethishasonlyafactor D D2insidetheloopat eachvertex,itcannotcontributetothequadraticdivergenceorthesupertrace.

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5208.BREAKDOWNb.Spontaneousbreaking Intheexamplesoftheprecedingsubsection,weusedratherelaboratemethodsto deriveresultsthatcanbefoundmoreeasilybyexplicitcomputationofthemasses;here wew illapplyt hesemethodstoderiveresultsthatrequiredthesomewhatlengthycalculationsofsec.8.3.We“rstconsidertheaction(8.3.12).Weexpand S tos econdorder inquantum“eldsi,withtheco ecientsevaluatedatthebackgroundclassicalvalues < i> : S(2)= d4xd4 jIKi ji+ d4xd2 Xijij+ h c .(8. 4.4) where Xij= Pij+ D2IKij(8.4.5) Incompleteanalogywith(8.4.1),thequadraticallydivergenttermintheone-loopeectiveactionis = d4p (2 )4p2 d4 tr [ ln ( IKi j)](8.4.6) where IKi jŠ i jplaystheroleof U and,asabove,thec hiralvertex(here Xij)doesnot contribute.Thesupertraceistherefore strM2= Š 2 d4 tr [ ln ( IKi j)]= Š 2 { D2 D2[ trln ( IKi j)] }| = Š 2[ trln ( IKi j)]k l flfk(8.4.7a) andhence,using(4.1.30b), strM2= Š 2 Rk l flfk,(8. 4.7b) inagreementwith(8.3.23). Fo rt he casewithgaugeinteractions(sec.8.3. c),weagainobtainthesupertraceby examiningthequadraticdivergenceintheone-loopeectiveaction.Tosecondorderin quantum“el dswehave S(2)= d4xd4 [ IKi j k( eV)k ji+1 4 ( QAB+ QAB) VAD D2DVB],(8.4 .8)

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8.4Traceformulaefromsuperspace521wherewehavedroppedtermsthatdonotcontributetothequadraticdivergence(that is,chiralinteractionsortermscorrespondingto(8.2.10)).Now( eV)k jIKi jŠ i kplays theroleof U above,and1 2 ( QABŠ AB)playstheroleof .The “nalresultis strM2= Š 2 d4 { tr [( VATA)i j+ ln ( IKi j)] Š trln (1 2 [ QAB+ QAB]) } = Š 2[ dA( TA)i i+ dA( TA)i j aji+ Rk l flfk+ tr ( Qk1 Q + Q Ql1 Q + Q ) flfkŠ tr ( Ql1 Q + Q )( TA)l i aidA](8. 4.9) wherewehavereplacedd4 2 2andused[ €, { €, } ]= Š 2 iW,and (8.3.29).Wethusrecovertheresult(8.3.39).(Wehavechosentoworkinthecoordina tesystemde“nedby(8.3.32)simplybecauseitismorefamiliar;thecomputationis eq ua llystraightforwardintermsofKillingvectors.)

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5228.BREAKDOWN8.5.Nonlinearrealizations Experiencewithspontaneouslybrokeni nternalsy mmetrieshasshownthatmuch usefulinsightcanbegainedbystudyingthegeneraltheoryofnonlinearrealizations. Themethodsthathavebeendevelopedcanbeappliedquitesuccessfullytosupersymmetry. Onewaytoformulateanonlinearrealizationofsupersymmetryistoconsider(nonlinearly) constrainedsuper“elds;however,itisfarfromobvioushowtochoosesuchconstraints,andsowewillreturntothisapproachafterwehavestudiednonlinearrealizationsdirectly. ThesimplestnonlinearrealizationistheVolkov-Akulovmodel.Itisfoundbyconsideringacovariantlytransformingsetofhypersurfacesinsuperspace.Let ( x )= (8.5.1) de“neahypersurfa ce;ittransformsas ( x)= (8.5.2) wherewerecallthat x= x Ši 2 ( + ), = + .Thisi mplies ( x)= + = ( x )+ =0(8.5 .3) andhence ( x Ši 2 [ ( x )+ ( x )])= ( x )+ (8.5.4) or ( x )= +i 2 ( €+ €) €.(8. 5.5) Thisgivesanonlinearrealizationofthealgebracarriedbythespinor“eld ( x );itisby nomeansunique,butothernonlinearrealizationsarerelatedtoitby“eldrede“nitions. Notethat (oranyotherequivalentnonlinearrealization)containsaconstanttermin itstransformationlaw,andisthereforeasui table“eldfor describingtheGoldstino. To “ndaninvariantaction,wereca llthattheone-form(3.3.31) s€( x , )= dx€+i 2 ( d €+ €d )(8. 5.6)

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8.5.Nonlinearrealizations523is in va riantundersupersymmetrytransformations.Ifweconstrainthisone-formtolie onthehypersurface ( x )= ,we “nd s€( x )= dx€+1 2 ( i m €) dx m dx mv m a,(8. 5.7) wherew ehavede“nedaninversevierbein v m a.Sincetheone-for misinvar iant,the vierbeinmusttransformcovariantly,i.e.,supersymmetrytransformationsof must i nducecoordinatetransformationsof v a m.Nowit iseasytowritedownaninvariant actionintermsofthedeterminant v = det ( v a m): S= d4xd4 vŠ 14( Š ( x ))= d4xvŠ 1.(8. 5.8) Wenote that cvŠ 14( Š ( x ))isascalarsuper“eldwhosecomponentsare functionsof ( c isanarbitrarydimensionalconstantthatsetsthescaleofsupersymmetrybreaking;seebelow).Wecanals oconstructachi ralsuper“eld= D2outof Since[ 4( )]2=0thesesup er“eldssat isfythenonlinearconstraints 2=2=0(8.5 .9a) = cŠ 1 D2 = cŠ m( D2 )m(8.5.9b) = cŠ 1D2 D2=1 2 cŠ 1 D D2D(8. 5.9c) etc.Thesolutiontotheconstraints(8.5. 9a,b)or(8.5.9a,c)ispreciselyorrespectively. Theexpectationvalues < > < > of,,thatfollowfrom <> =0are typicaloft heexpectationvalueofamultipletwithspontaneouslybrokensupersymmetry(asinsec.8.3):theauxiliarycomponents( 2or 2 2forandrespectively)get nonvanishingexpectationvalues c ,andallotherco mponentscanbetakentohavevanishingexpectationvalues.Thefermioncomponentsatone Š levellowerthantheauxiliary“elds,e.g., = D | or = D2D | ,havethisconst anttermintheirtransformationsas = c + ... ,con“ rmingouridenti“cationof c asthesupersymmetrybreakingscale. Thevierbein v canbeusedtowritedownotherinvariantactions; any expression covariantizedwith v issupersymmetric.For example,wecancouple toascala r“eld

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5248.BREAKDOWNA asfollows: SA= Š1 2 d4xvŠ 1( v a m mA )2(8.5.10a) where A transformsas A ( x )= A( x)= A( x Ši 2 ( + )).(8.5.10b) Wenowdiscussthepre scriptionfordescribingspontaneouslybrokentheoriesin termsof .Inaspontane ouslybrokentheory,theGoldstinoisone,oringenerala uniquelinearcombination,ofthefermioniccomponentsoftheordinarysuper“elds.We introducestandardvariablesbyreplaci ngtheGoldstinowiththeVolkov-Akulov“eld ,andthes uper“eldsbynewsuper“eldswhosecomponentstransformhomogeneouslyas in(8.5.10b),andinparticular,withnomixingofdierent -components. Webeginby constructingahomogeneouslytransformingsuper“eld outof andanordinarysuper“eld.Consider ( x , ) ( x +i 2 ( + ), Š Š )(8. 5.11) where( x , )is any super“eld.Undersupersy mmetrytransformations( x, ) ( x Ši 2 ( + ), + + )=( x , ),we“ndforthetransformationof : (x Ši 2 ( + ), )=(x Ši 2 ( + Š Š ), Š Š )=(x Ši 2 ( + Š Š ) Ši 2 [ ( Š )+ ( Š )], Š + Š + ); (8.5.12a) using(8. 5.4),wehave (x Ši 2 ( + ), )=(x +i 2 ( + ), Š Š )= ( x , ).(8.5 .12b)

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8.5.Nonlinearrealizations525Thusweseethatundersupersymmetrytransformations transfo rmshomogeneously (dier ent -componentsdonotmix)butnonlinearlywithrespectto ;the x -coordinate undergoesatranslation.Therefore,thewholesupersymmetrygroupisrealizedon by elementsofthePoincar egro up.Thisisageneralfeatureofn onlinearrealizations:Given agroup G (herethesupersymmetrygroup) andalinearlyrealizedsubgroup H (herethe Poincar egro up),thenonlinearrealizationsonsuitablyde“ned“eldsisperformedbyelementsof H .Consequently, wecanimpose anytranslationallyinvariantconstraint on withoutbreakingsupersymmetry.Forexample,ifweconstrainitentirelybysettingit equalto c 2 2(or c 2inthechiralcase),weexpressallthecomponentsofintermsof andrecoverthepreviousresult(8.5.9). Wenowconsi deramodelwithspontaneousbreakdownofsupersymmetry.Wecan describethemodelintermsofstandardcompon ents,i.e.,componentstransformingasin (8.5.4,10b),asfollows: (1)Foreachsuper“ eldsweconstructtheassociated (2)Weidentifythefermioniccomponent oftheappropriatelinearcombinationof sthatistheGoldstino,andisthereforeasuitablecandidatetobereplacedby andconstrainthecorrespondingcomponent inthecorrespondi nglinearcombinationof stozero.Th isgivesusthecombinationofthecomponentsofthes thattransformsas(8.5.4). (3)Weexpresstheremainingcomponentsofthesintermsoftheremainingcomponentsofthe s. Thisprocedureistheanalogofgoingtoradialandanglevariablesfornonlinearsigma models:theremainingcomponentsofthe scorrespondtotheradialvariables, whereas correspondstotheanglevariable. Asanexample,we considertheORaiferteaighmodelofsec.8.3.a,withthreechiralsuper“eldsi, i =0 ,1 ,2 .U si ng (8.3.18),we“ndthattheauxiliary“eld F0ofthe mult iplet0getsanonvanishingexpectationvalue F0= c .Todes cribethesystemin termsofstandardcomponents,we“rstde“ne homogeneouslytransformingsuper“elds ibyintro ducingtheVolkov -Akulov“eld asanextravariable;thenwerestorethe numberofde greesoffreedombyconstraining 0=0 (thefermioniccomponentof 0) andeliminating 0(thefermioniccomponentof0)infav orof .Wecandothis

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5268.BREAKDOWNb ecause,examining 0,we “nd 0= 0Š c + ... ,where c = < F0> .Ifspontaneous symmetrybreakingd idnotoccur(i.e.,if c =0),wecoulds tillde “nehomogeneouscomponents, butwecouldnotremovetheextradegreeoffreedom(thechangeofvariables from 0to wouldbesingu lar).Havingeliminated 0infavorof (thestandardangle va riable,whichtransformsas(8.5.4)),wecanproceedtoexpresstheremainingcomponentsofiintermsof andthecomponentsofthehomogeneoussuper“elds i(the standardradialvariables,wh ichtransformas(8.5.10b)).

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8.6.SuperHiggsmechanism5278.6.SuperHiggsmechanism Whensupersymmetrybreaksinasy stemcoupledtosupergravity,a superHiggs mechanismeliminatestheGoldstinoandgivesmasstothegravitino(theGoldstino b ecomesitslongitudinalcomponent).Toexam inethesuperHiggsmechanismindetail, west udythelocallysupersymmetr icanalogoftheconstrainedsuper“eldsoftheprevioussection(8.5.9). ThebasicingredientofthesuperHiggsmechanismisthetransformationlawofthe Goldst ino, = c + ... (seepr evioussection);whentheGoldstinoiscoupledto supergravity,thesupersymmetryparameter b ecomeslocal: = c ( x )+ ... (8.6.1) Consequently,theGoldstinocanbecompletelygaugedaway;sincethenumberof dynamicalmodesofthetheoryshouldnotchange,weexpecttheGoldstinotore-emerge somehow,anditdoessobygivingthegravitinoamassandbecomingitslongitudinal mode.Toseethisdirectly,wedescribetheGoldstinobyalocalconstrainedchiral“eld obeyingthelocalsuperspaceversionof(8.5.9a-b).Thuswetake2=0, = cŠ 1( 2+ R ) (hereweco nsider onlyminimal( n = Š1 3 )supergrav ity).These constraintshaveaconsistentsolutionintermsofasinglefermicomponent“eld(the Goldstino).Becauseoftheconstraintson,a nyloca llysupersymmetricaction(without explicitderivatives,see(5.5.15))reducesto Shi ggs= Š 2 d4xd23( + )+ h c .(8. 6.2) Theconstrainedsuper“eldisanonlinearfunctionoftheGoldstino;however,whenwe gaugetheGoldstinoaway(goto U-gauge) itsimpli“estobecome= Š 2c (alternativelyandequivalently,wecaneliminatetheGoldstinobyarede“nitionofthegravitino. Insuperspace,thiscorrespondstorescaling by(1+ )Š1 3 ;howev er,itissimplerto chooseUgauge).Theaction(8.6.2)becomes,using(5.6.60,64) Shi ggs= Š 2 d4x eŠ 1[ (3 S +1 2 (€ |€ |€ )€)+ c + h c .](8.6 .3) where S isthecomplexscalarauxiliary“eldofthesupergravitymultiplet;wealsohave the Š 3 Š 2| S |2termfromthesupergravityaction(5.6.63).Eliminating S byitsequ ation

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5288.BREAKDOWNofmotion,we“ndacosmologicalconstantandgravitinomassterms: Shi ggs= Š 2 d4x eŠ 1[ 1 2 (€ |€ |€ )€+ h c .+3 | |2+ ( c + c )](8.6 .4) Asdiscussedinsec.5.7,grav itinomasstermswhena ccompaniedbyacosmological constantdonotingeneralmeanthatthegravitinoismassive.However,if ( c + c )= Š 3 | |2(8.6.5) thenthecosmologicaltermcancels,andwecanunambiguouslyidentifythe termsas massterms. Sinceanyspontaneouslybrokentheorycanbedescribedintermsofstandardvariables,andinparticular,theGoldstinocanbedescribedintermsof,inanyspontaneouslybrokentheoryinwhich thecosmologicalconstantvanishesthegravitinomassis Re whenthe(superspace)kinetictermhastheusualnormalization Š 3 Š 2.Itisf o und byse ttingallmatter“eldstotheirvacuumexp ectationvalues.Moregenerally,when (8.6.5)isnotsatis“ed,wecanstill“ndtheapparent mass Re andthecosmological constantfromthetransforma tionoftheGoldstinoandbycomparingtothesuperspace action(8.6.2).

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8.7.Supergravityandsymmetrybreaking5298.7.Supergravityandsymmetrybreaking Supersymmetrybreakinginalocalcontextcanbestudieddirectly,usingthe componenttoolsofsection5.6.Wecandete rmineconditionsforsu persy mmetrybreakingandderiveamassformulaanalogousto(8.3.35).However,itismuchmoreecient torecastthep roblemasa global supersymmetryproblemthatcanbestudiedusingthe tec hniquesofsecs.8.3b-cand8.4.Weconsiderageneralsystemofinteractingscalar andvectormultipletscoupledto N =1superg ravity.Themattermultipletsare describedbychiraland(rea l)gaugescalarsuper“eldsi, VA,resp ectively;thesupergravitymulti pletisdescribedbytherealaxial-vectorsuper“eld H mand(for n = Š1 3 ) thechiralcompensator .Howev er, H mplaysnodirect roleinthesupersymmetry breakingmechanismorinthederivationofmassformulae.Thereforealltherelevant informationcanbeext ractedfromaglobal nonrenormalizable systemdescribedby ,i, and VA. Webeginbyre ducingthecoupledmatter-supergravitysystem.Theaxial-vector realgaugesuper“eldofsupergravity H mcontainsthegra vitonandgravitinophysical de greesoffreedom,aswellastheaxialvectorauxiliary“eld A m(5.2.8).Inthepresence ofthecompensator thesupergravitygaugegroupconsistsofthefullsuperconformal group,andwehaveatourdisposalallofthecomponentgaugetransformationsof (5.2.10).Consequently,wecangototheWe ss-Zuminogaugediscussedafter(5.2.10), andfurther,usetheremainingsuperconformaltransformationstoremovethegraviton trace,thegravitino -t race,andthelongitudinalpartoftheaxialvectorauxiliary“eld. Inthisgauge H mcontainsonlythetracelesscomponentsofthegravitonandthegravitino,andthetransversepartof A m;t he sp inzerocomplexauxiliary“eld S ,the -trace ofthe(left-handed)gravitino( )L= € €,thetraceo fthevierbein,o requivalently, itsdet erminant e = dete a mand Š 1 mA marecontainedin (theselasttwoarethereal andimaginary partsofthe -independentcomponentof ).Si nceonlythesequantities arerelevantforstudyingspontaneoussupersymmetrybreaking(e.g.,thespinzero bosonscang etvacuumexpectationvaluesandthe -tracecanmixwiththematter fermions),wecanignorethe H mdependenttermsintheLagran gianandworkentirely with andthemattersuper“eldsinaglobalse tting.Thissimpli“esthediscussion enormously;however,because Im | replacesthedivergenceof A m,therear esomesubtletiesassociatedwithitscontributiont othema ssesofthematter“elds(seesubsec.

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5308.BREAKDOWN8.7.a.4).Wetreatonly n = Š1 3 supergravity;analogousmethodscanbeusedforother n butsince n = Š1 3 allowsthemostgeneralcoupling,itisthemostinterestingcase. Thesupergravitymultipletitself(through )a ectsthepatternofsymmetry breaking.At“rstsight,thisseemsstrange:IntheusualHiggsmechanism,wedonot expectthepatternofsymmetrybreakingtodependonthecouplingstogauge“elds(at thetreelevel!).However,ananalogoussituationarisesinanonsupersymmetriccontext, whenscalar“eldsarecoupledtogravity.Weconsidertheaction S = d4x g [ Š 3 Š 2r ( g ) Š1 4 g m nGij( A ) mAi nAj+ rV1( A )+ V2( A )].(8.7.1) To “ndthevacuumexpectationvaluesofthesc alar“eldswecannotignorethegravitational“eld.IngeneraltheRicciscalar r willhaveanonzeroexpectationvaluethat aectsthemassesandscalarpotential.Howe ver,wen eednotconsiderthefullEinstein system;itissucienttolookforsolutionsoftheform g m n= 2 m nandtotreatthesystemofsca lar“elds Ai(subjecttothecondition =0at allpoints).(Thisisanalogoustokeeping andignoring H m.)Twopossiblesituationscanarise:If < V2> =0 wehaveanon zerocosmologicalconstant, Š 1Š 1 < V2>1 2 x2,andthevac uumvalues andthemassesof Aiareshiftedfromtheir”atspacevalues.If < V2> =0,thecosmologicalconstantvanishesandaconsistentsolutionis =const ant.Inthiscasethe gravitational“elddoesnotmo dify”atspaceresults.(Thisisnotthecaseinsupergravity:Evenifthecosmologicalconstantvanishes,thesupergravityauxiliary“eldsmodify globalresults.) Returningtothematter-supergravitysy stem,weconsidertheaction(5.5.32) S = d4xd4 EŠ 1( H ) {Š3 2 eŠ1 3 2( trV + G )+[ 1 R ( g +1 4 QABWAcovWB cov)+ h c .] } ,(8. 7.2) where EŠ 1isthesuper determinantofthevielbeinand R isthescalarcurvaturesuper“eld(seee.g.,( 5.2.74-6)).Thesupergravityactionisgivenbythe“rstterminthe expansionoftheexponential.Here G (i,j)isanarbit rarygaugeinvariantfunctionof chiralsuper“ elds,withde“nedby( 8.3.26), g (i)isachiralf unction,and WAcovisthe

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8.7.Supergravityandsymmetrybreaking531(supergravitycovariant)Yang-Mills“eldstrength.Thefunction G hasanaturalinterpretationasaK¨ ahlerpotentialwithgaugetransformations G G +(i)+ ( j) compensatedbyscalingsof : exp [1 3 2()] The exp [ Š1 3 2 trV ]factoristhelocalf ormoftheFayet-Iliopoulosterm(4.3.3). Itisgaugeinvariantbyvirtueofacombinedgaugetransformationof V andsuperscale transformationsof EŠ 1(5.3.8-10).Itspres enceseverelyrestrictstheformofthe g terms; theymustbe R -invariant(see(3.6.14)and(4.1.15))sothatthewholeactionisinvariant underthesuperscaletransformationsof EŠ 1(seebelow).Inthe 0 limittheaction (8.7.2)becomes(8.3.25),withtheidenti“cation G IK g P Asdisc ussedabove,wecansplitothetermsindependentof H m.Furthe rmore, accordingtothediscussionfollowing(5.5.28)the dependenceof Wcovcanbefactored out( Wcov Š3 2 W )sothattherele vantpartof(8 .7.2)becomes S = d4xd4 [ Š eŠ ( trV + G )] + d4xd2 [ 3g +1 4 QABWAWB]+ h c .(8. 7.3) Wehavesett hegravit ationalconstant1 3 2=1.Wew illrestoreitw hennecessary. Underthegaugetransformation trV tr [ V + i ( Š )], G G ,( D€=0),the actionisinvariantifwerescale eŠ i tr .Thustheloca lFayet-I liopoulostermacts asaconventionalgaugetermfor .If =0,asnot edabovetheformof g (i)is extremelyrestricted: 3g mustbe gaugeinvariant. Wenowa nalyzetheg lobalsystem(8.7.3)subjecttotheconditionthatthecosmologicalconstantvanishes.Wecanthenchoose Re <> | = =const ant.Withthe identi“cation Š eŠ trV eŠ G (i,j)= IK 3g (i)=P(8. 7.4) wehavetheactiono f(8. 3.25)withouta global Fayet-I liopoulosterm.We labelc omponentsof as

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5328.BREAKDOWN | = A D | = , D2 | = S (8.7.5) withexpectationvalues < A > =1, < S > = s .(8. 7.6) a.Massmatrices Webeginbyexp licitlycomputingthemassmatricesforthevarious“eldsinthe system.Thiscalculationis a littlelengthy,sowewillsimplifyitasmuchaspossible withoutlossofgenerality.Thus,werescale toremove g fromthechira lpartofthe action: 3g 3.Wealsor ede“ne G : G G +1 3 ln ( g ge3 trV).Thismakes inert undergaugetransformations,andabsorbstheFayet-Iliopoulosterminto G .Inthecase when < g > =0weca nnotperformthisresca ling.However,thiscaseisnotinteresting, sincethensupersymmetryisnotbrokeneveninthepresenceofaFayet-Iliopoulosterm (ifthecosmologicalconstantvanishes). a.1.Vacuumconditions Thesuper“eldequations(8.3.30)fortheaction(8.7.3)are: D2( eŠ G) Š 3 2=0(8.7 .7a) eŠ G 2Gi+3 3Gi+1 4 QAB, iWAWB=0(8.7 .7b) Š eŠ GkAiGi+1 2 ( QABWB) Š1 2 €( QABW€B)=0(8. 7.7c) wherewehaveused(8.7.7a)tosimplify(8.7.7b),andthrownawaysometermsthatlead tohigherorde rspi norand/orderivativeinteractionsthatdonotenterbelow.Wehave written(8.7.7)intermsofKillingvectorsbymakingthesubstitutions ( TA)i jjŠ ikAi,j( TA)j i ikAi.Thevac uumconditions(8.3.31)foundbyapplying spinorderivativesto(8.7.7)become s Š 3 eGŠ Gi fi=0, Gi j fj+3 eGGi=0,

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8.7.Supergravityandsymmetrybreaking5336 s 3=( QAB+ QAB) dAdB, 2eŠ GGi( ikAi)+( QAB+ QAB) dB=0, 3 eGGijfj+ Gij k fkfj+ Gi j( ikAj) dA+9 GieG( s Š eG)+1 2 Š 2eGQAB, idAdB=0.(8. 7.8) Theassumptionthatthecosmologicalconstantvanishesisequivalenttothecondition thattheseequationshaveasolutionforconstant .Wealsohav ethe gaugeinvariance conditions(8.3.c13)(theseholdforgeneralvaluesofthe“elds,notjustatthevacuum poin t): GikAi+ GikAi=0, Š ikCiQAB, i+( TC)B DQDA+( TC)A DQDB=0.(8. 7.9) Togivethegravi tationalactionthecorrectnormalization(5.2.72),weidentify 2=3 Š 2eG.(8. 7.10) a.2.Gravitinomass Asdisc ussedinsec.8.6,wecan“ndthespin3 2 massbysettingall matter “eldsto theirvacuumexpectationvalues,an dcompari ngthecoecientsofthed4xd4 andd4xd23terms.From(8.7.3),thekinetictermhasacoecient Š 2eŠ G= Š1 3 2, andthechiraltermhasacoecient 3(thefactorsof comefromthede“nitionsofthe dynamical“elds(8.7.5));hence,using(8.6.2),we“ndthatthespin3 2 massis m =3 eG(8.7.11) Wesimp lifyourcom putationfurtherbychoosingnormalcoordinates Gi j= i j, Gi j1...= Gi j1...=0.Using(8.7 .10,11),andnormalcoordinates,werewritethevacuumconditions (8.7.8)as s Š m Š Gi fi=0

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5348.BREAKDOWN fi+ mGi=0 6 m Š 2s Š ( Q + Q )ABdAdB=0 3 i Š 2kAiGi+( Q + Q )ABdB=0 mGijfj+ ikAidA+ mGi(3 s Š m )+1 6 2QAB, idAdB=0.(8. 7.12) a.3.Waveeq uations Wenow “ndthelinearizedwaveequations(8. 3.36)thatfollowfrom(8.7.7).Asin sec.8.3,weexpandthe“eldsinsmall”uctuationsabouttheirvacuumvalues.Forthe remainderofthissubsection,allquantities G... ...and QAB...areevaluatedati= aiand i= ai.We “nditusefultointroduceshiftedvariables A A Š GiAi, Š Gii, S S Š GiFi.(8. 7.13) From(8 .7.7a)wehave SŠ A( s Š m ) Š 2 m AŠ 2 mGiAiŠ ( Gijfj+ sGi) Ai=0 (8.7.14a) i € €Š kAiGiAŠ 2 m Š 2 mGi i=0.(8. 7.14b) From(8 .7.7b) ,we “nd Fi+ m [(2 AŠ A) Gi+(3 GiGj+ Gij) Aj+ Ai]=0(8. 7.14c) i €€ i+2 mGi + m (3 GiGj+ Gij) j+ kAiAŠi 6 2QAB, idAB=0.(8. 7.14d) From(8 .7.7c),we“nd ( Q + Q )ABDB+ QAB, idBAi+ QAB, idBAi+3 i Š 2[ kAiAiŠ kAiAi+ kAiGi( A+ A)]=0(8.7.14e)

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8.7.Supergravityandsymmetrybreaking535 1 2 ( Q + Q )ABi € B€+1 2 QAB, i( fiBŠ idB i) +3 Š 2kAi( i+ Gi )=0.(8.7 .14f) Weareleftwiththe equationsofthephysicalboson“elds.Thesesimplifygreatlyifwe us e( 8.7.14a,c,e);we“nd A=0 (8.7.14g) Ai+{3 Š 2( Q + Q )Š 1AB( kAiŠ1 3 i 2QAC, idC)( kBj+1 3 i 2 QBE, jdE) + ikAi jdAŠ m2( GikGkj+3 GiGkGkj+3 GikGkGjŠ Gik jlGkGl) + m [(3 s Š m ) i j+3(3 s Š 2 m ) GiGj]}Aj+{3 Š 2( Q + Q )Š 1AB( kAiŠ1 3 i 2QAC, idC)( kBj+1 3 i 2QBE, jdE) +1 6 2QAB, ijdAdBŠ m2(3 GikGkGj+3 GiGjkGk+ GijkGk) + m (3 s Š 2 m )( Gij+3 GiGj)}Aj=0(8.7 .14h) ( Q + Q )AB€fBŠ 3 Š 2( kAikBi+ kBikAi) AB€+3 Š 2kAiGiX€=0(8.7 .14i) where X€= €ImA Š ImGj€Aj= €ImA+( GjkBjŠ GjkBj) AB€.Here €is theYang-Millscovariantderivative. a.4.Bosemasses We no wd iscusstheseresults.From(8.7.14g)weseethatthecomplexscalar Ais massless.Fortherealpart,thisisnosurprise: ReAisthetraceofthegraviton,which ismasslessbecausethecosmologicaltermwasassumedtovanish.However,theimaginarypartrequiressomecare.Thepseudoscalar ImA isnotrecognizableasoneof the“eldsofthesupergravit ymulti plet;itstandsfor Š 1 A where A€isthe

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5368.BREAKDOWNdi ve rgenceoftheaxialvectorauxiliary“eld.Thustheequation ImA= ImA Š ImGiAi=0shouldbe replacedby A€Š ImGi€Ai=0(8.7 .15) Formany purposes,itmakeslittledierencewhetherweuse ImA orreplaceitwith Š 1 A .Bydimension alanalysisandLorentzinvariance, A€canenterthewave equationofthescalar“elds Aionlythroughitsdivergence: Ai+ cGi€A€+ ... =0.(8. 7.16) Substitutingin(8.7.15),we“nd ( Ai+ cGiImGjAj)+ ... =0.(8. 7.17) Whenwehave A insteadof A€,wegetthesameresult,si nceinsteadof (8.7.16)we have Ai+ cGi ImA + ... =0,(8. 7.18) andusingthe A waveequation,wer eobtain(8.7.17). However,ifgaugeinvarianceisbrokenthegauge“eldwaveequationcangetaspuriouscontributionfrom ImA thatisnotpresentwhen Š 1 A isusedinstea d.Indeed, substi tuting(8.7.15)intothe“rstformof X€(with €ImA replacedby A€)givesa zerocontributiontothespin1mass.Whengaugeinvarianceisunbroken,wegetno contributionfromtheformwith A aswell: €ImAdoesnotaectthespin1mass, andthevacuumexpectationvalueof kBjiszero.However,ifgaugeinvarianceisbroken, theexpectationvalueof kBjisnotzero(equivalently, €( GiAi) = Gi€Ai)and X givesas puriouscontributionthatmustberemovedbyhand. a.5.Fermimasses Thecomponent correspo ndsto the -traceofthegravitino;wede“netheGoldstinoasthatcombinationofmatter“eldsthatcouplesto (itmakesno essentia ldierencewhetherweuse or ,sincewearea lwaysfreetoaddtermstothegravitino). Thuswede“ne Gi i+1 2 m GikAiA.(8. 7.19)

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8.7.Supergravityandsymmetrybreaking537We alsode“netransverse“eldsthatareorthogonalto : iT iŠ Gi AT A+i m dA .(8. 7.20) (These satisfy GiiT+1 2 m GikAiAT=0.)Inte rmsofthese,thespinorwaveequations b ecome: i € €Š 2 m ( + )= 0( 8.7.21a) i € €+2 m ( + )=0(8. 7.21b) i €€ i T+ m ( Gij+ GiGj) jT+( kAiŠ kAjGjGiŠi 6 2QAB, idB) A T=0(8.7 .21c) i € B€ T+[6 Š 2( Q + Q )Š 1BA( kAjŠi 6 2QAC, jdC) Š 2 idB] jTŠ [ m ( Q + Q )Š 1BCQCA, iGi+i m dBkAlGl] A T=0.(8. 7.21d) Caremustbetakentoensurethatthemassoperatoron T, Tisrestrictedtothe  transversesubspace,i.e.,preservestheorthogonalityto Observethatsincethetraceofthegravitinoisa negativenorm state, i.e.,aghost, itskinetictermhasaminussignrelativetophysicalspinors(thesameistrueforthe traceofthegraviton;thewhole mult iplethasnegativenorm,ascanbeseenfromthe action(8.7.3)).Consequently,thoughthemassmatrixinthe system(whichis decoupledfromtheotherspinors)doesnotvanish,botheigenvaluesarezero(themass matrixisnothermitian).Actually,wedidnothavetoexplicitly“ndthewaveequation toarriveatthisresult:TheconditionthattheGoldstinocanbegaugedaway(thatwe cangotoaU-gauge)impliesthatboththeGoldstino andthe -traceofthegravitino musthave zeromassinthegaugethatweuse.

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5388.BREAKDOWNa.6.Supertrace Havingfoundthewaveequations(8.7.14gi,a8),andunderstood theirsigni“cance, wecanevaluatethesu pert race.Thespin0contributionis(recallthatwearestillin normalcoordinates): Š 2[ikAi idAŠ 3 Š 2( Q + Q )Š 1AB( kAiŠ1 3 i 2QAC, idC)( kBi+1 3 i 2 QBE, idE) Š m2( GijGij+3 GiGjGij+3 GijGiGjŠ Gik ijGkGj) + m (3 s Š m )N+3(3 s Š 2 m )( m Š s )](8.7.22a) whereN i iisthenumberofc hiralmultiplets.Thecombinedcontributionofthespin 0andspin1 2 “eldsis: Š 2[9 Š 2( Q + Q )Š 1ABkAikBi+ i ( Q + Q )Š 1ABdC( kAi QBC, iŠ kAiQBC, i) + ikAi idA+ Gij ik fkfjŠ ( N +1) m2+( N Š 1)3 ms + tr(1 Q + Q Qi1 Q + Q Qj) fjfi].(8. 7.22b) Thespin1contribution,omittingthe X€termisgivenbyt heexpression 3 3 Š 2( Q + Q )Š 1AB( kAikBi+ kBikAi)andca ncelsthe“rsttermof(8.7.22b).(Thenormalizationcomesfromthe3statesofaspin1particleandfromtheform(8.3.37)ofthe spin1waveequ ation.)Finally,thespin3 2 contributionisjust Š 4 m2.Thusweget (usingthegauge-invariancerelations(8.7.9)tosimplifysomeexpressions) strM2= Š 2[ikAi idA+ Gij ik fkfjŠ ( N Š 1)( m2Š1 2 2( Q + Q )ABdAdB) Š itr(1 Q + Q Qi)kAidA+ tr(1 Q + Q Qi1 Q + Q Qj) fjfi],(8. 7.23) innormalcoordinates,or,ingeneral,usingcoordinateinvariance,wehave strM2= Š 2[ikAi ; idA+ Ri j fjfiŠ ( N Š 1)( m2Š1 2 2( Q + Q )ABdAdB) Š itr(1 Q + Q Qi)kAidA+ tr(1 Q + Q Qi1 Q + Q Qj) fjfi].(8. 7.24)

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8.7.Supergravityandsymmetrybreaking539Were mindthereaderthathere strM23/2 J =0 ( Š 1)2 J(2 J +1) MJ 2.(8. 7.25) b.Super“eldcomputationofthesupertrace Ifouronlyinterestisthesupertraceformula(8.7.71),wecanobtainitwithfar lessworkusingthete c hniquedevelopedinsec.8.4.b.(Ofcourse,ingeneralweare interestedinthemassmatri cesthemselves,andnotjustthesupertrace).Westartwith lndet ( IKi j)= Š ( N +1) G + lndet ( Gi j)+ Nln ( Š eŠ trV )(8. 7.26) wheredierentiationiswithrespectto = eŠ trVandnot Beforea ddingcontributionsfromthegravitinomassandcorrectingfortheaxial v ect orauxiliary“eld(seebelow),thesupertracereadfrom(8.4.9)is strM2= Š 2[ikAi ; idAŠ ( N +1) trd + Rk l flfkŠ ( N +1)( Gi j fjfi+ iGikAidA) + tr ( Qk1 Q + Q Ql1 Q + Q ) flfkŠ itr ( Qi1 Q + Q ) kAi].(8. 7.27) whereweuse(4.1.29,30): Rk l=[ lndet ( Gi j)]k l,l=[ lndet ( Gi j)]l.(8. 7.28) Theexpression(8.7.27)hasnotmadeuseofthevacuumconditions(8.7.8)or (8.7.12),a nddoesnotincludeeitherthespin3 2 co nt ributionortheaxialvectorauxiliary “eldcorrectiontothe spin1massmatrixdiscussedinsubsec.8.7.a.4.Aswesawinthe previoussection,thespin3 2 contributionmustbeincludedseparately,sincethe -trace ofthegravitinocannotcontri butedirectly:theconditionforthesuperHiggsmechanism tooccurandforthegravitinotoabsorbtheGoldstinoinU-gaugerequirestheGoldstino-gr avit ino -tracesystemtobemassless.Thespin1correction,thoughsomewhat subtle,canalsobefoundwithoutextensive computation.Asdescribedinsec.8.7.a.4, wesimply subtract Š 2( Q + Q )Š 1ABkAiGi( kBjGjŠ kBjGj)= Š 2 2( Q + Q )ABdAdB(see

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5408.BREAKDOWNdiscussionfollowing(8 .7.22b)foranexplanationofthefactors). Onefurtherpointdeservescomment:Whenwerescaled toremovethepotential g (seethebeginningofsec.8.7.a),welostsightofthecontributionoftheFayet-Iliopoulosterm.Whenwemaketheshift G G +1 3 ln ( g ge3 trV),the trd terminequation (8.7.27)isabsorbedintothe iGikAidAtermasaconsequenceofR-invarianceof g ;itis moststraightforwardtoworkinthecoordinatesystemwheretheKillingvectorstakethe formofusual gaugetransformations: 3 gtr ( TA) Š gi( TA)i j aj=0(8.7 .29) andhence tr ( TA) Š1 3 [ ln ( g ge3 V)]i( TA)i j aj=0(8.7 .30) Usingthevacuumequations,wecansubstituteintothesupertrace(8.7.27). Includingthegravitinoandthespin1correctionterm,werecover(8.7.24). c.Examples Wecanusethesup ertraceformulaetostudymanycasesofinterest.Inparticular,inextende dsupergravit ytheorieswee ncounternonminimal G and Q terms.For example,in N =4superg ravity,whichcontainsonephysicalchiralmultiplet,threevectormultiplets,three(3 2 ,1)multi pletsandthesupergravitymultiplet, G Š ln (1 Š ), Q 1 Š 1+ .(8. 7.31) Weca nnottreattheactual N =4theorysincea descriptionoftheinteracting(3 2 ,1) mult ipletisnotavailable,but(8.7.31)suggest slookingatas ystemwithonescalarmultipletand n v ectormult iplets VA,co upledto N =1superg ravity,with G asaboveand QAB= 1 Š 1+ AB.(8. 7.32) We “nd,with G= 2 G = 1 (1 Š a a )2 (8.7.33)

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8.7.Supergravityandsymmetrybreaking541thesupertrace3/2 J =0 ( Š 1)2 J(2 J +1) MJ 2= Š 2( n +2) Gf f = Š 2( n +2) m2.(8. 7.34) Notethatthe Q and R termsin(8.7.17)combinebecause ( Q + Q )2= Š 4( G)Š 1[(1+)(1+ )]Š 2.Unlessascal arpotential g ()isint roduced, nosupersymmetrybreakingwilloccur.Ho wever,it ispo ssibletoaddsuchatermin N =1superg ravity,andthereexistmechanismstogeneratetermsthatactlikeapotentialev enin N =4superg ravity. For N > 4thean alogsof G and Q areexpressedintermsofanovercompletesetof “elds.Wemayexpecthoweverthat Q and G arerelatedsuchthat det ( Gi j) det ( Q + Q ) h (i) h ( i)where h (i)isaholomo rphicfunction.Inthatcase wemayalsoexp ectasimpleresultforthesupertrace. Wecanalsoconst ructmodelswithaFayet-Iliopoulostermandvanishingcosmologicalconstant.Forexample,consider G = eV+ 23 ln [ eV]+ +1 3 ln [( + )( + )](8.7.35) whereand ar ec hiral“elds,transformingunderthegaugetransformationwhile isinert,and ischosensoastomakethecosmologicalconstantvanish(thepotential andtheFayet-I liopoulostermar eincl udedin G asthe -term).We“ndasolutionto (8.7.8)with d =0forsom e “niterangeof (ascanbeveri“edbya pertur bationexpansionabout =0).

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542INDEXINDEX Action,component15,150,331 scalarmultiplet15,150,302 superconformal245,303,312 supergravity255,259,309 v ectormultiplet23,26,162,168,306 Actions,ingravity238 insupergravity299 Adler-Bardeentheorem407,495 Adler-Rosenbergmethod402,478,486 Algebra,superconformal65 super-deSitter67 super-Lie63 super-Poincar e63 supersymmetry9 Anholonomycoecients236,249 Anomalies,inYang-Millscurrents401 localsup ersymmetry489 (super)conformal474 Anomalycancellation494 Anomaly,chiral407 trace473,476,479 Antisymmetrictensor186 Auxiliary“eld16,151,162,252,326 Axial( n =0 )s up ergravity257,274,288 Axial-vectorauxiliary“eld246 Background“eldmethod373 Ba ck ground-quantumsplitting373,377,379,382,410.414 Backgroundtransformations379,412 Beta-function,vanishingof369 Bianchiidentities22,25,29,39,140,174,181,204,292 Bianchiidentities,solutionof25,40,176,184,294,296 Bisection120,123,126 Breakingandauxiliary“elds508 Breaking,radiative509 soft500 spontaneous496,506 BRSTtransformations342,345

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INDE X 543Casimiroperator72,87 Catalystghost426 Centralcharge64,72  Checkobjects39,251,277 Chiralspinorsuper“eld95,123,159,188 Chiralsuper“eld89 Cliordvacuum69 Commutatoralgebra320 Commutator,graded56 Compensator,conformal240,480 de ns ity242,250,255,259,267,286 tensor242,274 Compensators112,267 Compensators,gravitinomultiplet208 Components,auxiliary13,108 byexpa nsion 10,92 byproj ection11,94 covariant24,178 gauge108 ofscalarmultiplet94 ofsupergravitymultiplet38,245,261,322 ofvectormultiplet160 physic al108 Conformalinvariance65,80,240 Conjugation,hermitian57 rest-frame123 Connection,centralcharge86 gauge18,165,170 isospin86 Lorentz36,86,235,252 Constraints,conformalbreaking265,274,470 conformalsupergravity270 conventional21,35,171,237,270,276,410,470 Poincar esupergr avit y 274 representation-preserving172,270,278,470 solutionof172,276,279,470 Contortion41,115,273,289,298 Converter163 Cosetspace,and -models117 andsuperspace74

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544INDEXCosmologicalconstant528 Cosmologicalterm44,312,333 CovariantFeynmanrules382,446 Covariantfunctionalderivative384,447 Covariantization,ofactions43,300 Covariantlychiral172 CP(n)models113,179 CPT77 Curvature38,236,264 Degauged U (1)289,298 Degreeofdivergence393 Delta-function8,97 Densitycompensator250,267,269 Deriva tive, D -9,83 spinor8,56 superfunctional101,168 Derivatives,covariant18,24,35,165,170,235,249,269 DeSittersupersymmetry67,335 Determinant,vierbein238 Dilatationgenerator65,81,275 Divergences358,452 D -manipulation48,50,360 Doublingtrick386,449 Duality,forthegravitinomultiplet211 ofminimaland n = Š1 3 supergravity310 ofnonminimalandchi ralmultiplets200 oftensorandchiralmultiplets190 transformation190,204 Eectiveaction47,357,373,452 Energy,positivity64,497 Energy-momentumtensor473,481 Eulernumber476 Fa dd eev-Popovghost52,340,344,381,420,432 Fa ye t-Iliopoulosterm178,218,308,389,514 Fe rmi-Feynmangauge342,345 Fe ynmanrules46,53,348,438 Fieldequations153,169,313 Fieldstrength25,40,122,167

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INDE X 545Fieldstrength,conformal124 gravitinomultiplet206 supergravity244,266 Ya ng -Mills156,167,176 Fieldstrengths,o-shell147 -trace474,481 Gauge,normal156 supersymmetric37,338,415,440 Gaugeaveraging52,341,344 Gauge“xing52,341,343,428 Gauge-restoringtransformation115,161,164,173 Gaugetransformations159 GaugeWZmodel198 Generalcoordinatetransformations233 Ghostcounting420 Goldstino498,509,513,522,525,527 Gravitinomass333,533 Gravit inomulti plet 206  Hatobjects250,282,411 Hiddenghost424,432 HyperK¨ ahlermanifold158,222 Hypermul tiplet218 Improvedtensormultiplet191 Indexconventions7,54,542 Indices,”at35,234,252 isospin55 wo rld35,234,252 Integral,Berezin8,97 superfunctional103 Jacobiidentities22 K gaugegroup34,170,172,270 K¨ ahler,manifold155,511 po tential155,511,531 K illingvectors157,514 Lagrangemultiplier203

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546INDEX gaugegroup159,162,173,247,279 Legendretransform191 Liederivative232 Light-cone,basis55,108 fo rmalism108,142 Linearsuper“eld91 Localscaletransformations240 Loca lityin 48,357 Lorentzgenerators35,76,235,249 Lorentztransformations,local35,234 orbital233 Mass,gau geinvariant26 Massmatrices532 Measure, chir al301 general300 Minimal( n = Š1 3 )s up ergravity256,287 Mult iplet,gravitino206 N =2scal ar218 N =2tens or223 N =2v ector216 N =4 Ya ng -Mills228,369 nonminimal scalar199 scalar15,70,149 tensor186 3-form193 variantt ensor203 variantv ector201 v ector18,159,185 Nielsen-Kalloshghost53,376,381,434 Nonlinearrealizations117,522 Nonlinear models117,154,219 Nonminimal( n =0, Š1 3 )s up ergravity256,287 No-renormalizationtheorem358 Normalcoordinates157,533 ORaiferteai ghmodel507 Po we r-counting358,393,454,455

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INDE X 547Prepotential147,173 Prepotential,gravitinomultiplet206 supergravity244 Ya ng -Mills159,173 Projectionoperators120 Qu antumtransformations378,413,431 Rarita-Schwinger“eld246 Recursionrelations547 Reduction,productof D s85 Regulari zation393 Regulariza tion,bydimensionalreduction394 inconsistenciesin397,472 localdimensional469 Pa u li-Villars398,404 po int-splitting399,405 Re pr esentation,chiral79,165,174,284 irreducible120 o-shell13,108,143 on-shell13,69,138,143 superconformal80 super-deSitter82 super-Poincar e75 v ector79 Riccite nsor 237 R-transf ormations96,153 Rwe ight96,153,169 Scal arpotential153 Scalei nvaria nce 240 Se lf-energy49,390,443,460 Smatrix391,463 Softbreakingterms502 Spurion500 S -supersymmetry66,246 Stueckelbergformalism112 Superan omaly484 Supercoordinatetransformations34 Supercovarian tization324 Sup ercurrent473,480

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548INDEXSuperdeterminant99,254 Super“eld9,75 Super“eldstrength140 Superfo rm28,181 Superhelicity13,73 Sup erHiggseect498,527 Superpotential507 Sup erscaletransformations250,271,275 Sup ertrace100,513,518,538 Sup ertracemultiplet473,481,486 Supervector34 Symmetrizat ion7,56 Tangents pace 35,86 Tangent-sp acebasis183 Tensorca lcul us 326 Timebeing,the250,357,384,410,433,485 To rsion38,236,264 Torsion, ”atsupersp ace36,87 Transf ormationsuper“eld96 Transver segauge440 U (1)covariantderivatives269 U-gauge527 Variantrep resentation31,201 Variation,co varian t 168 Vielbeindeterminant42,254,255 Vielbein,”at28,86 supergravity34 Vierbein232,246 Volkov-Aku lovmodel522 We ss-Zuminogauge,supergravity38,246,261,317 Ya ng -Mills20,161,163 We ss-Zuminomodel150 Weylte nsor 237