University Press of Florida
Introduction to String Field Theory
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Title: Introduction to String Field Theory
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Subjects / Keywords: physics, science, Interacting closed string field theory, Conformal algebra, Poincare algebra, Covariantized light cone, Fermions, Gauge fixing, Lorentz gauge, Becchi, Rouet, Stora, Tyutin, OGT+ isbn: 9781616100490
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Abstract: Introduction to string field theory, including comprehensive index and exercises for learners. Contents: 1) Introduction. 2) General Light Cone. 3) General BRST. 4) General Gauge Theory. 5) Particle. 6) Classical Mechanics. 7) Light-Cone Quantum Mechanics. 8) BRST Quantum Mechanics. 9) Graphs. 10) Light-Cone Field Theory. 11) BRST Field Theory. 12) Gauge-Invariant Interactions.
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arXiv:hep-th/0107094v1 11 Jul 2001 INTRODUCTIONto STRINGFIELDTHEORY WarrenSiegel UniversityofMaryland CollegePark,Maryland Presentaddress:StateUniversityofNewYork,StonyBrookmailto:warren@wcgall.physics.sunysb.eduhttp://insti.physics.sunysb.edu/~siegel/plan.html

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CONTENTS Preface1.Introduction 1.1.Motivation11.2.Knownmodels(interacting)31.3.Aspects41.4.Outline6 2.Generallightcone 2.1.Actions82.2.Conformalalgebra102.3.Poincarealgebra132.4.Interactions162.5.Graphs192.6.Covariantizedlightcone20Exercises23 3.GeneralBRST 3.1.Gaugeinvarianceand constraints25 3.2.IGL(1)293.3.OSp(1,1 j 2)35 3.4.Fromthelightcone383.5.Fermions453.6.Moredimensions46Exercises51 4.Generalgaugetheories 4.1.OSp(1,1 j 2)52 4.2.IGL(1)624.3.Extramodes674.4.Gaugexing684.5.Fermions75Exercises79 5.Particle 5.1.Bosonic815.2.BRST845.3.Spinning865.4.Supersymmetric955.5.SuperBRST110Exercises118 6.Classicalmechanics 6.1.Gaugecovariant1206.2.Conformalgauge1226.3.Lightcone125Exercises127 7.Light-conequantummechanics 7.1.Bosonic1287.2.Spinning1347.3.Supersymmetric137Exercises145 8.BRSTquantummechanics 8.1.IGL(1)1468.2.OSp(1,1 j 2)157 8.3.Lorentzgauge160Exercises170 9.Graphs 9.1.Externalelds1719.2.Trees1779.3.Loops190Exercises196 10.Light-coneeldtheory197 Exercises203 11.BRSTeldtheory 11.1.Closedstrings20411.2.Components207Exercises214 12.Gauge-invariantinteractions 12.1.Introduction21512.2.Midpointinteraction217Exercises228 References230Index241

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PREFACE First,I'dliketoexplainthetitleofthisbook.Ialwayshat edbookswhosetitles began\Introductionto..."Inparticular,whenIwasagrads tudent,bookstitled \IntroductiontoQuantumFieldTheory"werethemostdicul tandadvancedtextbooksavailable,andIalwaysfearedwhataquantumeldtheo rybookwhichwas notintroductorywouldlooklike.Thereisnowastandardref erenceonrelativistic stringtheorybyGreen,Schwarz,andWitten, SuperstringTheory [0.1],whichconsistsoftwovolumes,isover1,000pageslong,andyetadmits tohavingsomemajor omissions.NowthatIsee,fromanauthor'spointofview,how mucheortisnecessarytoproduceanon-introductorytext,thewords\Intro ductionto"takeamore tranquilizingcharacter.(Ihaveworkedonaone-volume,no n-introductorytexton anothertopic,butthatwasinassociationwiththreecoauth ors.)Furthermore,these wordsleavemetheoptionofomittingtopicswhichIdon'tund erstand,oratleast beingmoreheuristicintheareaswhichIhaven'tstudiedind etailyet. Therestofthetitleis\StringFieldTheory."Thisisthenew estapproach tostringtheory,althoughtheolderapproachesarecontinu ouslydevelopingnew twistsandimprovements.Themainalternativeapproachist hequantummechanical (/analog-model/path-integral/interacting-string-pic ture/Polyakov/conformal-\eldtheory")one,whichnecessarilytreatsaxednumberofeld s,correspondingto homogeneousequationsintheeldtheory.(Forexample,the reisnoanaloginthe mechanicsapproachofeventhenonabeliangaugetransforma tionoftheeldtheory, whichincludessuchfundamentalconceptsasgeneralcoordi nateinvariance.)Itisalso anS-matrixapproach,andcanthuscalculateonlyquantitie swhicharegauge-xed (althoughlimitedbackground-eldtechniquesallowtheca lculationof1-loopeective actionswithonlysomecoecientsgauge-dependent).Inthe oldS-matrixapproach toeldtheory,thebasicideawastostartwiththeS-matrix, andthenanalytically continuetoobtainquantitieswhichareo-shell(andperha psinmoregeneralgauges). However,inthelongrun,itturnedouttobemorepracticalto workdirectlywith eldtheoryLagrangians,evenforsemiclassicalresultssu chasspontaneoussymmetry breakingandinstantons,whichchangethemeaningof\on-sh ell"byredeningthe vacuumtobeastatewhichisnotasobviousfromlookingatthe unphysical-vacuum S-matrix.Ofcourse,S-matrixmethodsarealwaysvaluablef orperturbationtheory,

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buteveninperturbationtheoryitisfarmoreconvenienttos tartwiththeeldtheory inordertodeterminewhichvacuumtoperturbabout,whichga ugestouse,andwhat power-countingrulescanbeusedtodeterminedivergencest ructurewithoutspecic S-matrixcalculations.(Moredetailsonthiscomparisonar eintheIntroduction.) Unfortunately,stringeldtheoryisinaratherprimitives taterightnow,andnot evenclosetobeingaswellunderstoodasordinary(particle )eldtheory.Ofcourse, thisisexactlythereasonwhythepresentisthebesttimetod oresearchinthisarea. (Anyonewhocanhonestlysay,\I'lllearnitwhenit'sbetter understood,"shouldmark adateonhiscalendarforreturningtograduateschool.)Iti sthereforesimultaneously thebesttimeforsomeonetoreadabookonthetopicandthewor sttimeforsomeone towriteone.Ihavetriedtocompensateforthisproblemsome whatbyexpandingon themoreintroductorypartsofthetopic.Severaloftheearl ychaptersareactually onthetopicofgeneral(particle/string)eldtheory,bute xplainedfromanewpoint ofviewresultingfrominsightsgainedfromstringeldtheo ry.(Amorestandard courseonquantumeldtheoryisassumedasaprerequisite.) Thisincludestheuse ofauniversalmethodfortreatingfreeeldtheories,which allowsthederivationof asingle,simple,free,local,Poincare-invariant,gauge -invariantactionthatcanbe applieddirectlytoanyeld.(Previously,onlysomespecia lcaseshadbeentreated, andeachinadierentway.)Asaresult,eventhoughthefactt hatIhavetriedto makethisbookself-containedwithregardtostringtheoryi ngeneralmeansthatthere issignicantoverlapwithothertreatments,withinthisov erlaptheapproachesare sometimesquitedierent,andperhapsinsomewayscompleme ntary.(Thetreatments ofref.[0.2]arealsoquitedierent,butforquitedierent reasons.) Exercisesaregivenattheendofeachchapter(excepttheint roduction)toguide thereadertoexampleswhichillustratetheideasinthechap ter,andtoencourage himtoperformcalculationswhichhavebeenomittedtoavoid makingthelengthof thisbookdiverge. ThisworkwasdoneattheUniversityofMaryland,withpartia lsupportfrom theNationalScienceFoundation.Itispartlybasedoncours esIgaveinthefallsof 1985and1986.IreceivedvaluablecommentsfromAleksandar Mikovic,Christian Preitschopf,AntonvandeVen,andHaroldMarkWeiser.Iespe ciallythankBarton Zwiebach,whocollaboratedwithmeonmostoftheworkonwhic hthisbookwas based.June16,1988 WarrenSiegel Originallypublished1988byWorldScienticPublishingCo PteLtd. ISBN9971-50-731-5,9971-50-731-3(pbk)July11,2001: liberated,corrected,bookmarksadded(topdf)

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1.1.Motivation 1 1.INTRODUCTION1.1.Motivation Theexperimentswhichgaveusquantumtheoryandgeneralrel ativityarenow quiteold,butasatisfactorytheorywhichisconsistentwit hbothofthemhasyet tobefound.Althoughtheimportanceofsuchatheoryisunden iable,theurgency ofndingitmaynotbesoobvious,sincethequantumeectsof gravityarenot yetaccessibletoexperiment.However,recentprogressint heproblemhasindicated thattherestrictionsimposedbyquantummechanicsonaeld theoryofgravitation aresostringentasto require thatitalsobeauniedtheoryofallinteractions,and thusquantumgravitywouldleadtopredictionsforotherint eractionswhichcanbe subjectedtopresent-dayexperiment.Suchindicationswer egivenbysupergravity theories[1.1],wherenitenesswasfoundatsomehigher-or derloopsasaconsequence ofsupersymmetry,whichrequiresthepresenceofmatterel dswhosequantumeects canceltheultravioletdivergencesofthegravitoneld.Th us,quantumconsistencyled tohighersymmetrywhichinturnledtounication.However, eventhissymmetrywas foundinsucienttoguaranteenitenessatallloops[1.2]( unlessperhapsthegraviton werefoundtobeabound-stateofatrulynitetheory).Inter estthenreturnedto theorieswhichhadalreadypresentedthepossibilityofcon sistentquantumgravity theoriesasaconsequenceofevenlarger(hidden)symmetrie s:theoriesofrelativistic strings[1.3-5].Stringsthusoerapossibilityofconsist entlydescribingallofnature. However,evenifstringseventuallyturnouttodisagreewit hnature,ortobetoo intractabletobeusefulforphenomenologicalapplication s,theyarestilltheonly consistenttoymodelsofquantumgravity(especiallyforth etheoryofthegraviton asaboundstate),sotheirstudywillstillbeusefulfordisc overingnewpropertiesof quantumgravity. Thefundamentaldierencebetweenaparticleandastringis thataparticleisa0dimensionalobjectinspace,witha1-dimensionalworld-li nedescribingitstrajectory inspacetime,whileastringisa(nite,openorclosed)1-di mensionalobjectinspace, whichsweepsouta2-dimensionalworld-sheetasitpropagat esthroughspacetime:

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2 1.INTRODUCTION xx ( ) particle r # # # # # # # # c c c c c c c c X ( ) X ( ; ) string # # # # # # # # c c c c c c c c # # # # # # # # c c c c c c c c Thenontrivialtopologyofthecoordinatesdescribesinter actions.Astringcanbe eitheropenorclosed,dependingonwhetherithas2freeends (itsboundary)oris acontinuousring(noboundary),respectively.Thecorresp ondingspacetimegure istheneitherasheetoratube(andtheircombinations,andt opologicallymore complicatedstructures,whentheyinteract). Stringswereoriginallyintendedtodescribehadronsdirec tly,sincetheobserved spectrumandhigh-energybehaviorofhadrons(linearlyris ingReggetrajectories, whichinaperturbativeframeworkimpliesthepropertyofha dronicduality)seems realizableonlyinastringframework.Afteraquarkstructu reforhadronsbecame generallyaccepted,itwasshownthatconnementwouldnatu rallyleadtoastring formulationofhadrons,sincethetopologicalexpansionwh ichfollowsfromusing 1 =N color asaperturbationparameter(theonlydimensionlessoneinm asslessQCD, besides1 =N flavor ),aftersummationintheotherparameter(thegluoncouplin g,which becomesthehadronicmassscaleafterdimensionaltransmut ation),isthesameper-

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1.2.Knownmodels(interacting) 3 turbationexpansionasoccursintheoriesoffundamentalst rings[1.6].Certainstring theoriescanthusbeconsideredalternativeandequivalent formulationsofQCD,just asgeneraleldtheoriescanbeequivalentlyformulatedeit herintermsof\fundamental"particlesorintermsoftheparticleswhichariseas boundstates.However, inpracticecertaincriteria,inparticularrenormalizabi lity,canbesimplyformulated onlyinoneformalism:Forexample,QCDiseasiertousethana theorywheregluons aretreatedasboundstatesofself-interactingquarks,the latterbeinganonrenormalizabletheorywhichneedsanunwieldycriterion(\asymp toticsafety"[1.7])to restricttheavailableinnitenumberofcouplingstoanit esubset.Ontheother hand,atomicphysicsiseasiertouseasatheoryofelectrons ,nuclei,andphotons thanaformulationintermsofeldsdescribingself-intera ctingatomswhoseexcitationslieonReggetrajectories(particularlysinceQEDi snotconning).Thus, thechoiceofformulationisdependentonthedynamicsofthe particulartheory,and perhapsevenontheregioninmomentumspaceforthatparticu larapplication:perhapsquarksforlargetransversemomentaandstringsforsma ll.Inparticular,the runningofthegluoncouplingmayleadtononrenormalizabil ityproblemsforsmall transversemomenta[1.8](whereaninnitenumberofarbitr arycouplingsmayshow upasnonperturbativevacuumvaluesofoperatorsofarbitra rilyhighdimension),and thusQCDmaybebestconsideredasaneectivetheoryatlarge transversemomenta (inthesamewayasaperturbativelynonrenormalizabletheo ryatlowenergies,like theFermitheoryofweakinteractions,unlessasymptoticsa fetyisapplied).Hence,a stringformulation,wheremesonsarethefundamentalelds (andbaryonsappearas skyrmeon-typesolitons[1.9])maybeunavoidable.Thus,st ringsmaybeimportant forhadronicphysicsaswellasforgravityanduniedtheori es;however,thepresently knownstringmodelsseemtoapplyonlytothelatter,sinceth eycontainmassless particlesandhave(maximum)spacetimedimension D =10(whereasconnementin QCDoccursfor D 4). 1.2.Knownmodels(interacting) Althoughmanystringtheorieshavebeeninventedwhicharec onsistentatthe treelevel,mosthaveproblemsattheone-looplevel.(There arealsotheorieswhich arealreadysocomplicatedatthefreelevelthattheinterac tingtheorieshavebeen toodiculttoformulatetotestattheone-looplevel,andth esewillnotbediscussed here.)Theseone-loopproblemsgenerallyshowupasanomali es.Itturnsoutthat theanomaly-freetheoriesareexactlytheoneswhichareni te.Generally,topologi-

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4 1.INTRODUCTION calargumentsbasedonreparametrizationinvariance(the\ stretchiness"ofthestring worldsheet)showthatanymultiloopstringgraphcanberepr esentedasatreegraph withmanyone-loopinsertions[1.10],soalldivergencessh ouldberepresentableasjust one-loopdivergences.Thefactthatone-loopdivergencess houldgenerateoverlapping divergencesthenimpliesthatone-loopdivergencescausea nomaliesinreparametrizationinvariance,sincetheresultantmulti-loopdivergenc esareinconrictwiththe one-loop-insertionstructureimpliedbytheinvariance.T herefore,nitenessshould beanecessaryrequirementforstringtheories(evenpurely bosonicones)inorderto avoidanomaliesinreparametrizationinvariance.Further more,theabsenceofanomaliesinsuchglobaltransformationsdeterminesthedimensi onofspacetime,whichin allknownnonanomaloustheoriesis D =10.(Thisisalsoknownasthe\critical,"or maximum,dimension,sincesomeofthedimensionscanbecomp actiedorotherwise madeunobservable,althoughthenumberofdegreesoffreedo misunchanged.) Infact,thereareonlyfoursuchtheories: I:N=1supersymmetry,SO(32)gaugegroup,open[1.11]IIA,B:N=2nonchiralorchiralsupersymmetry[1.12]heterotic:N=1supersymmetry,SO(32)orE 8 n E 8 [1.13] or brokenN=1supersymmetry,SO(16) n SO(16)[1.14] Allexcepttherstdescribeonlyclosedstrings;therstde scribesopenstrings,which produceclosedstringsasboundstates.(Therearealsomany casesofeachofthese theoriesduetothevariouspossibilitiesforcompacticat ionoftheextradimensions ontotoriorothermanifolds,includingsomewhichhavetach yons.)However,forsimplicitywewillrstconsidercertaininconsistenttheorie s:thebosonicstring,whichhas globalreparametrizationanomaliesunless D =26(andforwhichthelocalanomalies describedaboveevenfor D =26havenotyetbeenexplicitlyderived),andthespinningstring,whichisnonanomalousonlywhenitistruncated totheabovestrings. Theheteroticstringsareactuallyclosedstringsforwhich modespropagatinginthe clockwisedirectionarenonsupersymmetricand26-dimensi onal,whilethecounterclockwiseonesare N =1(perhaps-broken)supersymmetricand10-dimensional,o r viceversa.1.3.Aspects Thereareseveralaspectsof,orapproachesto,stringtheor ywhichcanbestbe classiedbythespacetimedimensioninwhichtheywork: D =2 ; 4 ; 6 ; 10.The2D

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1.3.Aspects 5 approachisthemethodofrst-quantizationinthetwo-dime nsionalworldsheetswept outbythestringasitpropagates,andisapplicablesolelyt o(second-quantized)perturbationtheory,forwhichitistheonlytractablemethodo fcalculation.Sinceit discussesonlythepropertiesofindividualgraphs,itcan' tdiscusspropertieswhich involveanunxednumberofstringelds:gaugetransformat ions,spontaneoussymmetrybreaking,semiclassicalsolutionstothestringeld equations,etc.Also,itcan describeonlythegauge-xedtheory,andonlyinalimitedse tofgauges.(However, byintroducingexternalparticleelds,alimitedamountof informationonthegaugeinvarianttheorycanbeobtained.)Recentlymostoftheeor tinthisareahasbeen concentratedonapplyingthisapproachtohigherloops.How ever,inparticleeld theory,particularlyforYang-Mills,gravity,andsupersy mmetrictheories(allofwhich arecontainedinvariousstringtheories),signicant(and sometimesindispensable) improvementsinhigher-loopcalculationshaverequiredte chniquesusingthegaugeinvarianteldtheoryaction.Sincesuchtechniques,whose stringversionshavenot yetbeenderived,coulddrasticallyaecttheS-matrixtech niquesofthe2Dapproach, wedonotgivethemostrecentdetailsofthe2Dapproachhere, butsomeofthebasic ideas,andtheoneswesuspectmostlikelytosurvivefuturer eformulations,willbe describedinchapters6-9. The4Dapproachisconcernedwiththephenomenologicalappl icationsofthe low-energyeectivetheoriesobtainedfromthestringtheo ry.Sincethesetheoriesare stillverytentative(andstilltooambiguousformanyappli cations),theywillnotbe discussedhere.(See[1.15,0.1].) The6Dapproachdescribesthecompactications(orequival enteliminations)of the6additionaldimensionswhichmustshrinkfromsightino rdertoobtainthe observeddimensionalityofthemacroscopicworld.Unfortu nately,thisapproachhas severalproblemswhichinhibitausefultreatmentinabook: (1)Sofar,nojustication hasbeengivenastowhythecompacticationoccurstothedes iredmodels,orto 4dimensions,oratall;(2)thestyleofcompactication(Ka lu_za-Klein,Calabi-Yau, toroidal,orbifold,fermionization,etc.)deemedmostpro misingchangesfromyear toyear;and(3)thestringmodelchosentocompactify(seepr evioussection)also changeseveryfewyears.Therefore,the6Dapproachwon'tbe discussedhere,either (see[1.16,0.1]). Whatisdiscussedhereisprimarilythe10Dapproach,orseco ndquantization, whichseekstoobtainamoresystematicunderstandingofstr ingtheorythatwould allowtreatmentofnonperturbativeaswellasperturbative aspects,anddescribethe

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6 1.INTRODUCTION enlargedhiddengaugesymmetrieswhichgivestringtheorie stheirnitenessandother unusualproperties.Inparticular,itwouldbedesirableto haveaformalisminwhich allthesymmetries(gauge,Lorentz,spacetimesupersymmet ry)aremanifest,niteness followsfromsimplepower-countingrules,andallpossible models(includingpossible 4Dmodelswhoseexistenceisimpliedbythe1 =N expansionofQCDandhadronic duality)canbestraightforwardlyclassied.Inordinary( particle)supersymmetric eldtheories[1.17],suchaformalism( superelds or superspace )hasresultedinmuch simplerrulesforconstructinggeneralactions,calculati ngquantumcorrections( supergraphs ),andexplainingallnitenessproperties(independentfr om,butveriedby, explicitsupergraphcalculations).Thenitenessresults makeuseofthebackground eldgauge,whichcanbedenedonlyinaeldtheoryformulat ionwhereallsymmetriesaremanifest,andinthisgaugedivergencecancellati onsareautomatic,requiring noexplicitevaluationofintegrals.1.4.Outline Stringtheorycanbeconsideredaparticularkindofparticl etheory,inthatits modesofexcitationcorrespondtodierentparticles.Allt heseparticles,whichdier inspinandotherquantumnumbers,arerelatedbyasymmetryw hichrerectsthe propertiesofthestring.Asdiscussedabove,quantumeldt heoryisthemostcompleteframeworkwithinwhichtostudythepropertiesofpart icles.Notonlyisthis frameworknotyetwellunderstoodforstrings,butthestudy ofstringeldtheoryhas broughtattentiontoaspectswhicharenotwellunderstoode venforgeneraltypesof particles.(Thisisanotherrespectinwhichthestudyofstr ingsresemblesthestudy ofsupersymmetry.)Wethereforedevotechapts.2-4toagene ralstudyofeldtheory. Ratherthantryingtodescribestringsinthelanguageofold quantumeldtheory, werecasttheformalismofeldtheoryinamoldprescribedby techniqueslearned fromthestudyofstrings.Thislanguageclariestherelati onshipbetweenphysical statesandgaugedegreesoffreedom,aswellasgivingagener alandstraightforward methodforwritingfreeactionsforarbitrarytheories. Inchapts.5-6wediscussthemechanicsoftheparticleandst ring.Asmentioned above,thisapproachisausefulcalculationaltoolforeval uatinggraphsinperturbationtheory,includingtheinteractionverticesthemselve s.Thequantummechanics ofthestringisdevelopedinchapts.7-8,butitisprimarily discusseddirectlyasan operatoralgebrafortheeldtheory,althoughitfollowsfr omquantizationoftheclassicalmechanicsofthepreviouschapter,andviceversa.Ing eneral,theprocedureof

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1.4.Outline 7 rst-quantizationofarelativisticsystemservesonlytoi dentifyitsconstraintalgebra, whichdirectlycorrespondstoboththeeldequationsandga ugetransformationsof thefreeeldtheory.However,asdescribedinchapts.2-4,s ucharst-quantization proceduredoesnotexistforgeneralparticletheories,but theconstraintsystemcan bederivedbyothermeans.Thefreegauge-covarianttheoryt henfollowsinastraightforwardway.Stringperturbationtheoryisdiscussedincha pt.9. Finally,themethodsofchapts.2-4areappliedtostringsin chapts.10-12,where stringeldtheoryisdiscussed.Thesechaptersarestillra therintroductory,since manyproblemsstillremaininformulatinginteractingstri ngeldtheory,eveninthe light-coneformalism.However,amorecompleteunderstand ingoftheextensionofthe methodsofchapts.2-4tojustparticleeldtheoryshouldhe lpintheunderstanding ofstrings. Chapts.2-5canbeconsideredalmostasanindependentbook, anattemptata generalapproachtoallofeldtheory.Forthosefewhighene rgyphysicistswhoare notintenselyinterestedinstrings(ordonothavehighenou ghenergytostudythem), itcanbereadasanewintroductiontoordinaryeldtheory,a lthoughfamiliaritywith quantumeldtheoryasitisusuallytaughtisassumed.Strin gscanthenbeleftfor laterasanexample.Ontheotherhand,forthosewhowantjust abriefintroduction tostrings,astraightforward,thoughlesselegant,treatm entcanbefoundviathe lightconeinchapts.6,7,9,10(withperhapssomehelpfroms ects.2.1and2.5).These chaptersoverlapwithmostothertreatmentsofstringtheor y.Theremainderofthe book(chapts.8,11,12)isbasicallythesynthesisoftheset wotopics.

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8 2.GENERALLIGHTCONE 2.GENERALLIGHTCONE2.1.Actions Beforediscussingthestringwerstconsidersomegeneralp ropertiesofgauge theoriesandeldtheories,startingwiththelight-conefo rmalism. Ingeneral,light-coneeldtheory[2.1]lookslike non relativisticeldtheory.Using light-conenotation,forvectorindices a andtheMinkowskiinnerproduct A B = ab A b B a = A a B a a =(+ ; ;i ) ;A B = A + B + A B + + A i B i ; (2 : 1 : 1) weinterpret x + asbeingthe\time"coordinate(eventhoughitpointsinalig htlike direction),intermsofwhichtheevolutionofthesystemisd escribed.Themetric canbediagonalizedby A 2 1 = 2 ( A 1 A 0 ).Forpositiveenergy E (= p 0 = p 0 ), wehaveonshell p + 0and p 0(correspondingtopathswith x + 0and x 0),withtheoppositesignsfornegativeenergy(antipartic les).Forexample, forarealscalareldthelagrangianisrewrittenas 1 2 ( p 2 + m 2 ) = p + p + p i 2 + m 2 2 p + = p + ( p + H ) ; (2 : 1 : 2) wherethemomentum p a i@ a p = i@=@x + withrespecttothe\time" x + ,and p + appearslikeamassinthe\hamiltonian" H .(Inthelight-coneformalism, p + isassumedtobeinvertible.)Thus,theeldequationsarer st-orderinthesetime derivatives,andtheeldsatisesanonrelativistic-styl eSchrodingerequation.The eldequationcanthenbesolvedexplicitly:Inthefreetheo ry, ( x + )= e ix + H (0) : (2 : 1 : 3) p canthenbeeectivelyreplacedwith H .Notethat,unlikethenonrelativistic case,thehamiltonian H ,althoughhermitian,isimaginary(incoordinatespace),d ue tothe i in p + = i@ + .Thus,(2.1.3)isconsistentwitha(coordinate-space)rea lity conditionontheeld.

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2.1.Actions 9 Foraspinor,halfthecomponentsareauxiliary(nonpropaga ting,sincetheeld equationisonlyrst-orderinmomenta),andallauxiliaryc omponentsareeliminated inthelight-coneformalismbytheirequationsofmotion(wh ich,bydenition,don't involveinvertingtimederivatives p ): 1 2 ( =p + im ) = 1 2 2 1 = 4 ( + y y ) p 2 p i p i + im i p i im p 2 p + 2 1 = 4 + = + y p + + y p + 1 p 2 y ( i p i im ) + 1 p 2 + y ( i p i + im ) + y ( p + H ) + ; (2 : 1 : 4) where H isthe same hamiltonianasin(2.1.2).(Thereisanextraoverallfactor of2 in(2.1.4)forcomplexspinors.Wehaveassumedreal(Majora na)spinors.) ForthecaseofYang-Mills,thecovariantactionis S = 1 g 2 Z d D xtr L ; L = 1 4 F ab 2 ; (2 : 1 : 5 a ) F ab [ r a ; r b ] ; r a p a + A a ; r a 0 = e i r a e i : (2 : 1 : 5 b ) (Contractionwithamatrixrepresentationofthegroupgene ratorsisimplicit.)The light-conegaugeisthendenedas A + =0 : (2 : 1 : 6) Sincethegaugetransformationofthegaugeconditiondoesn 'tinvolvethetimederivative @ ,theFaddeev-Popovghostsarenonpropagating,andcanbeig nored.Theeld equationof A containsnotimederivatives,so A isanauxiliaryeld.Wetherefore eliminateitbyitsequationofmotion: 0=[ r a ;F + a ]= p + 2 A +[ r i ;p + A i ] A = 1 p + 2 [ r i ;p + A i ] : (2 : 1 : 7) Theonlyremainingeldsare A i ,correspondingtothephysicaltransversepolarizations.Thelagrangianisthen L = 1 2 A i 2 A i +[ A i ;A j ] p i A j + 1 4 [ A i ;A j ] 2 +( p j A j ) 1 p + [ A i ;p + A i ]+ 1 2 1 p + [ A i ;p + A i ] 2 : (2 : 1 : 8) Infact,for arbitrary spin,aftergauge-xing( A + =0)andeliminatingauxiliary elds( A = ),wegetforthefreetheory L = y ( p + ) k ( p + H ) ; (2 : 1 : 9)

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10 2.GENERALLIGHTCONE where k =1forbosonsand0forfermions. Thechoiceoflight-conegaugesinparticlemechanicswillb ediscussedinchapt.5, andforstringmechanicsinsect.6.3andchapt.7.Light-con eeldtheoryforstrings willbediscussedinchapt.10.2.2.Conformalalgebra Sincethefreekineticoperatorofanylight-coneeldisjus t 2 (uptofactorsof @ + ),theonlynontrivialpartofanyfreelight-coneeldtheor yistherepresentation ofthePoincaregroupISO(D 1,1)(see,e.g.,[2.2]).Inthenextsectionwewill derivethisrepresentationforarbitrarymasslesstheorie s(andwilllaterextendit tothemassivecase)[2.3].Theserepresentationsarenonli nearinthecoordinates, andareconstructedfromalltheirreducible(matrix)repre sentationsofthelightcone'sSO(D 2)rotationsubgroupofthespinpartoftheSO(D 1,1)Lorentzgroup. Onesimplemethodofderivationinvolvestheuseoftheconfo rmalgroup,whichis SO(D,2)forD-dimensionalspacetime(for D> 2).WethereforeuseSO(D,2)notation bywriting(D+2)-dimensionalvectorindiceswhichtakethe values aswellasthe usualD a 's: A =( ;a ).Themetricisasin(2.1.1)forthe indices.(These 's shouldnotbeconfusedwiththelight-coneindices ,whicharerelatedbutarea subsetofthe a 's.)Wethenwritetheconformalgroupgeneratorsas J AB =( J + a = ip a ;J a = iK a ;J + = ;J ab ) ; (2 : 2 : 1) where J ab aretheLorentzgenerators,isthedilatationgenerator,a nd K a are theconformalboosts.Anobviouslinearcoordinatereprese ntationintermsofD+2 coordinatesis J AB = x [ A @ B ] + M AB ; (2 : 2 : 2) where[]meansantisymmetrizationand M AB istheintrinsic(matrix,orcoordinateindependent)part(withthesamecommutationrelationstha tfollowdirectlyforthe orbitalpart).TheusualrepresentationintermsofDcoordi natesisobtainedby imposingtheSO(D,2)-covariantconstraints x A x A = x A @ A = M A B x B + d x A =0(2 : 2 : 3 a ) forsomeconstant d (thecanonicaldimension,orscaleweight).Corresponding to theseconstraints,whichcanbesolvedforeverythingwitha \ "index,arethe \gaugeconditions"whichdetermineeverythingwitha\+"in dexbutno\ "index: @ + = x + 1= M + a =0 : (2 : 2 : 3 b )

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2.2.Conformalalgebra 11 Thisgaugecanbeobtainedbyaunitarytransformation.Thes olutionto(2.2.3)is then J + a = @ a ;J a = 1 2 x b 2 @ a + x a x b @ b + M a b x b + d x a ; J + = x a @ a + d ;J ab = x [ a @ b ] + M ab : (2 : 2 : 4) Thisrealizationcanalsobeobtainedbytheusualcosetspac emethods(see,e.g., [2.4]),forthespaceSO(D,2)/ISO(D-1,1) n GL(1).Thesubgroupcorrespondstoallthe generators except J + a .Onewaytoperformthisconstructionis:Firstassigntheco set spacegenerators J + a tobepartialderivatives @ a (sincetheyallcommute,according tothecommutationrelationswhichfollowfrom(2.2.2)).We nextequatethisrstquantizedcoordinaterepresentationwithasecond-quanti zedeldrepresentation:In general, 0= D x E = D Jx E + D x ^ J E J D x E = D Jx E = ^ J D x E = D x ^ J E ; (2 : 2 : 5) where J (whichactsdirectlyon h x j )isexpressedintermsofthecoordinatesandtheir derivatives(plus\spin"pieces),while ^ J (whichactsdirectlyon j i )isexpressedin termsofthe elds andtheir functional derivatives.Theminussignexpressesthe usualrelationbetweenactiveandpassivetransformations .Thestructureconstants ofthesecond-quantizedalgebrahavethesamesignasthers t-quantizedones.We canthensolvethe\constraint" J + a = ^ J + a on h x j i as D x E ( x )= U (0)= e x a ^ J + a (0) : (2 : 2 : 6) Theothergeneratorscanthenbedeterminedbyevaluating J ( x )= ^ J ( x ) U 1 JU (0)= U 1 ^ JU (0) : (2 : 2 : 7) Ontheleft-handside,theunitarytransformationreplaces any @ a witha ^ J + a (the @ a itselfgettingkilledbythe(0)).Ontheright-handside,t hetransformationgives termswith x dependenceandother ^ J 's(asdeterminedbythecommutatoralgebra). (Thecalculationsareperformedbyexpressingthetransfor mationasasumofmultiple commutators,whichinthiscasehasanitenumberofterms.) Thenetresultis (2.2.4),where d is ^ J + on(0), M ab is ^ J ab ,and J a canhavetheadditionalterm ^ J a .However, ^ J a on(0)canbesettozeroconsistentlyin(2.2.4),anddoes vanishforphysicallyinterestingrepresentations. Fromnowon,weuse asinthelight-conenotation,notSO(D,2)notation.

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12 2.GENERALLIGHTCONE Theconformalequationsofmotionareallthosewhichcanbeo btainedfrom p a 2 =0byconformaltransformations(or,equivalently,theirr educibletensoroperatorquadraticinconformalgeneratorswhichincludes p 2 asacomponent).Since conformaltheoriesareasubsetofmasslessones,themassle ssequationsofmotionare asubsetoftheconformalones(i.e.,themasslesstheoriess atisfyfewerconstraints). Inparticular,sincemasslesstheoriesarescaleinvariant butnotalwaysinvariantunderconformalboosts,theequationswhichcontainthegener atorsofconformalboosts mustbedropped. Thecompletesetofequationsofmotionforanarbitrarymass lessrepresentation ofthePoincaregrouparethusobtainedsimplybyperformin gaconformalbooston thedeningequation, p 2 =0[2.5,6]: 0= 1 2 [ K a ;p 2 ]= 1 2 f J a b ;p b g + 1 2 f ;p a g = M a b p b + d D 2 2 p a : (2 : 2 : 8) d isdeterminedbytherequirementthattherepresentationbe nontrivial(forother valuesof d thisequationimplies p =0).Fornonzerospin( M ab 6 =0)thisequation implies p 2 =0byitself.Forexample,forscalarstheequationimplieso nly d = ( D 2) = 2.ForaDiracspinor, M ab = 1 4 [ r a ;r b ]implies d =( D 1) = 2andtheDirac equation(intheform r a r p =0).Forasecond-rankantisymmetrictensor,we nd d = D= 2andMaxwell'sequations.Inthiscovariantapproachtosol vingthese equations,allthesolutionsareintermsofeldstrengths, notgaugeelds(sincethe latterarenotunitaryrepresentations).Wecansolvethese equationsinlight-cone notation :Choosingareferenceframewheretheonlynonvanishingcom ponentofthe momentumis p + ,(2.2.8)reducestotheequations M i =0and M + = d ( D 2) = 2. Theequation M i =0saysthattheonlynonvanishingcomponentsaretheoneswi th asmany(lower)\+"indicesaspossible(andforspinors,pro jectwith r + ),andno \ "indices.IntermsofYoungtableaux,thismeans1\+"foreac hcolumn. M + thenjustcountsthenumberof\+"'s(plus1/2fora r + -projectedspinorindex),so wendthat d ( D 2) = 2=thenumberofcolumns(+1/2foraspinor).Wealso ndthatthe on-shell gaugeeldistherepresentationfoundbysubtractingonebo x fromeachcolumnoftheYoungtableau,andintheeldstrengt hthosesubtracted indicesareassociatedwithfactorsofmomentum. Theseresultsformasslessrepresentationscanbeextended tomassiverepresentationsbythestandardtrickofaddingonespatialdimensio nandconstrainingthe extramomentumcomponenttobethemass(operator):Writing a ( a;m ) ;p m = M; (2 : 2 : 9)

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2.3.Poincarealgebra 13 wheretheindex m takesonevalue, p 2 =0becomes p 2 + M 2 =0,and(2.2.8)becomes M a b p b + M am M + d D 2 2 p a =0 : (2 : 2 : 10) Theelds(orstates)arenowrepresentationsofanSO(D,1)s pingroupgenerated by M ab and M am (insteadoftheusualSO(D-1,1)ofjust M ab forthemasslesscase). Theeldsadditionaltothoseobtainedinthemasslesscase( on-shelleldstrengths) correspondtotheon-shellgaugeeldsinthemasslesslimit ,resultinginarst-order formalism.Forexample,forspin1theadditionaleldisthe usualvector.Forspin 2,theextraeldscorrespondtotheon-shell,andthustrace less,partsoftheLorentz connectionandmetrictensor. Foreldtheory,we'llbeinterestedinrealrepresentation s.Forthemassivecase, since(2.2.9)forcesustoworkinmomentumspacewithrespec tto p m ,thereality conditionshouldincludeanextrafactorofthererectionop eratorwhichreversesthe \ m "direction.Forexample,fortensorelds,thosecomponent swithanoddnumber of m indicesshouldbeimaginary(andthosewithanevennumberre al). Inchapt.4we'llshowhowtoobtaintheo-shellelds,andth usthetraceparts, byworkingdirectlyintermsofthegaugeelds.Themethodis basedonthelight-cone representationofthePoincarealgebradiscussedinthene xtsection. 2.3.Poincarealgebra Incontrasttotheabovecovariantapproachtosolving(2.2. 8,10),wenowconsider solvingtheminunitarygauges(suchasthelight-conegauge ),sinceinsuchgauges thegaugeeldsareessentiallyeldstrengthsanywaybecau sethegaugehasbeen xed:e.g.,forYang-Mills A a = r + 1 F + a ,since A + =0.Insuchgaugeswework intermsofonlythephysicaldegreesoffreedom(asinthecas eoftheon-shelleld strengths),whichsatisfy p 2 =0(unliketheauxiliarydegreesoffreedom,whichsatisfy algebraicequations,andthegaugedegreesoffreedom,whic hdon'tappearinanyeld equations). Inthelight-coneformalism,theobjectistoconstructallt hePoincaregenerators fromjustthemanifestonesofthe( D 2)-dimensionalPoincaresubgroup, p + ,and thecoordinatesconjugatetothesemomenta.Thelight-cone gaugeisimposedbythe condition M + i =0 ; (2 : 3 : 1)

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14 2.GENERALLIGHTCONE which,whenactingontheindependentelds(thosewithonly i indices),saysthat alleldswith+indiceshavebeensettovanish.Theeldswit h indices(auxiliary elds)arethendeterminedasusualbytheeldequations:by solving(2.2.8)for M i Thesolutiontothe i ,+,and partsof(2.2.8)gives M i = 1 p + ( M i j p j + kp i ) ; M + = d D 2 2 k; kp 2 =0 : (2 : 3 : 2) If(2.2.8)issolvedwithoutthecondition(2.3.1),then M + i canstillberemoved(and (2.3.2)regained)byaunitarytransformation.(Inarst-q uantizedformalism,this correspondstoagaugechoice:seesect.5.3forspin1/2.)Th eappearanceof k is relatedtoorderingambiguities,andwecanalsochoose M + =0bya non unitary transformation(arescalingoftheeldbyapowerof p + ).Ofcourse,wealsosolve p 2 =0as p = p i 2 2 p + : (2 : 3 : 3) Theseequations,togetherwiththegaugeconditionfor M + i ,determineallthePoincare generatorsintermsof M ij p i p + x i ,and x .Intheorbitalpiecesof J ab x + canbe settovanish,since p isnolongerconjugate:i.e.,weworkat\time" x + =0forthe \hamiltonian" p ,orequivalentlyintheSchrodingerpicture.(Ofcourse,t hisalso correspondstoremoving x + byaunitarytransformation,i.e.,atimetranslationvia p .Thisisalsoagaugechoiceinarst-quantizedformalism:s eesect.5.1.)The nalresultis p i = i@ i ;p + = i@ + ;p = p i 2 2 p + ; J ij = ix [ i p j ] + M ij ;J + i = ix i p + ;J + = ix p + + k; J i = ix p i ix i p j 2 2 p + + 1 p + ( M i j p j + kp i ) : (2 : 3 : 4) Thegeneratorsare(anti)hermitianforthechoice k = 1 2 ;otherwise,theHilbertspace metricmustincludeafactorof p + 1 2 k ,withrespecttowhichallthegeneratorsare pseudo(anti)hermitian.Inthislight-coneapproachtoPoi ncarerepresentations,where weworkwiththefundamentaleldsratherthaneldstrength s, k =0forbosonsand 1 2 forfermions(givingtheusualdimensions d = 1 2 ( D 2)forbosonsand 1 2 ( D 1)for fermions),andthusthemetricis p + forbosonsand1forfermions,sothelight-cone kineticoperator(metric) 2( i@ p ) 2 forbosonsand 2 =p + forfermions.

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2.3.Poincarealgebra 15 ThisconstructionoftheD-dimensionalPoincarealgebrai ntermsofD 1coordinatesisanalogoustotheconstructionintheprevioussec tionoftheD-dimensional conformalalgebraSO(D,2)intermsof D coordinates,exceptthatintheconformal case(1)westartwithD+2coordinatesinsteadofD,(2) x 'sand p 'sareswitched, and(3)thefurtherconstraint x p =0andgaugecondition x + =1areused.Thus, J ab of(2.3.4)becomes J AB of(2.2.4)if x isreplacedwith (1 =p + ) x j p j p + isset to1,andwethenswitch p x;x p .Justastheconformalrepresentation (2.2.4)canbeobtainedfromthePoincarerepresentation( in2extradimensions,by i a )(2.3.4)byeliminatingonecoordinate( x ),(2.3.4)canbereobtainedfrom (2.2.4)byreintroducingthiscoordinate:Firstchoose d = ix p + + k .Thenswitch x i p i p i x i .Finally,makethe(almostunitary)transformationgenera tedby exp [ ip i x i ( lnp + )],whichtakes x i p + x i p i p i =p + x x + p i x i =p + Toextendtheseresultstoarbitraryrepresentations,weus ethetrick(2.2.9),or directlysolve(2.2.10),givingthelight-coneformoftheP oincarealgebraforarbitrary representations:(2.3.4)becomes p i = i@ i ;p + = i@ + ;p = p i 2 + M 2 2 p + ; J ij = ix [ i p j ] + M ij ;J + i = ix i p + ;J + = ix p + + k; J i = ix p i ix i p j 2 + M 2 2 p + + 1 p + ( M i j p j + M im M + kp i ) : (2 : 3 : 5) Thus,masslessirreduciblerepresentationsofthePoincar egroupISO(D 1,1)areirreduciblerepresentationsofthespinsubgroupSO(D 2)(generatedby M ij )which alsodependonthecoordinates( x i ;x ),andirreduciblemassiveonesareirreducible representationsofthespinsubgroupSO(D 1)(generatedby( M ij ;M im ))forsome nonvanishingconstant M .Noticethattheintroductionofmasseshasmodiedonly p and J i .Thesearealsotheonlygeneratorsmodiedwheninteractio nsareintroduced,wheretheybecomenonlinearintheelds. Thelight-conerepresentationofthePoincarealgebrawil lbeusedinsect.3.4 toderiveBRSTalgebras,usedforenforcingunitarityincov ariantformalisms,which inturnwillbeusedextensivelytoderivegauge-invarianta ctionsforparticlesand stringsinthefollowingchapters.Thegenerallight-conea nalysisofthissectionwill beappliedtothespecialcaseofthefreestringinchapt.7.

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16 2.GENERALLIGHTCONE 2.4.Interactions Forinteractingtheories,thederivationofthePoincarea lgebraisnotsogeneral, butdependsonthedetailsoftheparticulartypeofinteract ionsinthetheory.We againconsiderthecaseofYang-Mills.Sinceonly p and J i obtaininteracting contributions,weconsiderthederivationofonlythoseope rators.Theexpressionfor p A i isthengivendirectlybytheeldequationof A i 0=[ r a ;F ai ]=[ r j ;F ji ]+[ r + ;F i ]+[ r ;F + i ]=[ r j ;F ji ]+2[ r + ;F i ]+[ r i ;F + ] p A i =[ r i ;A ] 1 2 p + [ r j ;F ji ]+[ r i ;p + A ] ; (2 : 4 : 1) wherewehaveusedtheBianchiidentity[ r [+ ;F i ] ]=0.Thisexpressionfor p is alsousedintheorbitalpieceof J i A j .Inthespinpiece M i westartwiththe covariant-formalismequation M i A j = ij A ,substitutethesolutionto A 'seld equation,andthenaddagaugetransformationtocancelthec hangeofgaugeinduced bythecovariant-formalismtransformation M i A + = A i .Thenetresultisthatin thelight-coneformalism J i A j = i ( x p i x i p ) A j ij A +[ r j ; 1 p + A i ] ; (2 : 4 : 2) with A givenby(2.1.7)and p A j by(2.4.1).Intheabeliancase,theseexpressions agreewiththoseobtainedbyadierentmethodin(2.3.4).Al ltransformationscan thenbewritteninfunctionalsecond-quantizedformas = Z d D 2 x i dx tr ( A i ) A i [ ;A i ]= ( A i ) : (2 : 4 : 3) Theminussignisasin(2.2.5)forrelatingrst-andsecondquantizedoperators. Asanalternative,wecanconsidercanonicalsecond-quanti zation,whichhascertainadvantagesinthelightcone,andhasaninterestinggen eralizationinthecovariant case(seesect.3.4).Fromthelight-conelagrangian L = i Z y p + H () ; (2 : 4 : 4) where isthe\time"-derivative i@=@x + ,wendthattheeldshaveequal-time commutatorssimilartothoseinnonrelativisticeldtheor y: [ y (1) ; (2)]= 1 2 p +2 (2 1) ; (2 : 4 : 5)

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2.4.Interactions 17 wherethe -functionisoverthetransversecoordinatesand x (andmayincludea Kronecker inindices,ifhascomponents).Unlikenonrelativistice ldtheory,the eldssatisfyarealitycondition,incoordinatespace: =n ; (2 : 4 : 6) wherenistheidentityorsomesymmetric,unitarymatrix(th e\chargeconjugation" matrix; hereisthehermitianconjugate,oradjoint,intheoperator sense,i.e.,unlike y ,itexcludesmatrixtransposition).Asinquantummechanic s(orthePoissonbracket approachtoclassicalmechanics),thegeneratorscanthenb ewrittenasfunctionsof thedynamicalvariables: V = X n 1 n Z dz 1 dz n V ( n ) ( z 1 ;:::;z n )( z 1 ) ( z n ) ; (2 : 4 : 7) wherethearguments z standforeithercoordinatesormomentaandthe V 'sarethe vertexfunctions,whicharejustfunctionsofthecoordinat es(notoperators).Without lossofgeneralitytheycanbechosentobecyclicallysymmet ricintheelds(ortotally symmetric,ifgroup-theoryindicesarealsopermuted).(An yasymmetricpiececan beseentocontributetoalower-pointfunctionbytheuseof( 2.4.5,6).)Inlight-cone theoriesthecoordinate-spaceintegralsareoverallcoord inatesexcept x + .Theaction ofthesecond-quantizedoperator V oneldsiscalculatedusing(2.4.5): [ V; ( z 1 ) y ]= 1 2 p +1 X n 1 ( n 1)! Z dz 2 dz n V ( n ) ( z 1 ;:::;z n )( z 2 ) ( z n ) : (2 : 4 : 8) Aparticularcaseoftheaboveequationsisthefreecase,whe retheoperator V is quadraticin.Wewillthengenerallywritethesecond-quan tizedoperator V in termsofarst-quantizedoperator V withasingleintegration: V = Z dz y p + V [ V; ]= V : (2 : 4 : 9) Thiscanbecheckedtorelateto(2.4.7)as V (2) ( z 1 ;z 2 )=2n 1 p +1 V 1 (2 1)(with thesymmetryof V (2) imposingcorrespondingconditionsontheoperator V ).Inthe interactingcase,thegeneralizationof(2.4.9)is V = 1 N Z dz y 2 p + ( V ) ; (2 : 4 : 10) where N isjustthenumberofeldsinanyparticularterm.(Inthefre ecase N =2, giving(2.4.9).)

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18 2.GENERALLIGHTCONE Forexample,forYang-Mills,wend p = Z 1 4 ( F ij ) 2 + 1 2 ( p + A ) 2 ; (2 : 4 : 11 a ) J i = Z ix ( p + A j )( p i A j )+ ix i h 1 4 ( F jk ) 2 + 1 2 ( p + A ) 2 i A i p + A : (2 : 4 : 11 b ) (Theothergeneratorsfollowtriviallyfrom(2.4.9).) p isminusthehamiltonian H (asinthefreecase(2.1.2,4,9)),asalsofollowsfromperfo rmingtheusualLegendre transformationonthelagrangian. Ingeneral,alltheexplicit x i -dependenceofallthePoincaregeneratorscanbedeterminedfromthecommutationrelationswiththemomenta(t ranslationgenerators) p i .Furthermore,sinceonly p and J i getcontributionsfrominteractions,weneed consideronlythose.Let'srstconsiderthe\hamiltonian" p .Sinceitcommutes with p i ,itistranslationinvariant.Intermsofthevertexfunctio ns,thistranslates intothecondition: ( p 1 + + p n ) e V ( n ) ( p 1 ;:::;p n )=0 ; (2 : 4 : 12) wherethe f indicatesFouriertransformationwithrespecttothecoord inate-space expression,implyingthatmostgenerally e V ( n ) ( p 1 ;:::;p n )= ~ f ( p 1 ;:::;p n 1 ) ( p 1 + + p n ) ; (2 : 4 : 13) orincoordinatespace V ( n ) ( x 1 ;:::;x n )= ~ f i @ @x 1 ;:::;i @ @x n 1 ( x 1 x n ) ( x n 1 x n ) = f ( x 1 x n ;:::;x n 1 x n ) : (2 : 4 : 14) Inthiscoordinaterepresentationonecanseethatwhen V isinsertedbackin(2.4.7) wehavetheusualexpressionforatranslation-invariantve rtexusedineldtheory. Namely,eldsatthesamepointincoordinatespace,withder ivativesactingonthem, aremultipliedandintegratedovercoordinatespace.Inthi sformitisclearthatthere isnoexplicitcoordinatedependenceinthevertex.Ascanbe seenin(2.4.14),themost generaltranslationallyinvariantvertexinvolvesanarbi traryfunctionofcoordinate dierences,denotedas f above.Forthecaseofbosoniccoordinates,thefunction ~ f maycontaininversederivatives(thatis,translationalin variancedoesnotimply locality.)Forthecaseofanticommutingcoordinates(sees ect.2.6)thesituationis simpler:Thereisnolocalityissue,sincethemostgeneralf unction f canalwaysbe obtainedfromafunction ~ f polynomialinderivatives,actingon -functions.

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2.5.Graphs 19 Wenowconsider J i .Fromthecommutationrelationswend: [ p i ;J j g = ij p [ J i ; ]= ix i [ p ; ]+[ J i ; ] ; (2 : 4 : 15) where J i istranslationallyinvariant(commuteswith p i ),andcanthereforebe representedwithoutexplicit x i 's.FortheYang-Millscase,thiscanbeseentoagree with(2.4.2)or(2.4.11). Thislight-coneanalysiswillbeappliedtointeractingstr ingsinchapt.10. 2.5.Graphs Feynmangraphsforanyinteractinglight-coneeldtheoryc anbederivedasin covarianteldtheory,butanalternativenotavailablethe reistouseanonrelativistic styleofperturbation(i.e.,justexpanding e iHt in H INT ),sincetheeldequationsare nowlinearinthetimederivative p = i@=@x + = i@=@ .(Asinsect.2.1,butunlike sects.2.3and2.4,wenowuse p torefertothispartialderivative,asincovariant formalisms,while H referstothecorrespondinglight-conePoincaregenerato r,the twobeingequalonshell.)Thisformalismcanbederivedstra ightforwardlyfromthe usualFeynmanrules(afterchoosingthelight-conegaugean deliminatingauxiliary elds)bysimplyFouriertransformingfrom p to x + = (butkeepingallother momenta): Z 1 1 dp 2 e ip 1 2 p + p + p i 2 + m 2 + i = i ( p + ) 1 2 j p + j e i ( p i 2 + m 2 ) = 2 p + : (2 : 5 : 1) (( u )=1for u> 1,0for u< 1.)Wenowdrawallgraphstorepresentthe coordinate,sothatgraphswithdierent -orderingsoftheverticesmustbeconsidered asseparatecontributions.Thenwedirectallthepropagato rstowardincreasing ,so thechangein betweentheendsofthepropagator(asappearsin(2.5.1))is always positive(i.e.,theorientationofthemomentaisdenedtob etowardincreasing ). WenextWickrotate i .Wealsointroduceexternallinefactorswhichtransform H backto p onexternallines.Theresultingrulesare: (a)Assigna toeachvertex,andorderthemwithrespectto (b)Assign( p ;p + ;p i )toeachexternalline,butonly( p + ;p i )toeachinternalline,all directedtowardincreasing .Enforceconservationof( p + ;p i )ateachvertex,and totalconservationof p (c)Giveeachinternallineapropagator ( p + ) 1 2 p + e ( p i 2 + m 2 ) = 2 p +

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20 2.GENERALLIGHTCONE forthe( p + ;p i )ofthatlineandthepositivedierence inthepropertimebetween theends. (d)Giveeachexternallineafactor e p forthe p ofthatlineandthe ofthevertextowhichitconnects. (e)Readotheverticesfromtheactionasusual. (f)Integrate Z 1 0 d foreach dierencebetweenconsecutive(thoughnotnecessarilycon nected)vertices.(Performingjustthisintegrationgivestheusualol d-fashionedperturbation theoryintermsofenergydenominators[2.1],exceptthatou rexternal-linefactors dieroshellinordertoreproducetheusualFeynmanrules. ) (g)Integrate Z 1 1 dp + d D 2 p i (2 ) D 1 foreachloop.Theuseofsuchmethodsforstringswillbediscussedinchapt .10. 2.6.Covariantizedlightcone Thereisacovariantformalismforanyeldtheorythathasth einterestingpropertythatitcanbeobtaineddirectlyandeasilyfromtheligh t-coneformalism,without anyadditionalgauge-xingprocedure[2.7].Althoughthis covariantgaugeisnotas generalorconvenientastheusualcovariantgauges(inpart icular,itsometimeshas additionalo-shellinfrareddivergences),itbearsstron grelationshiptoboththelightconeandBRSTformalisms,andcanbeusedasaconceptualbrid ge.Thebasicidea oftheformalismis:Consideracovarianttheoryin D dimensions.Thisisequivalent toacovarianttheoryin( D +2) 2dimensions,wherethenotationindicatestheadditionof2extracommutingcoordinates(1space,1time)and 2(real)anticommuting coordinates,withasimilarextensionofLorentzindices[2 .8].(AsimilaruseofOSp groupsingauge-xedtheories,butappliedtoonlytheLoren tzindicesandnotthecoordinates,appearsin[2.9].)ThisextendsthePoincaregr oupISO(D 1,1)toagraded analogIOSp(D,2 j 2).Inpractice,thismeanswejusttakethelight-conetrans verseindicestobegraded,watchingoutforsignsintroducedbythec orrespondingchangein

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2.6.Covariantizedlightcone 21 statistics,andreplacetheEuclideanSO(D-2)metricwitht hecorrespondinggraded OSp(D-1,1 j 2)metric: i =( a; ) ; ij ij =( ab ;C ) ; (2 : 6 : 1) where ab istheusualLorentzmetricand C = C = 2 (2 : 6 : 2) istheSp(2)metric,whichsatisestheusefulidentity C C r = [ r ] A [ B ] = C C r A r B : (2 : 6 : 3) TheOSpmetricisusedtoraiseandlowergradedindicesas: x i = ij x j ;x i = x j ji ; ik jk = j i : (2 : 6 : 4) Thesignconventionsarethatadjacentindicesarecontract edwiththecontravariant (up)indexrst.Theequivalencefollowsfromthefactthat, formomentum-space Feynmangraphs,thetreeswillbethesameifweconstrainthe 2 2extra\ghost" momentatovanishonexternallines(sincethey'llthenvani shoninternallinesby momentumconservation);andtheloopsarethenthesamebeca use,whenthemomentumintegrandsarewrittenasgaussians,thedeterminan tfactorscomingfromthe 2extraanticommutingdimensionsexactlycancelthosefrom the2extracommuting ones.Forexample,usingtheproper-timeform(\Schwingerp arametrization")ofthe propagators(cf.(2.5.1)), 1 p 2 + m 2 = Z 1 0 de ( p 2 + m 2 ) ; (2 : 6 : 5) allmomentumintegrationstaketheform 1 Z d D +2 pd 2 p e f (2 p + p + p a p a + p p + m 2 ) = Z d D pe f ( p a p a + m 2 ) = f D= 2 e fm 2 ; (2.6.6) where f isafunctionoftheproper-timeparameters. Thecovarianttheoryisthusobtainedfromthelight-coneon ebythesubstitution ( p ;p + ; p i ) ( p ;p + ; p a ;p ) ; (2 : 6 : 7 a )

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22 2.GENERALLIGHTCONE where p = p =0(2 : 6 : 7 b ) onphysicalstates.It'snotnecessarytoset p + =0,sinceitonlyappearsinthecombination p p + inOSp(D,2 j 2)-invariantproducts.Thus, p + canbechosenarbitrarilyon externallines(butshouldbenonvanishingduetotheappear anceoffactorsof1 =p + ). Wenowinterpret x and x astheunphysicalcoordinates.Vectorindicesonelds aretreatedsimilarly:Havingbeenreducedtotransverseon esbythelight-coneformalism,theynowbecomecovariantvectorindiceswith2addi tionalanticommuting values((2.6.1)).Forexample,inYang-Millsthevectorel dbecomestheusualvector eldplustwoanticommutingscalars A ,correspondingtoFaddeev-Popovghosts. Thegraphicalrulesbecome: (a)Assigna toeachvertex,andorderthemwithrespectto (b)Assign( p + ;p a )toeachexternalline,but( p + ;p a ;p )toeachinternalline,all directedtowardincreasing .Enforceconservationof( p + ;p a ;p )ateachvertex (with p =0onexternallines). (c)Giveeachinternallineapropagator ( p + ) 1 2 p + e ( p a 2 + p p + m 2 ) = 2 p + forthe( p + ;p a ;p )ofthatlineandthepositivedierence inthepropertimebetween theends. (d)Giveeachexternallineafactor 1 : (e)Readotheverticesfromtheactionasusual. (f)Integrate Z 1 0 d foreach dierencebetweenconsecutive(thoughnotnecessarilycon nected)vertices. (g)Integrate Z d 2 p

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Exercises 23 foreachloop(rememberingthatforanyanticommutingvaria ble R d 1=0, R d =1, 2 =0). (h)Integrate 2 Z 1 1 dp + foreachloop. (i)Integrate Z d D p (2 ) D foreachloop.Fortheorieswithonlyscalars,integratingjust(f-h)give stheusualFeynman graphs(althoughitmaybenecessarytoaddseveralgraphsdu etothe -orderingof non-adjacentvertices).Besidesthecorrespondenceofthe parameterstotheusual Schwingerparameters,afterintegratingoutjusttheantic ommutingparametersthe p + parametersresembleFeynmanparameters. Thesemethodscanalsobeappliedtostrings(chapt.10). Exercises (1)Findthelight-coneformulationofQED.Comparewiththe Coulombgaugeformulation. (2)Derivethecommutationrelationsoftheconformalgroup from(2.2.2).Check that(2.2.4)satisesthem.Evaluatethecommutatorsimpli citin(2.2.7)foreach generator. (3)FindtheLorentztransformation M ab ofavector(consistentwiththeconventions of(2.2.2)).(Hint:Lookatthetransformationsof x and p .)Findtheexplicit formof(2.2.8)forthatcase.Solvetheseequationsofmotio n.Towhatsimpler representationisthisequivalent?Studythisequivalence withthelight-coneanalysisgivenbelow(2.2.8).Generalizetheanalysistototall yantisymmetrictensors ofarbitraryrank. (4)Repeatproblem(3)forthemassivecase.Lookingatthese parateSO(D-1,1) representationscontainedintheSO(D,1)representations ,showthatrst-order formalismsintermsoftheusualeldshavebeenobtained,an dndthecorrespondingsecond-orderformulations.

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24 2.GENERALLIGHTCONE (5)CheckthattheexplicitformsofthePoincaregenerator sgivenin(2.3.5)satisfy thecorrectalgebra(seeproblem(2)).Findtheexplicittra nsformationsacting onthevectorrepresentationofthespingroupSO(D-1).Comp arewith(2.4.1-2). (6)Derive(2.4.11).Comparethat p withthelight-conehamiltonianwhichfollows from(2.1.5). (7)Calculatethe4-pointamplitudein 3 theorywithlight-conegraphs,andcomparewiththeusualcovariantFeynmangraphcalculation.Ca lculatethe1-loop propagatorcorrectioninthesametheoryusingthe covariantized light-conerules, andagaincomparewithordinaryFeynmangraphs,payingspec ialattentionto Feynmanparameters.

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3.1.Gaugeinvarianceandconstraints 25 3.GENERALBRST3.1.Gaugeinvarianceandconstraints Inthepreviouschapterwesawthatagaugetheorycanbedescr ibedeitherina manifestlycovariantwaybyusinggaugedegreesoffreedom, orinamanifestlyunitary way(withonlyphysicaldegreesoffreedom)withPoincaret ransformationswhichare nonlinear(inbothcoordinatesandelds).Inthegauge-cov ariantformalismthereisa D -dimensionalmanifestLorentzcovariance,andinthelight -coneformalisma D 2dimensionalone,andineachcaseacorrespondingnumberofd egreesoffreedom. Thereisalsoanintermediateformalism,morefamiliarfrom nonrelativistictheory: Thehamiltonianformalismhasa D 1-dimensionalmanifestLorentzcovariance(rotations).Asinthelight-coneformalism,thenotationalse parationofcoordinates intotimeandspacesuggestsaparticulartypeofgaugecondi tion:temporal(timelike)gauges,wheretime-componentsofgaugeeldsaresett ovanish.Inchapt.5, thisformalismwillbeseentohaveaparticularadvantagefo rrst-quantizationof relativistictheories:Intheclassicalmechanicsofrelat ivistictheories,thecoordinates aretreatedasfunctionsofa\propertime"sothattheusualt imecoordinatecanbe treatedonanequalfootingwiththespacecoordinates.Thus ,canonicalquantization withrespecttothisunobservable(proper)\time"coordina tedoesn'tdestroymanifest Poincarecovariance,souseofahamiltonianformalismcan beadvantageous,particularlyinderivingBRSTtransformations,andthecorrespon dingsecond-quantized theory,wheretheproper-timedoesn'tappearanyway. We'llrstconsiderYang-Mills,andthengeneralizetoarbi trarygaugetheories. Inordertostudythetemporalgauge,insteadofthedecompos ition(2.1.1)wesimply separateintotimeandspatialcomponents a =(0 ;i ) ;A B = A 0 B 0 + A i B i : (3 : 1 : 1) Thelagrangian(2.1.5)isthen L = 1 4 F ij 2 1 2 ( p 0 A i [ r i ;A 0 ]) 2 : (3 : 1 : 2)

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26 3.GENERALBRST Thegaugecondition A 0 =0(3 : 1 : 3) transformsunderagaugetransformationwithatimederivat ive:Underaninnitesimaltransformationabout A 0 =0, A 0 @ 0 ; (3 : 1 : 4) sotheFaddeev-Popovghostsarepropagating.Furthermore, thegaugetransformation (3.1.4)doesnotallowthegaugechoice(3.1.3)everywhere: Forexample,ifwechoose periodicboundaryconditionsintime(tosimplifytheargum ent),then Z 1 1 dx 0 A 0 0 : (3 : 1 : 5) A 0 canthenbexedbyanappropriateinitialcondition,e.g., A 0 j x 0 =0 =0,butthen thecorrespondingeldequationislost.Therefore,wemust impose 0= S A 0 = [ r i ;F 0 i ]= [ r i ;p 0 A i ] atx 0 =0(3 : 1 : 6) asaninitialcondition.Anotherwaytounderstandthisisto notethatgaugexing eliminatesonlydegreesoffreedomwhichdon'toccurinthel agrangian,andthus caneliminateonlyredundantequationsofmotion:Since[ r i ;F 0 i ]=0followedfrom thegauge-invariantaction,thefactthatitdoesn'tfollow aftersetting A 0 =0means somepieceof A 0 can'ttrulybegaugedaway,andsowemustcompensatebyimpos ing theequationofmotionforthatpiece.Duetotheoriginalgau geinvariance,(3.1.6) thenholdsforalltimefromtheremainingeldequations:In thegauge(3.1.3),the lagrangian(3.1.2)becomes L = 1 2 A i 2 A i 1 2 ( p i A i ) 2 +[ A i ;A j ] p i A j + 1 4 [ A i ;A j ] 2 ; (3 : 1 : 7) andthecovariantdivergenceoftheimpliedeldequationsy ieldsthe timederivative of(3.1.6).(Thisfollowsfromtheidentity[ r b ; [ r a ;F ab ]]=0uponapplyingthe eldequations[ r a ;F ia ]=0.Inunitarygauges,thecorrespondingconstraintcanbe derivedwithouttimederivatives,andhenceisimpliedbyth eremainingeldequations undersuitableboundaryconditions.)Equivalently,ifwen oticethat(3.1.4)doesnot xthegaugecompletely,butleavestime-independentgauge transformations,weneed toimposeaconstraintontheinitialstatestomakethemgaug einvariant.Butthe generatoroftheresidualgaugetransformationsontherema iningelds A i is G ( x i )= r i ;i A i # ; (3 : 1 : 8)

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3.1.Gaugeinvarianceandconstraints 27 whichisthesameastheconstraint(3.1.6)undercanonicalq uantizationof(3.1.7). Thus,thesameoperator(1)givestheconstraintwhichmustb eimposedinadditionto theeldequationsbecausetoomuchof A 0 wasdropped,and(2)(itstranspose)gives thegaugetransformationsremainingbecausetheylefttheg auge-xingfunction A 0 invariant.Thefactthattheseareidenticalisnotsurprisi ng,sinceinFaddeev-Popov quantizationthelattercorrespondstotheFaddeev-Popovg hostwhiletheformer correspondstotheantighost. Thesepropertiesappearverynaturallyinahamiltonianfor mulation:Westart againwiththegauge-invariantlagrangian(3.1.2).Since A 0 hasnotime-derivative terms,weLegendretransformwithrespecttojust A i .Theresultis S H = 1 g 2 Z d D xtr L H ; L H = A i i H ; H = H 0 + A 0 i G ; H 0 = 1 2 i 2 1 4 F ij 2 ; G =[ r i ; i ] ; (3 : 1 : 9) where = @ 0 .Asinordinarynonrelativisticclassicalmechanics,elim inatingthe momentum i fromthehamiltonianformoftheaction(rstorderintimede rivatives)byitsequationofmotiongivesbackthelagrangianfo rm(secondorderintime derivatives).Notethat A 0 appearslinearly,asaLagrangemultiplier. Thegauge-invarianthamiltonianformalismof(3.1.9)canb egeneralized[3.1]: Consideralagrangianoftheform L H = z M e M A ( z ) A H ; H = H 0 ( z; )+ i i G i ( z; ) ; (3 : 1 : 10) where z ,and arethevariables,representing\coordinates,"covariant \momenta," andLagrangemultipliers,respectively.Theydependonthe time,andalsohave indices(whichmayincludecontinuousindices,suchasspat ialcoordinates). e ,which isafunctionof z ,hasbeenintroducedtoallowforcaseswithasymmetry(such assupersymmetry)underwhich dz M e M A (butnot dz itself)iscovariant,sothat willbecovariant,andthusamoreconvenientvariableinter msofwhichtoexpress theconstraints G .When H 0 commuteswith G (quantummechanically,orinterms ofPoissonbracketsforaclassicaltreatment),thisaction hasagaugeinvariance generatedby G ,forwhich isthegaugeeld: ( z; )=[ i G i ; ( z; )] ; @ @t i G i =0 ( i ) G i = i G i +[ j G j ; i G i ] ; (3 : 1 : 11)

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28 3.GENERALBRST wherethegaugetransformationof hasbeendeterminedbytheinvarianceofthe \total"time-derivative d=dt = @=@t + i H .(Moregenerally,if[ i G i ; H 0 ]= f i i G i ,then i hasanextraterm f i .)Usingthechainrule(( d=dt )on f ( t;q k ( t ))equals @=@t + q k ( @=@q k ))toevaluatethetimederivativeof G ,wendthelagrangiantransformsas atotalderivative L H = d dt h ( z M ) e M A A i i G i i ; (3 : 1 : 12) whichistheusualtransformationlawforanactionwithloca lsymmetrygenerated bythecurrent G .When H 0 vanishes(asinrelativisticmechanics),thespecialcase i = i ofthetransformationsof(3.1.11)are reparametrizations,generatedbythe hamiltonian i G i .Ingeneral,aftercanonicalquantization,thewavefuncti onsatises theSchrodingerequation @=@t + i H 0 =0,aswellastheconstraints G =0(andthus @=@t + i H =0inanygaugechoicefor ).Since[ H 0 ; G ]=0, G =0at t =0implies G =0forall t Insomecases(suchasYang-Mills),theLorentzcovariantfo rmoftheactioncan beobtainedbyeliminatingallthe 's.Acovariantrst-orderformcangenerallybe obtainedbyintroducingadditionalauxiliarydegreesoffr eedomwhichenlarge to makeitLorentzcovariant.Forexample,forYang-Millsweca nrewrite(3.1.9)as L H = 1 2 G 0 i 2 G 0 i F 0 i + 1 4 F ij 2 !L 1 = 1 4 G ab 2 + G ab F ab ; (3 : 1 : 13) where G 0 i = i i ,andtheindependent(auxiliary)elds G ab alsoinclude G ij ,which havebeenintroducedtoput 1 4 F ij 2 intorst-orderformandthusmakethelagrangian manifestlyLorentzcovariant.Eliminating G ij bytheireldequationsgivesbackthe hamiltonianform. Manyexampleswillbegiveninchapts.5-6forrelativistic rst-quantization, where H 0 vanishes,andthustheSchrodingerequationimpliesthewa vefunctionis proper-time-independent(i.e.,werequire H 0 =0becausethepropertimeisnot physicallyobservable).Herewegiveaninterestingexampl einD=2whichwillalso beusefulforstrings.Considerasingleeld A withcanonicalmomentum P and choose i G = 1 4 ( P + A 0 ) 2 ; H 0 = 1 4 ( P A 0 ) 2 ; (3 : 1 : 14) where 0 isthederivativewithrespecttothe1spacecoordinate(whi chactsasthe index M or i fromabove).Fromthealgebraof P A 0 ,it'seasytocheck,atleast atthePoissonbracketlevel,thatthe G algebraclosesand H 0 isinvariant.(This

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3.2.IGL(1) 29 algebra,withparticularboundaryconditions,willbeimpo rtantinstringtheory:See chapt.8.Notethat P + A 0 doesnotformanalgebra,soitssquaremustbeused.) Thetransformationlaws(3.1.11)arefoundtobe A = 1 2 ( P + A 0 ) ; = @ 1 : (3 : 1 : 15) Inthegauge =1theactionbecomestheusualhamiltonianoneforamassles s scalar,buttheconstraintimplies P + A 0 =0,whichmeansthatmodespropagate onlytotherightandnottheleft.Thelagrangianformagainr esultsfromeliminating P ,andaftertheredenitions ^ =2 1 1+ ; ^ = p 2 1 1+ ; (3 : 1 : 16) wend[3.2] L = ( @ + A )( @ A )+ 1 2 ^ ( @ A ) 2 ; A = ^ @ A; ^ =2 @ + ^ + ^ @ ^ ;(3 : 1 : 17) where @ aredenedasinsect.2.1. Thegaugexing(includingFaddeev-Popovghosts)andiniti alconditioncanbe describedinaveryconcisewaybytheBRSTmethod.Thebasici deaistoconstruct asymmetryrelatingtheFaddeev-Popovghoststotheunphysi calmodesofthegauge eld.Forexample,inYang-Millsonly D 2Lorentzcomponentsofthegaugeeld arephysical,sotheLorentz-gauge D -componentgaugeeldrequires2Faddeev-Popov ghostswhilethetemporal-gauge D 1-componenteldrequiresonly1.TheBRST symmetryrotatestheadditionalgauge-eldcomponentsint otheFPghosts,andvice versa.SincetheFPghostsareanticommuting,thegenerator ofthissymmetrymust be,also.3.2.IGL(1) WewillndthatthemethodsofBecchi,Rouet,Stora,andTyut in[3.3]arethe mostusefulwaynotonlytoperformquantizationinLorentzcovariantandgeneral nonunitarygauges,butalsotoderivegauge-invarianttheo ries.BRSTquantizationis amoregeneralwayofquantizinggaugetheoriesthaneitherc anonicalorpath-integral (Faddeev-Popov),becauseit(1)allowsmoregeneralgauges ,(2)givestheSlavnovTayloridentities(conditionsforunitarity)directly(th ey'rejusttheWardidentities forBRSTinvariance),and(3)canseparatethegauge-invari antpartofagauge-xed

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30 3.GENERALBRST action.Itisdenedbytheconditions:(1)BRSTtransformat ionsformaglobalgroup withasingle(abelian)anticommutinggenerator Q .Thegrouppropertythenimplies Q 2 =0forclosure.(2) Q actsonphysicaleldsasagaugetransformationwiththe gaugeparameterreplacedbythe(real)ghost.(3) Q onthe(real)antighostgivesa BRSTauxiliaryeld(necessaryforclosureofthealgebrao shell).Nilpotenceof Q thenimpliesthattheauxiliaryeldisBRSTinvariant.Phys icalstatesaredened tobethosewhichareBRSTinvariant(modulonullstates,whi chcanbeexpressed as Q onsomething)andhavevanishingghostnumber(thenumberof ghostsminus antighosts). TherearetwotypesofBRSTformalisms:(1)rst-quantizedstyleBRST,originallyfoundinstringtheory[3.4]butalsoapplicabletoord inaryeldtheory,which containsalltheeldequationsaswellasthegaugetransfor mations;and(2)secondquantized-styleBRST,theoriginalformofBRST,whichcont ainsonlythegauge transformations,correspondinginahamiltonianformalis mtothoseeldequations (constraints)foundfromvaryingthetimecomponentsofthe gaugeelds.However, we'llnd(insect.4.4)that,afterrestrictiontoacertain subsetoftheelds,BRST1is equivalenttoBRST2.(It'stheBRSTvariationoftheadditio naleldsofBRST1that leadstotheeldequationsforthephysicalelds.)TheBRST 2transformationswere originallyfoundfromYang-Millstheory.Wewillrstderiv etheYMBRST2transformations,andbyasimplegeneralizationndBRSToperatorsf orarbitrarytheories, applicabletoBRST1orBRST2andtolagrangianorhamiltonia nformalisms. Inthegeneralcase,therearetwoformsfortheBRSToperator s,correspondingtodierentclassesofgauges.Thegaugescommonlyusedi neldtheoryfall intothreeclasses:(1)unitary(Coulomb,Arnowitt-Fickle r/axial,light-cone)gauges, wheretheghostsarenonpropagating,andtheconstraintsar esolvedexplicitly(since theycontainnotimederivatives);(2)temporal/timelikeg auges,wheretheghostshave equationsofmotionrst-orderintimederivatives(making themcanonicallyconjugatetotheantighosts);and(3)Lorentz(Landau,Fermi-Fey nman)gauges,where theghostequationsaresecond-order(soghostsareindepen dentofantighosts),and theNakanishi-Lautrupauxiliaryelds[3.5](Lagrangemul tipliersforthegaugeconditions)arecanonicallyconjugatetotheauxiliarytime-c omponentsofthegauge elds.Unitarygaugeshaveonlyphysicalpolarizations;te mporalgaugeshavean additionalpairofunphysicalpolarizationsofoppositest atisticsforeachgaugegenerator;Lorentzgaugeshavetwopairs.InunitarygaugestheBR SToperatorvanishes identically;intemporalgaugesitisconstructedfromgrou pgenerators,orconstraints,

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3.2.IGL(1) 31 multipliedbythecorrespondingghosts,plustermsfornilp otence;inLorentzgauges ithasanextra\abelian"termconsistingoftheproductsoft hesecondsetofunphysicalelds.Temporal-gaugeBRSTisdenedintermsofaghost numberoperatorin additiontotheBRSToperator,whichitselfhasghostnumber 1.Wethereforerefer tothisformalismbythecorrespondingsymmetrygroupwitht wogenerators,IGL(1). Lorentz-gaugeBRSThasalsoanantiBRSToperator[3.6],and thisandBRSTtransformasan\isospin"doublet,givingthelargergroupISp(2) ,whichcanbeextended furthertoOSp(1,1 j 2)[2.3,3.7].AlthoughtheBRST2OSpoperatorsaregenerall y oflittlevalue(onlytheIGLisrequiredforquantization), theBRST1OSpgivesa powerfulmethodforobtainingfreegauge-invariantformal ismsforarbitrary(particle orstring)eldtheories.Inparticular,forarbitraryrepr esentationsofthePoincare groupacertainOSp(1,1 j 2)canbeextendedtoIOSp(D,2 j 2)[2.3],whichisderived from(butdoesnotdirectlycorrespondtoquantizationin)t helight-conegauge. Onesimplewaytoformulateanticommutingsymmetries(such assupersymmetry)isthroughtheuseofanticommutingcoordinates[3.8]. Wethereforeextend spacetimetoincludeoneextra,anticommutingcoordinate, correspondingtotheone anticommutingsymmetry: a ( a; )(3 : 2 : 1) forallvectorindices,includingthoseoncoordinates,wit hFermistatisticsforall quantitieswithanoddnumberofanticommutingindices.( takesonlyonevalue.) Covariantderivativesandgaugetransformationsarethend enedbythecorrespondinggeneralizationof(2.1.5b),andeldstrengthswithgra dedcommutators(commutatorsoranticommutators,accordingtothestatistics).H owever,unlikesupersymmetry,theextracoordinatedoesnotrepresentextraphysic aldegreesoffreedom,and soweconstrainalleldstrengthswithanticommutingindic estovanish[3.9]:For Yang-Mills, F a = F =0 ; (3 : 2 : 2 a ) sothatgauge-invariantquantitiescanbeconstructedonly fromtheusual F ab .When Yang-Millsiscoupledtomatterelds ,wesimilarlyhavetheconstraints r = r r a =0 ; (3 : 2 : 2 b ) andtheseinfactimply(3.2.2a)(consider fr ; r g and[ r ; r a ]actingon ).These constraintscanbesolvedeasily: F a =0 p A a =[ r a ;A ] ;

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32 3.GENERALBRST F =0 p A = 1 2 f A ;A g = A A ; r =0 p = A : (3 : 2 : 3) (Inthesecondlinewehaveusedthefactthat takesonlyonevalue.)Dening\ j tomean j x =0 ,wenowinterpret A a j astheusualgaugeeld, iA j astheFPghost, andtheBRSToperator Q as Q ( j )=( p ) j .(Similarly, j istheusualmatter eld.)Then @ @ =0(since takesonlyonevalueand @ isanticommuting)implies nilpotence Q 2 =0 : (3 : 2 : 4) Inahamiltonianapproach[3.10]thesetransformationsare sucienttoperformquantizationinatemporalgauge,butforthelagrangianapproac horLorentzgaugeswe alsoneedtheFPantighostandNakanishi-Lautrupauxiliary eld,whichwedenein termsofanunconstrainedscalareld e A : e A j istheantighost,and B =( p i e A ) j (3 : 2 : 5) istheauxiliaryeld. TheBRSTtransformations(3.2.3)canberepresentedinoper atorformas Q = C i G i + 1 2 C j C i f ij k @ @C k iB i @ @ e C i ; (3 : 2 : 6 a ) where i isacombinedspace(time)/internal-symmetryindex, C istheFPghost, e C is theFPantighost, B istheNLauxiliaryeld,andtheactiononthephysicalelds is givenbytheconstraint/gauge-transformation G satisfyingthealgebra [ G i ; G j g = f ij k G k ; (3 : 2 : 6 b ) wherewehavegeneralizedtogradedalgebraswithgradedcom mutator[ ; g (commutatororanticommutator,asappropriate).Inthiscase, G = r ; i A # ; (3 : 2 : 7) wherethestructureconstantsin(3.2.6b)aretheusualgrou pstructureconstants times -functionsinthecoordinates. Q of(3.2.6a)isantihermitianwhen C e C ,and B arehermitianand G isantihermitian,andisnilpotent(3.2.4)asaconsequence of (3.2.6b).Since e C and B appearonlyinthelasttermin(3.2.6a),theseproperties alsoholdifthattermisdropped.(Inthenotationof(3.2.15),theelds A and e A are independent.)

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3.2.IGL(1) 33 When[ G i ;f jk l g6 =0,(3.2.6a)stillgives Q 2 =0.However,whenthegauge invariancehasagaugeinvarianceofitsown,i.e., i G i =0forsomenontrivial dependingonthephysicalvariablesimplicitin G ,then,although(3.2.6a)isstill nilpotent,itrequiresextratermsinordertoallowgaugex ingthisinvarianceofthe ghosts.Insomecases(seesect.5.4)thisrequiresaninnit enumberofnewterms(and ghosts).Ingeneral,theprocedureofaddingintheaddition alghostsandinvariances canbetedious,butinsect.3.4we'llndamethodwhichautom aticallygivesthem allatonce. Thegauge-xedactionisrequiredtobeBRST-invariant.The gauge-invariant partalreadyis,since Q onphysicaleldsisaspecialcaseofagaugetransformation.Thegauge-invariantlagrangianisquantizedbyaddin gtermswhichare Q on something(correspondingtointegrationover x ),andthusBRST-invariant(since Q 2 =0):Forexample,rewriting(3.2.3,5)inthepresentnotati on, QA a = i [ r a ;C ] ; QC = iC 2 ; Q ~ C = iB; QB =0 ; (3 : 2 : 8) wecanchoose L GF = iQ n e C [ f ( A )+ g ( B )] o = B [ f ( A )+ g ( B )] e C @f @A a [ r a ;C ] ; (3 : 2 : 9) whichgivestheusualFPtermforgaugecondition f ( A )=0withgauge-averaging function Bg ( B ).However,gaugesmoregeneralthanFPcanbeobtainedbyput ting morecomplicatedghost-dependenceintothefunctiononwhi ch Q acts,givingterms morethanquadraticinghosts.Inthetemporalgauge f ( A )= A 0 (3 : 2 : 10) and g containsnotimederivativesin(3.2.9),souponquantizati on B iseliminated (it'snonpropagating)and e C iscanonicallyconjugateto C .Thus,inthehamiltonian formalism(3.2.6a)givesthecorrectBRSTtransformations withoutthelastterm, wheretheeldsarenowfunctionsofjustspaceandnottime,t hesumin(3.2.7)runs overjustthespatialvaluesofthespacetimeindexasin(3.1 .8),andthederivatives correspondtofunctionalderivativeswhichgive functionsinjustspatialcoordinates. Ontheotherhand,inLorentzgaugestheghostandantighosta reindependenteven

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34 3.GENERALBRST afterquantization,andthelasttermin Q isneededinbothlagrangianandhamiltonianformalisms;buttheproductin(3.2.7)andtheargume ntsoftheeldsand functionsareasinthetemporalgauge.Therefore,inthelag rangianapproach Q isgaugeindependent,whileinthehamiltonianapproachthe onlygaugedependence isthesetofunphysicalelds,andthusthelasttermin Q .Specically,forLorentz gaugeswechoose f ( A )= @ A;g ( B )= 1 2 B L GF = 1 2 B 2 + B@ A e C@ [ r ;C ] = 1 1 2 ( @ A ) 2 + 1 2 e B 2 e C@ [ r ;C ] ; e B = B + 1 @ A; (3 : 2 : 11) using(3.2.9). Themainresultisthat(3.2.6a)givesageneralBRSToperato rforarbitraryalgebras(3.2.6b),forhamiltonianorlagrangianformalisms,f orarbitrarygauges(includingtemporalandLorentz),wherethelasttermcontainsarbi trarynumbers(perhaps 0)ofsetsof( e C B )elds.Since G =0istheeldequation(3.1.6),physicalstates mustsatisfy Q =0.Actually, G =0issatisedonlyasaGupta-Bleulercondition,butstill Q =0becauseinthe C i G i termin(3.2.6a)positive-energyparts of C i multiplynegative-energypartsof G i ,andviceversa.Thus,foranyvalueof anappropriateindex i ,either C i j i = h jG i =0or G i j i = h j C i =0,modulo contributionsfromthe C 2 @=@C term.However,since G isalsothegeneratorofgauge transformations(3.1.8),anystateoftheform + Q isequivalentto .Thephysical statesarethereforesaidtobelongtothe\cohomology"of Q :thosesatisfying Q =0 modulogaugetransformations = Q .(\Physical"hasamorerestrictivemeaning inBRST1thanBRST2:InBRST2thephysicalstatesarejustthe gauge-invariant ones,whileinBRST1theymustalsobeonshell.)Inaddition, physicalstatesmust haveaspeciedvalueoftheghostnumber,denedbytheghost numberoperator J 3 = C i @ @C i e C i @ @ e C i ; (3 : 2 : 12 a ) where [ J 3 ;Q ]= Q; (3 : 2 : 12 b ) andthelattertermin(3.2.12a)isdroppedifthelasttermin (3.2.6a)is.Thetwo operators Q and J 3 formthealgebraIGL(1),whichcanbeinterpretedasatransl ation

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3.3.OSp(1,1 j 2) 35 andscaletransformation,respectively,withrespecttoth ecoordinate x (i.e.,the conformalgroupin1anticommutingdimension). Fromthegaugegenerators G i ,whichactononlythephysicalvariables,wecan deneIGL(1)-invariantgeneralizationswhichtransforma lso C ,astheadjointrepresentation: b G i = ( Q; @ @C i ) = G i + C j f ji k @ @C k : (3 : 2 : 13) The b G 'saregauge-xedversionsofthegaugegenerators G Typesofgaugesforrst-quantizedtheorieswillbediscuss edinchapt.5forparticlesandchapt.6andsect.8.3forstrings.Gaugexingfor generaleldtheories usingBRSTwillbedescribedinsect.4.4,andforclosedstri ngeldtheoryinsect. 11.1.IGL(1)algebraswillbeusedforderivinggeneralgaug e-invariantfreeactionsin sect.4.2.Thealgebrawillbederivedfromrst-quantizati onfortheparticleinsect. 5.2andforthestringinsect.8.1.However,inthenextsecti onwe'llndthatIGL(1) canalwaysbederivedasasubgroupofOSp(1,1 j 2),whichcanbederivedinamore generalwaythanbyrst-quantization.3.3.OSp(1,1 j 2) AlthoughtheIGL(1)algebraissucientforquantizationin arbitrarygauges,in thefollowingsectionwewillndthelargerOSp(1,1 j 2)algebrausefulfortheBRST1 formalism,sowegiveaderivationhereforBRST2andagainge neralizetoarbitrary BRST.ThebasicideaistointroduceasecondBRST,\antiBRST ,"correspondingto theantighost.Wethereforerepeattheprocedureof(3.2.17)with2anticommuting coordinates[3.11]bysimplylettingtheindex runover2values(cf.sect.2.6).The solutionto(3.2.2)isnow F a =0 p A a =[ r a ;A ] ; F =0 p A = 1 2 f A ;A g iC B ; r =0 p = A ;(3 : 3 : 1 a ) where A nowincludesbothghostandantighost.Theappearanceofthe NLeldis duetotheambiguityintheconstraint F = p ( A ) + .Theremaining(anti)BRST transformationthenfollowsfromfurtherdierentiation: f p ;p g A r =0 p B = 1 2 [ A ;B ]+ i 1 12 h A ; f A ;A g i : (3 : 3 : 1 b )

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36 3.GENERALBRST Thegeneralizationof(3.2.6a)isthen[3.12],dening Q ( j )=( @ ) j (andrenaming C = A ), Q = C i G i + 1 2 C j C i f ij k @ @C k B i @ @C i + 1 2 C j B i f ij k @ @B k 1 12 C k C j C i f ij l f lk m @ @B m ; (3 : 3 : 2) andof(3.2.12a)is J = C i ( @ @C i ) ; (3 : 3 : 3) where()meansindexsymmetrization.Theseoperatorsforma nISp(2)algebra consistingofthetranslations Q androtations J onthecoordinates x : f Q ;Q g =0 ; [ J ;Q r ]= C r ( Q ) ; [ J ;J r ]= C ( r ( J ) ) : (3 : 3 : 4) InordertorelatetotheIGL(1)formalism,wewrite Q =( Q; e Q ) ;C =( C; e C ) ;J = J + iJ 3 iJ 3 J ; (3 : 3 : 5) andmaketheunitarytransformation lnU = 1 2 C j e C i f ij k i @ @B k : (3 : 3 : 6) Then UQU 1 is Q of(3.2.6a)and UJ 3 U 1 = J 3 is J 3 of(3.2.12a).However,whereas thereisanarbitrarinessintheIGL(1)algebrainredening J 3 byaconstant,there isnosuchambiguityintheOSp(1,1 j 2)algebra(sinceitis\simple"). UnliketheIGLcase,theNLeldsnowareanessentialpartoft healgebra. Consequently,thealgebracanbeenlargedtoOSp(1,1 j 2)[3.7]: J = Q ;J + =2 C i @ @B i ; J = C i ( @ @C i ) ;J + =2 B i @ @B i + C i @ @C i ; (3 : 3 : 7) with Q asin(3.3.2),satisfy [ J ;J r ]= C ( r ( J ) ) ; [ J ;J r ]= C r ( J ) ; f J ;J + g = C J + J ;

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3.3.OSp(1,1 j 2) 37 [ J + ;J ]= J ; rest =0 : (3 : 3 : 8) Thisgroupistheconformalgroupfor x ,withtheISp(2)subgroupbeingthecorrespondingPoincare(orEuclidean)subgroup: J = @ ;J + =2 x 2 @ + x M + x d J = x ( @ ) + M ;J + = x @ + d : (3 : 3 : 9) (WedenethesquareofanSp(2)spinoras( x ) 2 1 2 x x .) J arethetranslations, J theLorentztransformations(rotations), J + thedilatations,and J + the conformalboosts.Asaresultofconstraintsanalogousto(3 .3.1a),thetranslations arerealizednonlinearlyin(3.3.7)insteadoftheboosts.T hisshouldbecompared withtheusualconformalgroup(2.2.4).Theactionofthegen erators(3.3.9)have beenchosentohavetheoppositesignofthoseof(3.3.8),sin ceitisacoordinaterepresentationinsteadofaeldrepresentation(seesect.2.2 ).Inlatersectionswewill actuallybeapplying(3.3.7)tocoordinates,andhence(3.3 .9)shouldbeconsidereda \zeroth-quantized"formalism. Fromthegaugegenerators G i ,wecandeneOSp(1,1 j 2)-invariantgeneralizations whichtransformalso C and B ,asadjointrepresentations: b G i = 1 2 ( J ; J ; @ @B i #) = G i + C j f ji k @ @C k + B j f ji k @ @B k : (3 : 3 : 10) The b G 'saretheOSp(1,1 j 2)generalizationoftheoperators(3.2.13). TheOSp(1,1 j 2)algebra(3.3.7)canbeextendedtoaninhomogeneousalgeb ra IOSp(1,1 j 2)whenoneofthegenerators,whichwedenoteby G 0 ,isdistinguished [3.13].Wethendene p + = s 2 i @ @B 0 ; p = 1 p + i @ @C 0 + 1 2 C i f i 0 j @ @B j ; p = 1 p + i b G 0 + p 2 : (3 : 3 : 11) (The i indicesstillincludethevalue0.) b G 0 isthentheIOSp(1,1 j 2)invariant i 1 2 (2 p + p + p p ).Thisalgebraisusefulforconstructinggaugeeldtheory forclosedstrings.

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38 3.GENERALBRST OSp(1,1 j 2)willplayacentralroleinthefollowingchapters:Inchap t.4itwill beusedtoderivefreegauge-invariantactions.Amoregener alformwillbederived inthefollowingsections,butthemethodsofthissectionwi llalsobeusedinsect.8.3 todescribeLorentz-gaugequantizationofthestring.3.4.Fromthelightcone InthissectionwewillderiveageneralOSp(1,1 j 2)algebrafromthelight-cone Poincarealgebraofsect.2.3,usingconceptsdevelopedin sect.2.6.We'llusethis generalOSp(1,1 j 2)toderiveageneralIGL(1),andshowhowIGL(1)canbeexten ded toincludeinteractions. TheIGL(1)andOSp(1,1 j 2)algebrasoftheprevioussectioncanbeconstructed fromanarbitraryalgebra G ,whetherrst-quantizedorsecond-quantized,andlagrangianorhamiltonian.Thatalreadygives8dierenttype sofBRSTformalisms. Furthermore,arbitrarygauges,moregeneralthanthoseobt ainedbytheFPmethod, andgradedalgebras(wheresomeofthe G 'sareanticommuting,asinsupersymmetry)canbetreated.However,thereisaninthBRSTformali sm,similartothe BRST1OSp(1,1 j 2)hamiltonianformalism,whichstartsfromanIOSp(D,2 j 2)algebra [2.3]whichcontainstheOSp(1,1 j 2)asasubgroup.Thisapproachisuniqueinthat, ratherthanstartingfromthegaugecovariantformalismtod erivetheBRSTalgebra, itstartsfromjusttheusualPoincarealgebraandderivesb oththegaugecovariant formalismandBRSTalgebra.Inthissection,insteadofderi vingBRST1fromrstquantization,wewilldescribethisspecialformofBRST1,a ndgivetheOSp(1,1 j 2) subalgebraofwhichspecialcaseswillbefoundinthefollow ingchapters. ThebasicideaoftheIOSpformalismistostartfromthelight -coneformalism ofthetheorywithitsnonlinearrealizationoftheusualPoi ncaregroupISO(D-1,1) (withmanifestsubgroupISO(D-2)),extendthisgrouptoIOS p(D,2 j 2)(withmanifest IOSp(D-1,1 j 2))byadding2commutingand2anticommutingcoordinates,a ndtake theISO(D-1,1) n OSp(1,1 j 2)subgroup,wherethisISO(D-1,1)isnowmanifestandthe nonlinearOSp(1,1 j 2)isinterpretedasBRST.SincetheBRSToperatorsofBRST1 containalltheeldequations,thegauge-invariantaction canbederived.Thus,not onlycanthelight-coneformalismbederivedfromthegaugeinvariantformalism, buttheconverseisalsotrue.Furthermore,forgeneraleld theoriesthelight-cone formalism(atleastforthefreetheory)iseasiertoderive( althoughmoreawkward touse),andtheIOSpmethodthereforeprovidesaconvenient methodtoderivethe gaugecovariantformalism.

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3.4.Fromthelightcone 39 Wenowperformdimensionalcontinuationasinsect.2.6,but set x + =0asinsect. 2.3.Oureldsarenowfunctionsof( x a ;x ;x ),andhaveindicescorrespondingto representationsofthespinsubgroupOSp(D 1,1 j 2)inthemasslesscaseorOSp(D,1 j 2) inthemassive.OfthefullgroupIOSp(D,2 j 2)(obtainedfromextending(2.3.5))we arenowonlyinterestedinthesubgroupISO(D 1,1) n OSp(1,1 j 2).Theformerfactor istheusualPoincaregroup,actingonlyinthephysicalspa cetimedirections: p a = i@ a ;J ab = ix [ a p b ] + M ab : (3 : 4 : 1) ThelatterfactorisidentiedastheBRSTgroup,actinginon lytheunphysical directions: J = ix ( p ) + M ;J + = ix p + + k;J + = ix p + ; J = ix p + 1 p + h ix 1 2 ( p b p b + M 2 + p p )+ M p + kp + Q i ; fQ ; Q g = M ( p a p a + M 2 );(3 : 4 : 2 a ) Q = M b p b + M m M: (3 : 4 : 2 b ) We'llgenerallyset k =0. InordertorelatetotheBRST1IGLformalismobtainedfromor dinaryrstquantization(anddiscussedinthefollowingchaptersfort heparticleandstring), weperformananalysissimilartothatof(3.3.5,6):Makingt he(almost)unitary transformation[2.3] lnU =( lnp + ) c @ @c + M 3 ; (3 : 4 : 3 a ) where x =( c; ~ c ), M a =( M + a ;M a ),and M m =( M + m ;M m ),weget Q ic 1 2 ( p a 2 + M 2 )+ M + i @ @c +( M + a p a + M + m M )+ x i @ @ ~ c ; J 3 = c @ @c + M 3 ~ c @ @ ~ c : (3 : 4 : 3 b ) (Cf.(3.2.6a,12a).)Asinsect.3.2,theextratermsin x and~ c (analogousto B i and e C i )canbedroppedintheIGL(1)formalism.Afterdroppingsuch terms, J 3 y =1 J 3 (Orwecansubtract 1 2 tomakeitsimplyantihermitian.However,weprefernotto, sothatphysicalstateswillstillhavevanishingghostnumb er.) Since p + isamomentum,thisredenitionhasafunnyeectonreality( butnot hermiticity)properties:Inparticular, c isnowamomentumratherthanacoordinate(becauseithasbeenscaledby p + ,maintainingitshermiticitybutmakingit

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40 3.GENERALBRST imaginaryincoordinatespace).However,wewillavoidchan gingnotationorFourier transformingtheelds,inordertosimplifycomparisontot heOSp(1,1 j 2)formalism. Theeectof(3.4.3a)onaeldsatisfying=n isthatitnowsatises =( 1) c@=@c + M 3 n (3 : 4 : 4) duetothe i in p + = i@ + Theseresultscanbeextendedtointeractingeldtheory,an dweuseYang-Mills asanexample[2.3].Lorentz-covariantizingthelight-con eresult(2.1.7,2.4.11),wend p = Z 1 4 F ij F ji + 1 2 ( p + A ) 2 ; J = Z ix ( p A i )( p + A i )+ ix h 1 4 F ij F ji + 1 2 ( p + A ) 2 i A p + A ; A = 1 p + 2 [ r i ;p + A i g : (3 : 4 : 5) WhenworkingintheIGL(1)formalism,it'sextremelyuseful tointroducea Lorentzcovarianttypeofsecond-quantizedbracket[3.14] .Thisbracketcanbepostulatedindependently,orderivedbycovariantizationoft helight-conecanonicalcommutator,plustruncationofthe~ c;x + ,and x coordinates.Thelatterderivationwill proveusefulforthederivationofIGL(1)fromOSp(1,1 j 2).Uponcovariantizationof thecanonicallight-conecommutator(2.4.5),theargument softheeldsandofthe -functionontheright-handsideareextendedaccordingly. Wenowhavetotruncate. Thetruncationof x + isautomatic:Sincetheoriginalcommutatorwasanequal-ti me one,thereisno x + -functionontheright-handside,anditthereforesucesto deletethe x + argumentsoftheelds.Atthisstage,inadditionto x dependence, theeldsdependonboth c and~ c andtheright-handsidecontainsboth -functions. (ThiscommutatormaybeusefulforOSpapproachestoeldthe ory.)Wenowwish toeliminatethe~ c dependence.Thiscannotbedonebystraightforwardtruncat ion, sinceexpansionoftheeldinthisanticommutingcoordinat eshowsthatonecannot eliminateconsistentlytheeldsinthe~ c sector.Wethereforeproceedformallyand justdeletethe~ c argumentfromtheeldsandthecorresponding -function,obtaining [ y (1) ; (2)] c = 1 2 p +2 ( x 2 x 1 ) D ( x 2 x 1 ) ( c 2 c 1 ) ; (3 : 4 : 6) whichisabracketwithunusualstatisticsbecauseoftheant icommuting -functionon theright-handside.Thetransformation(3.4.3a)isperfor mednext;itsnonunitarity

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3.4.Fromthelightcone 41 causesthe p + dependenceof(3.4.6)todisappear,enablingonetodeletet he x argumentfromtheeldsandthecorresponding -functiontond(using( c@=@c ) c = c ) [ y (1) ; (2)] c = 1 2 D ( x 2 x 1 ) ( c 2 c 1 ) : (3 : 4 : 7) Thisisthecovariantbracket.Theargumentsoftheeldsare ( x a ;c ),namely,theusual D bosoniccoordinatesofcovarianttheoriesandthesinglean ticommutingcoordinate oftheIGL(1)formalism.Thecorresponding -functionsappearontheright-hand side.(3.4.6,7)aredenedforcommuting(scalar)elds,bu tgeneralizestraightforwardly:Forexample,forYang-Mills,where A i includesbothcommuting( A a )and anticommuting( A )elds,[ A i y ;A j ]hasanextrafactorof ij .Itmightbepossible todenethebracketbyacommutator[ A;B ] c = A B B A .Classicallyitcanbe denedbyaPoissonbracket: [ A y ;B ] c = 1 2 Z dz ( z ) A y ( z ) B ; (3 : 4 : 8) where z areallthecoordinatesof(inthiscase, x a and c ).For A = B =,the resultofequation(3.4.7)isreproduced.Theaboveequatio nimpliesthatthebracket isaderivation: [ A;BC ] c =[ A;B ] c C +( 1) ( A +1) B B [ A;C ] c ; (3 : 4 : 9) wherethe A 'sand B 'sintheexponentofthe( 1)are0ifthecorrespondingquantity isbosonicand1ifit'sfermionic.Thisdiersfromtheusual gradedLeibnitzruleby a( 1) B duetotheanticommutativityofthe dz inthefrontof(3.4.8),whichalso givesthebrackettheoppositeoftheusualstatistics:Weca nwrite( 1) [ A;B ] c = ( 1) A + B +1 toindicatethatthebracketof2bosonicoperatorsisfermio nic,etc.,a directconsequenceoftheanticommutativityofthetotal -functionin(3.4.7).One canalsoverifythatthisbracketsatisestheotherpropert iesofa(generalized)Lie bracket: [ A;B ] c =( 1) AB [ B;A ] c ; ( 1) A ( C +1) [ A; [ B;C ] c ] c +( 1) B ( A +1) [ B; [ C;A ] c ] c +( 1) C ( B +1) [ C; [ A;B ] c ] c =0 : (3 : 4 : 10) Thusthebrackethastheoppositeoftheusualgradedsymmetr y,beingantisymmetric forobjectsofoddstatisticsandsymmetricotherwise.This propertyfollowsfrom thehermiticitycondition(3.4.4):( 1) c@=@c gives( 1) ( @=@c ) c = ( 1) c@=@c upon integrationbyparts,whichgivestheeectofusinganantis ymmetricmetric.The Jacobiidentityhasthesameextrasignsasin(3.4.9).These propertiesaresucient toperformthemanipulationsanalogoustothoseusedinthel ightcone.

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42 3.GENERALBRST Beforeapplyingthisbracket,wemakesomegeneralconsider ationsconcerning thederivationofinteractingIGL(1)fromOSp(1,1 j 2).Westartwiththeoriginal untransformedgenerators J 3 and J c = Q .Therststepistorestrictourattention tojusttheeldsat~ c =0.Killingalltheeldsatlinearorderin~ c isconsistent withthetransformationlaws,sincethetransformationsof thelattereldsinclude notermswhichinvolveonly~ c =0elds.Sincethelinear-in-~ c eldsarecanonically conjugatetothe~ c =0elds,theonlytermsinthegeneratorswhichcouldspoilt his propertywouldthemselveshavetodependononly~ c =0elds,which,becauseofthe d ~ c (= @=@ ~ c )integration,wouldrequireexplicit~ c -dependence.However,from(2.4.15), since cc = C cc =0,weseethat J c anticommuteswith p c ,andthushasnoexplicit ~ c -dependenceateitherthefreeorinteractinglevels.(Theo nlyexplicitcoordinate dependencein Q isfroma c term.) Theprocedureofrestrictingto~ c =0eldscanthenbeimplementedverysimply bydroppingall p c (= @=@ ~ c )'sinthegenerators.Asaconsequence,wealsoloseall explicit x termsin Q .(Thisfollowsfrom[ J c ;p + ]= p c .)Since~ c and @=@ ~ c now occurnowhereexplicitly,wecanalsokillallimplicitdepe ndenceon~ c :Alleldsare evaluatedat~ c =0,the d ~ c isremovedfromtheintegralinthegenerators,andthe (~ c 2 ~ c 1 )isremovedfromthecanonicalcommutator,producing(3.4. 6).Inthecase ofYang-Millselds A i =( A a ;A )=( A a ;A c ;A ~ c ),theBRSTgeneratoratthispoint isgivenby Q = Z ic [ 1 4 F ij F ji + 1 2 ( p + A ) 2 ] A c p + A ; (3 : 4 : 11) wheretheintegralsarenowoverjust x a x ,and c ,andsomeoftheeldstrengths simplify: F cc =2( A c ) 2 ;otherF ic =[ r i ;A c g : (3 : 4 : 12) Beforeperformingthetransformationswhicheliminate p + dependence,it'snow convenienttoexpandtheeldsover c as A a = A a + c a ; A c = iC + cB; A ~ c = i e C + cD; (3 : 4 : 13) wheretheeldsontheright-handsidesare x -independent.(The i 'shavebeen choseninaccordancewith(3.4.4)tomakethenaleldsreal .)Wenextperformthe dc integration,andthenperformasthersttransformationth erst-quantizedone

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3.4.Fromthelightcone 43 (3.4.3a), p + J 3 (usingtherst-quantized J 3 = c@=@c + M 3 ),whichgives iQ = Z 1 4 F ab 2 1 2 2 B + 1 p + i [ r a ;p + A a ] f C; ~ C g 2 p + ( p + 2 C; 1 p + ~ C )! 2 24 2 D + ~ C 2 2 p + 2 1 p + ~ C 2 35 C 2 + 2 a i [ r a ; ~ C ] 2 i p + A a ; 1 p + ~ C #! [ r a ;C ] : (3 : 4 : 14) Thistransformationalsoreplaces(3.4.6)with(3.4.7),wi thanextrafactorof ij for[ A i y ;A j ],butstillwiththe x -function.Expandingthebracketoverthe c 's, [ a ;A b ] c = 1 2 ab ; [ D;C ] c = 1 2 ; [ B; ~ C ] c = 1 2 ; (3 : 4 : 15) wherewehaveleftoallthe -functionfactors(nowincommutingcoordinatesonly). Notethat,by(3.4.10),allthesebracketsare symmetric Wemightalsodeneasecond-quantized J 3 = Z A i p + c @ @c + M 3 A i ; (3 : 4 : 16 a ) butthisformautomaticallykeepsjusttheantihermitianpa rtoftherst-quantized operator c@=@c = 1 2 [ c;@=@c ]+ 1 2 :Doingthe c integrationandtransformation(3.4.3a), J 3 = Z a A a ~ CB 3 CD: (3 : 4 : 16 b ) Asaresult,thetermsin Q ofdierentordersintheeldshavedierentsecondquantizedghostnumber.Therefore,weuseonlytherst-qua ntizedghostoperator (orsecond-quantizeitinfunctionalform). Ascanbeseenintheaboveequations,despitetherescalingo ftheeldsby suitablepowersof p + thereremainsafairlycomplicateddependenceon p + .Thereis noexplicit x dependenceanywherebut,ofcourse,theeldshave x asanargument. Itwouldseemthatthereshouldbeasimpleprescriptiontoge tridofthe p + 'sinthe transformations.Setting p + = constant doesnotwork,sinceitviolatestheLeibnitz ruleforderivatives( p + = a impliesthat p + 2 =2 a 2 andnot a 2 ).Evensetting p + i = i i doesnotwork.Anattemptthatcomesverycloseisthefollowi ng:Give theeldssomespecic x dependenceinsuchawaythatthe p + factorscanbe evaluatedandthatafterwardssuchdependencecanbecancel edbetweentherighthandsideandleft-handsideofthetransformations.Inthea bovecaseitseemsthat

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44 3.GENERALBRST theonlypossibilityistoseteveryeldproportionalto( x ) 0 butthenitishard todene1 =p + and p + .Onethentriessettingeacheldproportionalto( x ) and thenlet 0attheend.Infactthisprescriptiongivesthecorrectansw erforthe quadratictermsoftheYang-MillsBRSTtransformations.Un fortunatelyitdoesnot givethecorrectcubicterms. Itmightbepossibletoeliminate p + -dependencesimplybyapplying J + =0asa constraint.However,thiswouldrequireresolvingsomeamb iguitiesintheevaluation ofthenonlocal(in x )operator p + intheinteractionvertices. Wethereforeremovetheexplicit p + -dependencebyuseofanexplicittransformation.IntheYang-Millscase,thistransformationcanbecom pletelydeterminedby choosingittobetheonewhichredenestheauxiliaryeld B inawaywhicheliminatesinteractiontermsin Q involvingit,thusmaking B + i 1 2 p A BRST-invariant. Theresultingtransformation[3.14]redenesonlytheBRST auxiliaryelds: Q e L Q;L A B [ A;B ] c ;L 2 =0 ; = Z ~ C 1 p + i [ A a ;p + A a ] ~ C 2 C +2 1 p + ~ C 2 ( p + 2 C ) ; (3 : 4 : 17) simplyredenestheauxiliaryeldstoabsorbtheawkwardin teractiontermsin (3.4.14).(Wecanalsoeliminatethefreetermsaddedto B and byaddinga term R ~ Cip a A a totomaketherstterm ~ C (1 =p + ) i [ r a ;p + A a ].)Wethenndfor thetransformedBRSToperator iQ = Z 1 4 F ab 2 1 2 (2 B + ip A ) 2 2 DC 2 +(2 a ip a e C )[ r a ;C ] : (3 : 4 : 18) Theresultingtransformationsarethen QA a = i [ r a ;C ] ; Q a = i 1 2 [ r b ;F ba ]+ i f C; a i 1 2 p a e C g + p a ( B + i 1 2 p A ) ; Q e C = 2 i ( B + i 1 2 p A ) ; QD = i h r a ; a i 1 2 p a e C i + i [ C;D ] ; QC = iC 2 ; QB = 1 2 p a [ r a ;C ] : (3 : 4 : 19) Sinceallthe p + 'shavebeeneliminated,wecannowdropall x dependencefromthe elds,integration,and -functions.Ontheelds A a C e C B oftheusualBRST2

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3.5.Fermions 45 formalism,thisresultagreeswiththecorrespondingtrans formations(3.2.8),where this B = 1 2 e B of(3.2.11).Byworkingwiththesecond-quantizedoperator formof Q (andoftheredenition),wehaveautomaticallyobtaineda formwhichmakes Q integrablein,orequivalentlymakestheverticeswhichfo llowfromthisoperator cyclicinalltheelds(orsymmetric,ifonetakesgroup-the oryindicesintoaccount). Thesignicanceofthispropertywillbedescribedinthenex tchapter. Thisextended-light-coneformoftheOSp(1,1 j 2)algebrawillbeusedtoderive freegauge-invariantactionsinthenextchapter.Thespeci cformofthegenerators forthecaseofthefreeopenstringwillbegiveninsect.8.2, andthegeneralizationto thefreeclosedstringinsect.11.1.Apartialanalysisofth einteractingstringalong theselineswillbegiveninsect.12.1.3.5.Fermions Theseresultscanbeextendedtofermions[3.15].Thisrequi resaslightmodicationoftheformalism,sincetheSp(2)representationsr esultingfromtheabove analysisforspinorsdon'tincludesinglets.Thismodicat ionisanalogoustotheadditionofthe B@=@ e C termsto Q in(3.2.6a).WecanthinkoftheOSp(1,1 j 2)generators of(3.4.2)as\orbital"generators,andadd\spin"generato rswhichthemselvesgenerateOSp(1,1 j 2).Inparticular,sincewearehereconsideringspinors,we choosethe spingeneratorstobethoseforthesimplestspinorrepresen tation,thegradedgeneralizationofaDiracspinor,whosegeneratorscanbeexpre ssedintermsofgraded Dirac\matrices": f ~ r A ; ~ r B ]=2 AB ;S AB = 1 4 [~ r A ; ~ r B g ;J AB 0 = J AB + S AB ; (3 : 5 : 1) where f ; ]istheoppositeof[ ; g .These~ r matricesarenottobeconfusedwith the\ordinary" r matriceswhichappearin M ij fromthedimensionalcontinuation ofthetruespinoperators.The~ r A ,like r i ,arehermitian.(Thehermiticityof r i inthelight-coneformalismfollowsfrom( r i ) 2 =1foreach i andthefactthatall statesinthelight-coneformalismhavenonnegativenorm,s incethey'rephysical.) Thechoiceofwhetherthe~ r 's(andalsothegraded r i 's)commuteoranticommute withotheroperators(whichcouldbearbitrarilychangedby aKleintransformation) followsfromtheindexstructureasusual(bosonicforindic es ,fermionicfor ). (Thus,asusual,theordinary r matrices r a commutewithotheroperators,although theyanticommutewitheachother.)

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46 3.GENERALBRST InordertoputtheOSp(1,1 j 2)generatorsinaformmoresimilarto(3.4.2),we needtoperformunitarytransformationswhicheliminateth enewtermsin J + and J + (whilenotaecting J ,althoughchanging J ).Ingeneral,theappropriate transformations J AB 0 = UJ AB U 1 toeliminatesuchtermsare: lnU = ( lnp + ) S + (3 : 5 : 2 a ) torsteliminatethe S + termfrom J + ,andthen lnU = S + p (3 : 5 : 2 b ) todothesamefor J + .Thegeneralresultis J + = ix p + + k;J + = ix p + ;J = ix ( p ) + c M ; J = ix p + 1 p + h ix 1 2 ( p b p b + M 2 + p p )+ c M p + kp + b Q i ;(3 : 5 : 3 a ) c M = M + S ; b Q =( M b p b + M m M )+ h S + S + 1 2 ( p b p b + M 2 ) i : (3 : 5 : 3 b ) (3.5.3a)isthesameas(3.4.2a),butwith M and Q replacedby c M and b Q .In thiscasethelasttermin b Q is S + S + 1 2 ( p b p b + M 2 )= 1 2 ~ r h ~ r +~ r + 1 2 ( p b p b + M 2 ) i : (3 : 5 : 4) Wecanagainchoose k =0. Thisalgebrawillbeusedtoderivefreegauge-invariantact ionsforfermionsin sect.4.5.Thegeneralizationtofermionicstringsfollows fromtherepresentationof thePoincarealgebragiveninsect.7.2.3.6.Moredimensions Intheprevioussectionwesawthatfermionscouldbetreated inawaysimilarto bosonsbyincludinganOSp(1,1 j 2)Cliordalgebra.InthecaseoftheDiracspinor, thereisalreadyanOSp(D 1,1 j 2)Cliordalgebra(orOSp(D,1 j 2)inthemassive case)obtainedbyadding2+2dimensionstothelight-cone r -matrices,intermsof which M ij (andthereforetheOSp(1,1 j 2)algebra)isdened.Includingtheadditional r -matricesmakesthespinorarepresentationofanOSp(D,2 j 4)Cliordalgebra (OSp(D+1,2 j 4)formassive),andisthusequivalenttoadding4+4dimensi onstothe

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3.6.Moredimensions 47 originallight-conespinorinsteadof2+2,ignoringtheext raspacetimecoordinates. Thissuggestsanotherwayoftreatingfermionswhichallows bosonstobetreated identically,andshouldthusallowastraightforwardgener alizationtosupersymmetric theories[3.16]. Weproceedsimilarlytothe2+2case:Beginbyadding4+4dime nsionstothe light-conePoincarealgebra(2.3.5).Truncatetheresult ingIOSp(D+1,3 j 4)algebrato ISO(D 1,1) n IOSp(2,2 j 4).IOSp(2,2 j 4)contains(inparticular)2inequivalenttruncationstoIOSp(1,1 j 2),whichcanbedescribedby(dening-representationdire ctproduct)factorizationoftheOSp(2,2 j 4)metricintotheOSp(1,1 j 2)metrictimesthe metricofeitherSO(2)(U(1))orSO(1,1)(GL(1)): AB = AB ^ a ^ b ; A = A ^ a J AB = ^ b ^ a J A ^ a;B ^ b ; ^ a ^ b = BA J A ^ a;B ^ b ; (3 : 6 : 1) whereisthegeneratorofthe U (1)or GL (1)and ^ a ^ b = I or 1 .These2OSp(1,1 j 2)'s areWickrotationsofeachother.We'lltreatthe2casessepa rately. TheGL(1)casecorrespondstorsttakingtheGL(2 j 2)(=SL(2 j 2) n GL(1))subgroupofOSp(2,2 j 4)(asSU(N) SO(2N),orGL(1 j 1) OSp(1,1 j 2)),keepingalsohalf oftheinhomogeneousgeneratorstogetIGL(2 j 2).ThentakingtheOSp(1,1 j 2)subgroupoftheSL(2 j 2)(inthesamewayasSO(N) SU(N)),wegetIOSp(1,1 j 2) n GL(1), whichislikethePoincaregroupin(1,1 j 2)dimensionsplusdilatations.(Thereisalso anSL(1 j 2)=OSp(1,1 j 2)subgroupofSL(2 j 2),butthisturnsoutnottobeuseful.)The advantageofbreakingdowntoGL(2 j 2)isthatforthissubgroupthecoordinatesof thestring(sect.8.3)canberedenedinsuchawaythattheex trazero-modesare separatedoutinanaturalwaywhileleavingthegeneratorsl ocalin .ThisGL(2 j 2) subgroupcanbedescribedbywritingtheOSp(2,2 j 4)metricas AB = 0 AB 0 A 0 B 0 ; AB 0 =( 1) AB B 0 A ; A =( A;A 0 )(3 : 6 : 2 a ) J AB = ~ J AB ~ J AB 0 ~ J A 0 B ~ J A 0 B 0 ; ~ J AB 0 = ( 1) AB ~ J B 0 A ; (3 : 6 : 2 b ) where ~ J A 0 B aretheGL(2 j 2)generators,towhichweaddthe~ p A 0 halfof p A toform IGL(2 j 2).(Themetric AB 0 canbeusedtoeliminateprimedindices,leavingcovariant andcontravariantunprimedindices.)Inthisnotation,the original indicesofthe lightconearenow+ 0 and (whereas+and 0 are\transverse").Toreduceto

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48 3.GENERALBRST theIOSp(1,1 j 2) n GL(1)subgroupweidentifyprimedandunprimedindices;i.e .,we choosethesubgroupwhichtransformstheminthesameway: J AB = ~ J A 0 B + ~ J AB 0 ; p A =~ p A 0 ; = BA 0 ~ J A 0 B : (3 : 6 : 3) Wedistinguishthemomenta p A andtheirconjugatecoordinates x A ,whichwewish toeliminate,from p A =~ p A andtheirconjugates x A ,whichwe'llkeepastheusual onesofOSp(1,1 j 2)(includingthenonlinear p ).Atthispointthesegeneratorstake theexplicitform = i x A p A ix p + ix p + M 0 + + C M 0 ; J + = i p + x i x + p + ip + x + M + 0 ; J + = i x p + + i x + p ix p + + M 0 + ; J = i x ( p ) ix ( p ) + M 0 + M 0 ; J = i x p + i x p ix p + ix p + M 0 + 1 p + Q 0 ; p = 1 2 p + ( p a 2 + M 2 +2 p + p +2 p p ) ; Q 0 = M 0 a p a + M 0 m M + M 0 + p + M 0 0 p + M 0 p M 0 0 p : (3 : 6 : 4) (Allthe p 'sarelinear,beingunconstrainedsofar.)Wenowuse p andtoeliminate theextrazero-modes.Weapplytheconstraintsandcorrespo ndinggaugeconditions =0 x = 1 p + ( x p + + x p x + p x p + iM 0 + + iC M 0 ) ; gauge p + =1; p A p A =0 p = 1 2 p p ; gauge x + =0 : (3 : 6 : 5) Theseconstraintsaredirectlyanalogousto(2.2.3),which wereusedtoobtainthe usualcoordinaterepresentationoftheconformalgroupSO( D,2)fromtheusualcoordinaterepresentation(with2morecoordinates)ofthesa megroupasaLorentz group.Infact,aftermakingaunitarytransformationofthe type(3.5.2b), U = e ( ip + x + M + 0 ) p ; (3 : 6 : 6) theremainingunwantedcoordinates x completelydecouple: UJ AB U 1 = J AB + J 0 AB ; (3 : 6 : 7 a )

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3.6.Moredimensions 49 J + = i x ; J + = i x p ; J = i x ( p ) ; J = i x p p ; (3 : 6 : 7 b ) J 0 + =2( ip + x + M + 0 ) ;J 0 + =2( ix p + + M 0 + )+( ix p + C M 0 ) ; J 0 = ix ( p ) +( M 0 + M 0 ) ; J 0 =( ix p + M 0 ) + h ix 1 2 ( p a 2 + M 2 )+( M 0 a p a + M 0 m M + M 0 0 p + M 0 0 p ) i ; (3 : 6 : 7 c ) where J arethegeneratorsoftheconformalgroupin2anticommuting dimensions ((3.3.9),afterswitchingcoordinatesandmomenta),and J 0 arethedesiredOSp(1,1 j 2) generators. Toeliminatezero-modes,it'sconvenienttotransformthes eOSp(1,1 j 2)generators tothecanonicalform(3.4.2a).Thisisperformed[3.7]byth eredenition p + 1 2 p + 2 ; (3 : 6 : 8 a ) followedbytheunitarytransformations U 1 = p + i 1 2 [ x ;p ]+2 M 0 + + C M 0 ; U 2 = e 2 M + 0 p : (3 : 6 : 8 b ) Since p + isimaginary(thoughhermitian)in( x )coordinatespace, U 1 changesreality conditionsaccordingly(an i foreach p + ).Thegeneratorsarethen J + = ix p + ;J + = ix p + ;J = ix ( p ) + c M ; J = ix p + 1 p + h ix 1 2 ( p a 2 + M 2 + p p )+ c M p + b Q i ;(3 : 6 : 9 a ) c M = M 0 + M 0 ; b Q = M 0 1 2 M 0 0 +( M 0 a p a + M 0 m M )+ M + 0 ( p a 2 + M 2 ) : (3 : 6 : 9 b ) FortheU(1)casethederivationisalittlemorestraightfor ward.Itcorresponds torsttakingtheU(1,1 j 1,1)(=SU(1,1 j 1,1) n U(1))subgroupofOSp(2,2 j 4).From (3.6.1),insteadof(3.6.2a,3)wenowhave AB = AB 0 0 A 0 B 0 ; AB = A 0 B 0 ; A =( A;A 0 ) ;

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50 3.GENERALBRST J AB = ~ J AB + ~ J A 0 B 0 ; p A =~ p A 0 ; = BA ~ J A 0 B : (3 : 6 : 10) Theoriginallight-cone arenowstill (noprimes),sotheunwantedzero-modes canbeeliminatedbytheconstraintsandgaugechoices p A =0 gauge x A =0 : (3 : 6 : 11) Alternatively,wecouldinclude p A amongthegenerators,usingIOSp(1,1 j 2)asthe group(asfortheusualclosedstring:seesects.7.1,11.1). (Thesameresultcanbe obtainedbyreplacing(3.6.11)withtheconstraints=0and ^ b ^ a p A ^ a p B ^ b p [ A p B ) = 0.)TheOSp(1,1 j 2)generatorsarenow J + = ix p + + M + 0 0 ;J + = ix p + + M 0 + 0 ; J = ix ( p ) + M + M 0 0 ; J = ix p ix 1 2 p + ( p a 2 + M 2 + p p )+ 1 p + ( M a p a + M m M + M p )+ M 0 0 ; (3 : 6 : 12 a ) or,inotherwords(symbols),theseOSp(1,1 j 2)generatorsarejusttheusualonesplus thespinofasecondOSp(1,1 j 2),withthesamerepresentationasthespinoftherst OSp(1,1 j 2): J AB = ~ J AB + M A 0 B 0 : (3 : 6 : 12 b ) (However,forthestring M m M willcontainoscillatorsfrombothsetsof2+2dimensions,sothesesetsofoscillatorswon'tdecouple,eventho ugh ~ J AB commuteswith M A 0 B 0 .)Tosimplifytheformof J + and J ,wemaketheconsecutiveunitary transformations(3.5.2): U 1 = p + M 0 + 0 ;U 2 = e M + 0 0 p ; (3 : 6 : 13) afterwhichthegeneratorsagaintakethecanonicalform: J + = ix p + ;J + = ix p + ;J = ix ( p ) + c M ; J = ix p + 1 p + h ix 1 2 ( p a 2 + M 2 + p p )+ c M p + b Q i ;(3 : 6 : 14 a ) c M = M + M 0 0 ; b Q = M 0 0 +( M b p b + M m M )+ M + 0 0 1 2 ( p a 2 + M 2 ) : (3 : 6 : 14 b ) Becauseof U 1 ,formerlyrealeldsnowsatisfy y =( 1) M 0 + 0 Examplesandactionsofthis4+4-extendedOSp(1,1 j 2)willbeconsideredinsect. 4.1,itsapplicationtosupersymmetryinsect.5.5,anditsa pplicationtostringsin sect.8.3.

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Exercises 51 Exercises (1)Derivethetimederivativeof(3.1.6)from(3.1.7).(2)Derive(3.1.12).Comparewiththeusualderivationofth eNoethercurrentin eldtheory. (3)Derive(3.1.15,17).(4)Showthat Q of(3.2.6a)isnilpotent.Showthisdirectlyfor(3.2.8). (5)Derive(3.3.1b).(6)Use(3.3.6)torederive(3.2.6a,12a).(7)Use(3.3.2,7)toderivetheOSp(1,1 j 2)algebraforYang-Millsintermsofthe explicitindependentelds(inanalogyto(3.2.8)). (8)Performthetransformation(3.4.3a)toobtain(3.4.3b) .ChoosetheDiracspinor representationofthespinoperators(intermsof r -matrices).Comparewith (3.2.6),andidentifytheeldequations G andghosts C (9)Checkthatthealgebraof b Q and c M closesfor(3.5.3),(3.6.9),and(3.6.14), andcomparewith(3.4.2).

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52 4.GENERALGAUGETHEORIES 4.GENERALGAUGETHEORIES4.1.OSp(1,1 j 2) Inthischapterwewillusetheresultsofsects.3.4-6toderi vefreegauge-invariant actionsforarbitraryeldtheories,anddiscusssomepreli minaryresultsfortheextensiontointeractingtheories. The(free)gaugecovarianttheoryforarbitraryrepresenta tionsofthePoincare group(exceptperhapsforthosesatisfyingself-dualityco nditions)canbeconstructed fromtheBRST1OSp(1,1 j 2)generators[2.3].Fortheeldsdescribedinsect.3.4 whicharerepresentationsofOSp(D,2 j 2),considerthegaugeinvariancegeneratedby OSp(1,1 j 2)andtheobvious(butunusual)correspondinggauge-invar iantaction: = 1 2 J BA AB S = Z d D x a dx d 2 x 1 2 y p + ( J AB ) ; (4 : 1 : 1) where J AB for A =(+ ; ; )(gradedantisymmetricinitsindices)arethegenerators ofOSp(1,1 j 2),andwehaveset k =0,sothatthe p + factoristheHilbertspace metric.Inparticular,the J + and J + transformationsallowalldependenceonthe unphysicalcoordinatestobegaugedaway: = ix p + + + ix p + (4 : 1 : 2) impliesthatonlythepartofat x = x =0canbegaugeinvariant.Amore explicitformof ( J AB )isgivenby p + ( J AB )= p + ( J 2 ) i ( J + ) 2 ( J + ) 2 ( J ) = ( x ) 2 ( x ) ( M 2 ) p + 2 J 2 ; (4 : 1 : 3) wherewehaveused J + ( J + )= ( J + ) J + =0 ( J + )= i 1 p + ( x ) ; (4 : 1 : 4)

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4.1.OSp(1,1 j 2) 53 since p + 6 =0inlight-coneformalisms.Thegaugeinvarianceofthekin eticoperator followsfromthefactthatthe -functionscanbereorderedfairlyfreely: ( J 2 ) (whichisreallyaKronecker )commuteswithalltheothers,while ( J + + a ) 2 ( J )= 2 ( J ) ( J + + a 2) [ ( J + ) ; 2 ( J + ) 2 ( J )]=0 ; [ 2 ( J ) ; 2 ( J + )]=2 J + +( C J + + J ) 1 2 [ J ;J + ] ; (4 : 1 : 5) wherethe J + and J eachcanbefreelymovedtoeithersideofthe[ J ;J + ]. Afterintegrationoftheactionoverthetrivialcoordinate dependenceon x and x (4.1.1)reducesto(using(3.4.2,4.1.3)) S = Z d D x a 1 2 y ( M 2 )( 2 M 2 + Q 2 ) ; = i 1 2 Q + 1 2 M ; (4 : 1 : 6) where nowdependsonlyontheusualspacetimecoordinates x a ,andforirreducible Poincarerepresentations hasindiceswhicharetheresultofstartingwithanirreduciblerepresentationofOSp(D 1,1 j 2)inthemasslesscase,orOSp(D,1 j 2)inthe massivecase,andthentruncatingtotheSp(2)singlets.(Th istypeofactionwasrst proposedforthestring[4.1,2].) istheremainingpartofthe J transformations afterusingupthetransformationsof(4.1.2)(andabsorbin ga1 =@ + ),andcontains theusualcomponentgaugetransformations,while justgaugesawaytheSp(2) nonsinglets.Wehavethusderivedageneralgauge-covarian tactionbyadding2+2 dimensionstothelight-conetheory.Insect.4.4we'llshow thatgauge-xingtothe lightconegivesbacktheoriginallight-conetheory,provi ngtheconsistencyofthis method. IntheBRSTformalismtheeldcontainsnotonlyphysicalpol arizations,but alsoauxiliaryelds(nonpropagatingeldsneededtomaket heactionlocal,suchas thetraceofthemetrictensorforthegraviton),ghosts(inc ludingantighosts,ghosts ofghosts,etc.),andStueckelbergelds(gaugedegreesoff reedom,suchasthegauge partofHiggselds,whichallowmorerenormalizableandles ssingularformalismsfor massiveelds).Allofthesebuttheghostsappearinthegaug e-invariantaction.For example,foramasslessvectorwestartwith A i =( A a ;A ),whichappearsintheeld as E = i E A i ; D i j E = ij : (4 : 1 : 7) ReducingtoSp(2)singlets,wecantruncatetojust A a .Usingtherelations M ij k E = [ i E j ) k M a b E = ab E ;M a E = C a E ; (4 : 1 : 8)

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54 4.GENERALGAUGETHEORIES where[)isgradedantisymmetrization,wend 1 2 M b M c a E = 1 2 M b ac E = ac b E ; (4 : 1 : 9) andthusthelagrangian L = 1 2 D ( M 2 )[ 2 ( M b @ b ) 2 ] E = 1 2 A a ( 2 A a @ a @ b A b ) : (4 : 1 : 10) Similarly,forthegaugetransformation E = i E i A a = D a 1 2 M b @ b E = 1 2 @ a : (4 : 1 : 11) Asaresultofthe ( M 2 )actingon Q ,theonlypartofwhichsurvivesis thepartwhichisanoverallsingletinthematrixindicesand explicit index:in thiscase, i = i A a = @ a .Notethatthe Q 2 termcanbewrittenas a( Q ) 2 term:Thiscorrespondstosubtractingouta\gauge-xing"t ermfromthe \gauge-xed"lagrangian ( 2 M 2 ) .(Seethediscussionofgaugexinginsect. 4.4.) Foramasslessantisymmetrictensorwestartwith A [ ij ) =( A [ ab ] ;A a ;A ( ) )appearingas E = 1 2 ij E A A ji ; ij E A = 1 p 2 [ i E n j ) E (4 : 1 : 12) (andsimilarlyfor j i ),andtruncatetojust A [ ab ] .Then,from(4.1.8), 1 2 M c M d a E b E = 1 2 M c da E b E + db a E E = da c E b E + db a E c E + d ( a b ) c 1 2 E E ; (4 : 1 : 13) andwehave L = 1 4 A ab ( 2 A ab + @ c @ [ a A b ] c ) ;A ab = 1 2 @ [ a b ] : (4 : 1 : 14) Foramasslesstracelesssymmetrictensorwestartwith h ( ij ] =( h ( ab ) ;h b ;h [ ] ) satisfying h i i = h a a + h =0,appearingas E = 1 2 ij E S h ji ; ij E S = 1 p 2 ( i E j ] E ; (4 : 1 : 15) andtruncateto( h ( ab ) ;h [ ] ),where h [ ] = 1 2 C ab h ( ab ) ,leavingjustanunconstrainedsymmetrictensor.Then,using(4.1.13),aswellas 1 2 M ra M r b E E = 1 2 C ( a E b ) E ab E E ; (4 : 1 : 16)

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4.1.OSp(1,1 j 2) 55 andusingthecondition h = h a a ,wend L = 1 4 h ab 2 h ab 1 2 h ab @ b @ c h a c + 1 2 h c c @ a @ b h ab 1 4 h a a 2 h b b ; h ab = 1 2 @ ( a b ) : (4 : 1 : 17) ThisisthelinearizedEinstein-Hilbertactionforgravity Themassivecasescanbeobtainedbythedimensionalreducti ontechnique,as in(2.2.9),sincethat'showitwasdoneforthisentireproce dure,fromthelight-cone Poincarealgebradownto(4.1.6).(Forthestring,theOSpg eneratorsarerepresented intermsofharmonicoscillators,and M m M iscubicinthoseoscillatorsinsteadof quadratic,sotheoscillatorexpressionsforthegenerator sdon'tfollowfromdimensionalreduction,and(4.1.6)mustbeuseddirectlywiththe M m M terms.)Technically, p m = m makessenseonlyforcomplexelds.However,atleastforfre etheories, theresulting i 'sthatappearinthe p a p m crosstermscanberemovedbyappropriate redenitionsforthecomplexelds,afterwhichtheycanbec hosenreal.(Seethediscussionbelow(2.2.10).)Forexample,forthemassivevecto rwereplace A m iA m (andthentake A m real)toobtain E = a E A a + i m E A m + E A ; D = A a D a iA m D m + A D : (4 : 1 : 18) Thelagrangianandinvariancethenbecome L = 1 2 A a [( 2 m 2 ) A a @ a @ b A b ]+ 1 2 A m 2 A m + mA m @ a A a = 1 4 F ab 2 1 2 ( mA a + @ a A m ) 2 ; A a = @ a ;A m = m: (4 : 1 : 19) ThisgivesaStueckelbergformalismforamassivevector. Otherexamplesreproduceallthespecialcasesofhigher-sp ineldsproposedearlier[4.3](aswellascasesthathadn'tbeenobtainedprevio usly).Forexample,for totallysymmetrictensors,theusual\double-tracelessne ss"conditionisautomatic: Startingfromthelightconewithatotallysymmetricandtra celesstensor(intransverseindices),extending i ( a; )andrestrictingtoSp(2)singlets,directlygivesa totallysymmetricandtracelesstensor(inD-dimensionali ndices)ofthesamerank, andoneofrank2lower(butnolowerthanthat,duetothetotal antisymmetryin theSp(2)indices). ThemostimportantfeatureoftheBRSTmethodofderivinggau ge-invariant actionsfromlight-cone(unitary)representationsoftheP oincaregroupisthatit

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56 4.GENERALGAUGETHEORIES automaticallyincludesexactlytherightnumberofauxilia ryeldstomaketheaction local.InthecaseofYang-Mills,theauxiliaryeld( A )wasobvious,sinceitresults directlyfromaddingjust2commutingdimensions(andnot2a nticommuting)tothe lightcone,i.e.,frommaking D 2-dimensionalindices D -dimensional.Furthermore, thenecessityofthiseldforlocalitydoesn'toccuruntili nteractionsareincluded (seesect.2.1).Alesstrivialexampleisthegraviton:Naiv ely,atracelesssymmetric D -dimensionaltensorwouldbeenough,sincethiswouldautom aticallyincludethe analogof A .However,theBRSTmethodautomaticallyincludesthetrace ofthis tensor.Ingeneral,theextraauxiliaryeldswithanticomm uting\ghost-valued" Lorentzindicesarenecessaryforgauge-covariant,localf ormulationsofeldtheories [4.4,5].Inordertostudythisphenomenoninmoredetail,an dbecausethediscussion willbeusefullaterinthe2Dcaseforstrings,wenowgiveabr iefdiscussionofgeneral relativity. GeneralrelativityisthegaugetheoryofthePoincaregrou p.Sincelocaltranslations(i.e.,generalcoordinatetransformations)inclu detheorbitalpartofLorentz transformations(astranslationbyanamountlinearin x ),wechooseasthegroup generators @ m andtheLorentzspin M ab .Treating M ab assecond-quantizedoperators,weindicatehowtheyactbywritingexplicit\spin"vec torindices a;b;::: (or spinorindices)ontheelds,whileusing m;n;::: for\orbital"vectorindicesonwhich M ab doesn'tact,ason @ m .(Theactionofthesecond-quantized M ab followsfromthat oftherst-quantized:E.g.,from(4.1.8),(2.2.5),andthe factthat( M ij ) y = M ij wehave M ab A c = c [ a A b ] .)Thespinindices(butnottheorbitalones)canbecontractedwiththeusualconstanttensorsoftheLorentzgroup (theLorentzmetricand r matrices).The(antihermitian)generatorsofgaugetransf ormationsarethus = m ( x ) @ m + 1 2 ab ( x ) M ba ; (4 : 1 : 20) andthecovariantderivativesare r a = e a m @ m + 1 2 a bc M cb ; (4 : 1 : 21) wherewehaveabsorbedtheusualderivativeterm,sincederi vativesarethemselves generators,andtomakethecovariantderivativetransform covariantlyunderthe gaugetransformations r a 0 = e r a e : (4 : 1 : 22) Covarianteldstrengthsaredened,asusual,bycommutato rsofcovariantderivatives, [ r a ; r b ]= T ab c r c + 1 2 R ab cd M dc ; (4 : 1 : 23)

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4.1.OSp(1,1 j 2) 57 sincethatautomaticallymakesthemtransformcovariantly (i.e.,byasimilaritytransformation,asin(4.1.22)),asaconsequenceofthetransfor mationlaw(4.1.22)ofthe covariantderivativesthemselves.Withoutlossofgeneral ity,wecanchoose T ab c =0 ; (4 : 1 : 24) sincethisjustdetermines ab c intermsof e a m ,andanyother canalwaysbewritten asthis plusatensorthatisafunctionofjust T .(Thetheorycouldthenalwaysbe rewrittenintermsofthe T =0 r and T itself,making T anarbitraryextratensor withnospecialgeometricsignicance.)Tosolvethisconst raintwerstdene e a = e a m @ m ; [ e a ;e b ]= c ab c e c : (4 : 1 : 25) c ab c canthenbeexpressedintermsof e a m ,thematrixinverse e m a e a m e m b = a b ;e m a e a n = m n ; (4 : 1 : 26) andtheirderivatives.Thesolutionto(4.1.24)isthen abc = 1 2 ( c bca c a [ bc ] ) : (4 : 1 : 27) TheusualglobalLorentztransformations,whichincludeor bitalandspinpieces togetherinaspecicway,areasymmetryofthevacuum,dene dby hr a i = @ a $h e a m i = a m : (4 : 1 : 28) isanarbitraryconstant,whichwecanchoosetobeaunitofle ngth,sothat r isdimensionless.(In D =4it'sjusttheusualgravitationalcouplingconstant, proportionaltothesquarerootofNewton'sgravitationalc onstant.)Asaresultof generalcoordinateinvariance,anycovariantobject(i.e. ,acovariantderivativeor tensorwithonlyspinindicesuncontracted)willthenalsob edimensionless.The subgroupoftheoriginalgaugegroupwhichleavesthevacuum (4.1.28)invariant isjusttheusual(global)Poincaregroup,whichtreatsorb italandspinindicesin thesameway.Wecanalsotreattheseindicesinasimilarwayw ithrespectto thefullgaugegroupbyusingthe\vielbein" e a m anditsinversetoconvertbetween spinandorbitalindices.Inparticular,theorbitalindice sonalleldsexceptthe vielbeinitselfcanbeconvertedintospinones.Also,since integrationmeasuresare antisymmetric,converting dx m inton a = dx m e m a converts d D x inton D = d D x e 1

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58 4.GENERALGAUGETHEORIES wheree= det ( e a m ).Onsuchcovariantelds, r alwaysactscovariantly.Onthe otherhand,intheabsenceofspinors,allindicescanbeconv ertedintoorbitalones. Inparticular,insteadofthevielbeinwecouldworkwiththe metrictensorandits inverse: g mn = ab e m a e n b ;g mn = ab e a m e b n : (4 : 1 : 29) Then,insteadof r ,wewouldneedacovariantderivativewhichknowshowtotrea t uncontractedorbitalindicescovariantly. Theactionforgravitycanbewrittenas S = 1 2 Z d D x e 1 R;R = 1 2 R ab ba : (4 : 1 : 30) Thiscanberewrittenintermsof c abc as e 1 R = @ m (e 1 e am c a b b )+e 1 h 1 2 ( c ab b ) 2 + 1 8 c abc c abc 1 4 c abc c bca i (4 : 1 : 31) using e 1 e a f a = @ m (e 1 e am f a )+e 1 c ab b f a : (4 : 1 : 32) Expandingaboutthevacuum, e a m = a m + D= 2 h a m ; (4 : 1 : 33) wherewecanchoose e am (andthus h am )tobesymmetricbythe ab transformation, thelinearizedactionisjust(4.1.17).Asanalternativefo rmfortheaction,wecan considermakingtheeldredenition e a m 2 = ( D 2) e a m ; (4 : 1 : 34) whichintroducesthenewgaugeinvarianceof(Weyl)localsc aletransformations e a m 0 = e e a m ; 0 = e ( D 2) = 2 : (4 : 1 : 35) (Thegaugechoice = constant returnstheoriginalelds.)Undertheeldredenition(4.1.34),theaction(4.1.30)becomes S Z d D x e 1 h 2 D 1 D 2 ( r a ) 2 1 2 R 2 i : (4 : 1 : 36) Wehaveactuallystartedfrom(4.1.30)withoutthetotal-de rivativetermof(4.1.31), whichisthenafunctionofjust e a m anditsrstderivatives,andthuscorrectevenat boundaries.(Wealsodroppedatotal-derivativeterm @ m ( 1 2 2 e 1 e am c a b b )in(4.1.36),

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4.1.OSp(1,1 j 2) 59 whichwillbeirrelevantforthefollowingdiscussion.)Ifw eeliminate byitseld equation,butkeepsurfaceterms,thisbecomes S Z d D x e 1 2 D 1 D 2 r ( r ) = Z d D x e 1 D 1 D 2 2 2 = Z d D x e 1 D 1 D 2 2 h h i 2 +2 h i ( h i )+( h i ) 2 i (4 : 1 : 37) Wecansolvethe eldequationas = h i 1 1 4 D 1 D 2 2 + R R : (4 : 1 : 38) (Wecanchoose h i =1,ortakethe outof(4.1.28)andintroduceitinsteadthrough h i = ( D 2) = 2 byaglobal transformation.)Assuming fallsoto h i fastenough at 1 ,thelasttermin(4.1.37)canbedropped,and,using(4.1.38 ),theactionbecomes [4.6] S 1 2 Z d D x e 1 R R 1 4 D 1 D 2 2 + R R : (4 : 1 : 39) Sincethisactionhastheinvariance(4.1.35),wecangaugea waythetraceof h or, equivalently,gaugethedeterminantof e a m to1.Infact,thesameactionresultsfrom (4.1.30)ifweeliminatethisdeterminantbyitsequationof motion. Thus,weseethat,althoughgauge-covariant,Lorentz-cova riantformulationsare possiblewithouttheextraauxiliaryelds,theyarenonloc al.Furthermore,thenonlocalitiesbecomemorecomplicatedwhencouplingtononcon formalmatter(suchas massiveelds),inawayreminiscentofCoulombtermsorthen onlocalitiesinlightconegauges.Thus,theconstructionofactionsinsuchaform alismisnotstraightforward,andrequirestheuseofWeylinvarianceinawayanalogo ustotheuseofLorentz invarianceinlight-conegauges.Anotheralternativeisto eliminatethetraceofthe metricfromtheEinsteinactionbyacoordinatechoice,butt heremainingconstrained (volume-preserving)coordinateinvariancecausesdicul tiesinquantization[4.7]. Wehavealsoseenthatsomepropertiesofgravity(theonesre latingtoconformal transformations)becomemoretransparentwhenthescaleco mpensator isintroduced.(Thisisparticularlytrueforsupergravity.)Intro ducingsucheldsintothe OSpformalismrequiresintroducingnewdegreesoffreedom, tomaketherepresentationlarger(atleastintermsofgaugedegreesoffreedom).A lthoughsuchinvariances arehardtorecognizeatthefreelevel,theextensionsofsec t.3.6showsignsofperformingsuchgeneralizations.However,whiletheU(1)-typ eextensioncanbeapplied

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60 4.GENERALGAUGETHEORIES toarbitraryPoincarerepresentations,theGL(1)-typeha sdicultywithfermions. We'llrstdiscussthisdiculty,thenshowhowthe2typesdi erforbosonsevenfor thevector,andnallylookagainatgravity. TheU(1)caseofspin1/2reproducesthealgebraofsect.3.5, since M A 0 B 0 of (3.6.12b)isexactlytheextratermof(3.5.1): M ij = 1 2 r ij ( c M 2 )( 2 + b Q 2 ) = () e ir + 0 = p= 2 i 1 2 =pr 0 e ir + 0 = p= 2 = i 1 4 ( r 0 + i 1 2 r 0 r + 0 =p ) () r + 0 =p ( r 0 + i 1 2 r 0 r + 0 =p ) = i 1 2 ^ =pr 0 ^ ; (4 : 1 : 40) where r ij = 1 2 [ r i ;r j g ,andwehaveused 0= 1 8 ( r + r 0 0 )( r + r 0 0 )=( r r 0 ) 2 +4 r r 0 =2 i: (4 : 1 : 41) (Wecouldequallywellhavechosentheothersign.Thischoic e,withourconventions,correspondstoharmonic-oscillatorboundarycondi tions:Seesect.4.5.)After eliminating r + 0 bygaugechoiceor,equivalently,byabsorbingitinto r 0 byeldredenition,thisbecomesjust '=p' .However,intheGL(1)case,theanalogto(4.1.41) is 0= 1 8 ( f r ;r 0 g + f r 0 ;r g )( f r ;r 0 g + f r 0 ;r g )=( r r 0 )( r 0 r ) ; r r 0 + r 0 r = 4 ; (4 : 1 : 42) andto(4.1.40)is 2 + b Q 2 = p 2 + 1 8 ( r r 0 ) r 0 ( =p r + p 2 )+ 1 8 ( r 0 r )( =p r + p 2 ) r 0 : (4 : 1 : 43) Unfortunately, and musthaveoppositeboundaryconditions r r 0 =0or r 0 r = 0inordertocontributeinthepresenceof ( c M 2 ),asisevidencedbytheasymmetric formof(4.1.43)foreitherchoice.Consequently,theparts of and thatsurvive arenothermitianconjugatesofeachother,andtheactionis notunitary.(Properly speaking,ifwechooseconsistentboundaryconditionsforb oth and ,theaction vanishes.)Thus,theGL(1)-typeOSp(1,1 j 2)isunsuitableforspinorsunlessfurther modied.Inanycase,suchamodicationwouldnottreatboso nsandfermions symmetrically,whichisnecessaryfortreatingsupersymme try.(Fermionsintheusual OSpformalismwillbediscussedinmoredetailinsect.4.5.)

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4.1.OSp(1,1 j 2) 61 Forthecaseofspin1(generalizingthelight-coneHilberts pace,asin(4.1.7-8)), weexpand = j a i A a + i j 0 i A + + i j + i A ; y = A a h a j iA + h 0 j iA h + j ; (4 : 1 : 44) fortheGL(1)case,andthesameforU(1)with j + i!j + 0 i (Sp(2)-spinoreldsagain dropoutofthefull ).WendforGL(1)[3.7] L = 1 4 F ab 2 1 2 ( A + @ A ) 2 = 1 4 F ab 2 1 2 ^ A 2 ; (4 : 1 : 45 a ) where F ab = @ [ a A b ] ,andforU(1) L = 1 4 F ab 2 + 1 2 ( A + 1 2 2 A + ) 2 = 1 4 F ab 2 + 1 2 ^ A 2 : (4 : 1 : 45 b ) Inbothcases A + canbegaugedaway,and A isauxiliary.However,thesignfor U(1)-typeOSp(1,1 j 2)isthesameasforauxiliaryeldsinsupersymmetry(foro -shell irreduciblemultiplets),whereasthesignforGL(1)isoppo site.Thesigndierence isnotsurprising,consideringtheU(1)andGL(1)typesareW ickrotationsofeach other:Thisauxiliary-eldterm,togetherwiththeauxilia rycomponentof A a (the light-cone A ),appearwiththemetric ^ a ^ b of(3.6.1),andthuswiththesamesign forSO(2)(U(1)).Infact,(4.1.45b)isjustthepartofthe4D N=1super-Yang-Mills lagrangianforeldswhichareR-symmetryinvariant: A canbeidentiedwiththe usualauxiliaryeld,and A + withthe =0componentofthesupereld.Similarly, r + 0 forspin1/2canbeidentiedwiththelinear-inpartofthissupereld.This closeanalogystronglysuggeststhatthenonminimaleldso fthisformalismmay benecessaryfortreatingsupersymmetry.Notealsothatfor GL(1)theauxiliary automaticallymixeswiththespin-1\gauge-xing"functio n,likeaNakanishi-Lautrup eld,whileforU(1)thereisakindof\parity"symmetryofth eOSp(1,1 j 2)generators, j A 0 i!j A 0 i ,whichpreventssuchmixing,andcanbeincludedintheusual parity transformationstostrengthentheidenticationwithsupe rsymmetry. Forspin2,forU(1)wedene = 1 2 h ab 1 p 2 ( a E b ) E + iA a + 1 p 2 ( a E 0 ) E + iA a 1 p 2 ( a E + 0 ) E + ++ j 0 ij 0 i + + 1 p 2 (+ 0 E 0 ) E + j + 0 ij + 0 i + 1 p 2 j ij i + 0 1 2 E 0 E + 0 E E + 00 1 p 2 0 E 0 E ; (4 : 1 : 46 a ) subjecttothetracelessnesscondition( h i i =0) 1 2 h a a + + ' 00 =0 ; (4 : 1 : 46 b )

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62 4.GENERALGAUGETHEORIES andndthelagrangian L = 1 4 h ab ( p 2 h ab 2 p a p c h cb +2 p a p b h c c ab p 2 h c c ) + 1 2 ( A a 1 2 p 2 A a + + p a p b A b + i p 2 p a 0 ) 2 +( + 00 )( p 2 + p 2 + 2 p 2 00 + 1 2 p 2 p 4 ++ + p 2 h b b p b p c h bc ) =\R"+ 1 2 ^ A a 2 + p 2^ + ^ : (4 : 1 : 47) Thesecondtermisthesquareofanauxiliary\axial"vector( whichagainappears withsignoppositetothatinGL(1)[3.7]),whichresemblest heaxialvectorauxiliary eldofsupergravity(includingtermswhichcanbeabsorbed ,asforspin1).Inthe lastterm,theredenition ^ involvesthe(linearized)Ricciscalar.Although it'sdiculttotellfromthefreetheory,itmayalsobepossi bletoidentifysomeof thegaugedegreesoffreedomwithconformalcompensators: 0 withthecompensator forlocalR-symmetry,and + (or or 00 ;oneiseliminatedbythetracelessness conditionandoneisauxiliary)withthelocalscalecompens ator. Asimpleexpressionforinteractingactionsintermsofjust theOSp(1,1 j 2)group generatorshasnotyetbeenfound.(However,thisisnotthec aseforIGL(1):Seethe followingsection.)Theusualgauge-invariantinteractin geldequationscanbederivedbyimposing J = J =0,whicharerequiredina(anti)BRSTformalism, andndingtheequationssatisedbythe x = x =0sector.However,thisrequires useoftheothersectorsasauxiliaryelds,whereasintheap proachdescribedhere theywouldbegaugedegreesoffreedom. Theseresultsforgauge-invariantactionsfromOSp(1,1 j 2)willbeappliedtothe specialcaseofthestringinchapt.11.4.2.IGL(1) Wenowderivethecorrespondinggauge-invariantactionint heIGL(1)formalism andcomparewiththeOSp(1,1 j 2)results.Webeginwiththeformofthegenerators (3.4.3b)obtainedfromthetransformation(3.4.3a).Forth eIGL(1)formalismwecan thendropthezero-modes x and~ c ,andtheactionandinvariancethenare(using ( Q )= Q ) S = Z d D xdc y iQ ( J 3 ) ; = iQ + J 3 ^ : (4 : 2 : 1) ThisistheIGL(1)analogof(4.1.1).(Thisactionalsowasr stproposedforthe string[4.8,9].)The ( J 3 )killsthesignfactorin(3.4.4).However,eventhoughsome

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4.2.IGL(1) 63 unphysicalcoordinateshavebeeneliminated,theeldisst illarepresentationofthe spingroupOSp(D-1,1 j 2)(orOSp(D,1 j 2)),andthusthereisstilla\hidden"Sp(2) symmetrybrokenbythisaction(butonlybyauxiliaryelds: seebelow). Toobtaintheanalogof(4.1.6),werstexpandtheeldinthe singleghost zero-mode c : = + ic : (4 : 2 : 2) istheeldoftheOSp(1,1 j 2)formalismaftereliminationofallitsgaugezeromodes,and isanauxiliaryeld(identiedwiththeNakanishi-Lautrup auxiliary eldsinthegauge-xedformalism[4.4,5]).Ifweexpandthe action(4.2.1)in c ,using (3.4.3b),andtherealityconditionontheeldtocombinecr ossterms,weobtain,with Q =( Q + ; Q ), L = Z dc y iQ ( J 3 ) = 1 2 y ( 2 M 2 ) ( M 3 ) y M + ( M 3 +1) +2 y Q + ( M 3 ) : (4 : 2 : 3) Asanexampleofthisaction,weagainconsideramasslessvec tor.Inanalogyto (4.1.7), = i E i + ic i E i : (4 : 2 : 4 a ) Afterthe ( J 3 )projection,theonlysurvivingeldsare = a E A a + ic E B; (4 : 2 : 4 b ) where B istheauxiliaryeld.Then,usingtherelations(from(4.1. 8)) M 3 a E =0 ;M 3 E = E ; M + a E = M + + E =0 ;M + E =2 i + E ; M + a b E = ab + E ;M + a + E =0 ;M + a E = i a E ;(4 : 2 : 5) wendthelagrangianandinvariance L = 1 2 A a 2 A a 2 B 2 +2 B@ a A a ; A a = @ a ;B = 1 2 2 ;(4 : 2 : 6) whichyieldstheusualresultaftereliminationof B byitsequationofmotion.This isthesamelagrangian,includingsignsandauxiliary-eld redenitions,asforthe GL(1)-type4+4-extendedOSp(1,1 j 2),(4.1.45a).

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64 4.GENERALGAUGETHEORIES AnyIGL(1)actioncanbeobtainedfromacorrespondingOSp(1 ,1 j 2)action,and viceversa[3.13].Eliminating from(4.2.3)byitsequationofmotion, 0= S y M + ( M 3 +1) Q + ( M 3 ) !L 0 = 1 2 y ( 2 M 2 2 Q + M + 1 Q + ) ( M 3 ) ; (4 : 2 : 7) theOSp(1,1 j 2)action(4.1.6)isobtained: ( 2 M 2 2 Q + M + 1 Q + ) h ( M 3 ) ( M 2 ) i =( 2 M 2 2 Q + M + 1 Q + ) M + M + 1 ( M 3 ) = h ( 2 M 2 ) M + 2 Q + M + 1 M + Q + i M + 1 ( M 3 ) = h ( 2 M 2 ) M + 2 Q +2 i M + 1 ( M 3 ) =0 !L 0 = 1 2 y ( 2 M 2 2 Q + M + 1 Q + ) ( M 2 ) = 1 2 y ( 2 M 2 + Q 2 ) ( M 2 ) : (4 : 2 : 8) Wehavealsoused Q +2 =( 2 M 2 ) M + ,whichfollowsfromtheOSpcommutation relations,orfrom Q 2 =0. M + 1 isanSp(2)loweringoperatornormalizedsothat itistheinverseoftheraisingoperator M + ,exceptthatitvanishesonstateswhere M 3 takesitsminimumvalue[4.1].Itisn'taninverseinthestri ctsense,since M + vanishesoncertainstates,butit'ssucientforittosatis fy M + M + 1 M + = M + : (4 : 2 : 9) Wecanobtainanexplicitexpressionfor M + 1 usingfamiliarpropertiesofSO(3) (SU(2)).TheSp(2)operatorsarerelatedtotheconventiona llynormalizedSO(3) operatorsby( M 3 ;M )=2( T 3 ;T ).However,thesearereallySO(2,1)operators, andsohaveunusualhermiticityconditions: T + and T areeachhermitian,while T 3 isantihermitian.SinceforanySU(2)algebra ~ T thecommutationrelations [ T 3 ;T ]= T ; [ T + ;T ]=2 T 3 (4 : 2 : 10) imply ~ T 2 =( T 3 ) 2 + 1 2 f T + ;T g = T ( T +1) T T =( T T 3 )( T T 3 +1) ; (4 : 2 : 11)

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4.2.IGL(1) 65 wecanwrite T + 1 = 1 T T + T = 1 T 3 ;T ( T T 3 )( T + T 3 +1) T : (4 : 2 : 12) Wecanthenverify(4.2.9),aswellastheidentities T + 1 T + T + 1 = T + 1 ; T + 1 T + =1 T 3 ;T ;T + T + 1 =1 T 3 ; T : (4 : 2 : 13) Conversely,theIGL(1)actioncanbeobtainedbypartialgau ge-xingofthe OSp(1,1 j 2)action,bywriting L of(4.2.3)as L 0 of(4.2.7)plusapureBRSTvariation. Usingthecovariantlysecond-quantizedBRSToperatorofse ct.3.4,wecanwrite L = L 0 + h Q; y M + 1 [ Q; ( M 3 ) ] c i c : (4 : 2 : 14) Alternatively,wecanusefunctionalnotation,deningthe operator Q = Z dxdc ( Q ) : (4 : 2 : 15) Intermsof J + =( R; e R ),theextratermsxtheinvariancegeneratedby R ,which hadallowed c tobegaugedaway.ThisalsobreakstheSp(2)downtoGL(1),an d breakstheantiBRSTinvariance.Anotherwaytounderstandt hisisbyreformulating theIGL(1)intermsofaeldwhichhasallthezero-modesofth eOSp(1,1 j 2)eld. Considertheaction S = Z d D xd 2 x dx y p + ( J 3 ) i ( J + ) ( e R ) ( Q ) : (4 : 2 : 16) Thegaugeinvarianceisnowgivenbythe4generatorsappeari ngasargumentsofthe functions,andisreducedfromtheOSp(1,1 j 2)casebytheeliminationofthegenerators ( J R e Q ).ThisalgebraisthealgebraGL(1 j 1)of N =2supersymmetricquantum mechanics(alsoappearingintheIGL(1)formalismforthecl osedstring[4.10]:see sect.8.2):The2fermionicgeneratorsarethe\supersymmet ries," J 3 + J + isthe O(2)generatorwhichscalesthem,and J 3 J + isthe\momentum."Ifthegauge coordinates x and~ c areintegratedout,theaction(4.2.1)isobtained,ascanbe seen withtheaidof(3.4.3). Incontrasttothelightcone,wherethehamiltonianoperato r H (= p )is essentiallytheaction((2.4.4)),wendthatwiththenewco variant,second-quantized bracketof(3.4.7)thecovariantactionistheBRSToperator :Becausetheaction

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66 4.GENERALGAUGETHEORIES (2.4.8)ofagenerator(2.4.7)on y isequivalenttothegenerator'sfunctionalderivative (becauseof(3.4.7)),thegauge-invariantactionnowthoug htofasanoperatorsatises [ S; y ] c = 1 2 S : (4 : 2 : 17) Furthermore,sincethegauge-covariantequationsofmotio nofthetheoryaregiven bytheBRSTtransformationsgeneratedbytheoperator Q ,onehas S = 2 i [ Q; y ] c (4 : 2 : 18) S = iQ: (4 : 2 : 19) (Strictlyspeaking, S and Q maydierbyanirrelevant-independentterm.)This statementcanbeappliedtoanyformalismwitheldequation sthatfollowfromthe BRSToperator,independentofwhetheritoriginatesfromth elight-cone,anditholds ininteractingtheoriesaswellasfreeones.Inparticular, forthecaseofinteracting Yang-Mills,theactionfollowsdirectlyfrom(3.4.18).Aft errestrictingtheeldsto J 3 =0,thisgivestheinteractinggeneralizationoftheexampl eof(4.2.6).Theaction canalsobewrittenas S = 2 i R d Q ,where R d n 1 n +1 R y n Thisoperatorformalismisalsousefulforderivingthegaug einvariancesofthe interactingtheory,ineithertheIGL(1)orOSp(1,1 j 2)formalisms(althoughthecorrespondinginteractingactionisknowninthisformonlyfor IGL(1),(4.2.19).)Just astheglobalBRSTinvariancescanbewrittenasaunitarytra nsformation(inthe notationof(3.4.17)) U = e iL G ;G = O ; = constant; (4 : 2 : 20) where O isanyIGL(1)(orOSp(1,1 j 2))operator(incovariantsecond-quantizedform), thegaugetransformationscanbewrittensimilarlybutwith G =[ f; O ] c ; (4 : 2 : 21) where f islinearin( f = R )fortheusualgaugetransformation(and f 'shigherorderinmaygiveeld-dependentgaugetransformations). Thus, 0 = U U 1 and g () 0 = Ug () U 1 foranyfunctional g of.Inthefreecase,thisreproduces (4.1.1,4.2.1). ThisrelationbetweenOSp(1,1 j 2)andIGL(1)formalismsisimportantforrelating dierentrst-quantizationsofthestring,aswillbediscu ssedinsect.8.2.

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4.3.Extramodes 67 4.3.Extramodes Asdiscussedinsect.3.2,extrasetsofunphysicalmodescan beaddedtoBRST formalisms,suchasthosewhichLorentzgaugeshavewithres pecttotemporalgauges, orthoseinthe4+4-extendedformalismsofsect.3.6.Wenowp rovetheequivalence oftheOSp(1,1 j 2)actionsofformulationswithandwithoutsuchmodes[3.13 ].Given thatIGL(1)actionsandequationsofmotioncanbereducedto OSpforms,it'ssufcienttoshowtheequivalenceoftheIGLactionswithandwit houtextramodes. TheBRSTandghost-numberoperatorswithextramodes,after theredenitionof (3.3.6),dierfromthosewithoutbytheadditionofabelian terms.We'llprovethat theadditionofthesetermschangestheIGLaction(4.2.1)on lybyaddingauxiliary andgaugedegreesoffreedom.Toprovethis,weconsideraddi ngsuchterms2setsof modesatatime(anevennumberofadditionalghostmodesisre quiredtomaintain thefermionicstatisticsoftheintegrationmeasure): Q = Q 0 +( b y f f y b ) ; [ b ; g y ]=[ g ; b y ]= f c ; f y g = f f ; c y g =1 ; (4 : 3 : 1) intermsofthe\old"BRSToperator Q 0 andthe2newsetsofmodes b g c and f ,andtheirhermitianconjugates.Wealsoassumeboundaryco nditionsinthe newcoordinatesimpliedbytheharmonic-oscillatornotati on.(Otherwise,additional unphysicaleldsappear,andthenewactionisn'tequivalen ttotheoriginalone:see below.)Byanexplicitexpansionoftheneweldoverallthen ewoscillators, = 1 X m;n =0 ( A mn + iB mn c y + iC mn f y + iD mn f y c y ) 1 p m n ( b y ) m ( g y ) n j 0 i ; (4 : 3 : 2) wend y Q =( A y Q 0 A +2 B y Q 0 C D y Q 0 D )+2 B y ( b y D i b A ) = 1 X m;n =0 ( A y mn Q 0 A nm +2 B y mn Q 0 C nm D y mn Q 0 D nm ) +2 B y mn ( p nD n 1 ;m i 1 p m +1 A n;m +1 ) # : (4 : 3 : 3) (Thegroundstatein(4.3.2)andthematrixelementsevaluat edin(4.3.3)arewith respecttoonlythenewoscillators.)Wecanthereforeshift A mn bya Q 0 C m;n 1 term tocancelthe B y Q 0 C term(using Q 0 2 =0),andthen B mn bya Q 0 D m;n 1 termto cancelthe D y Q 0 D term.Wecanthenshift A mn by D m 1 ;n 1 tocancelthe B y D

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68 4.GENERALGAUGETHEORIES term,leavingonlythe A y Q 0 A and B y A terms.(Theseredenitionsareequivalentto thegaugechoices C = D =0usingtheusualinvariance = Q .)Finally,wecan eliminatetheLagrangemultipliers B bytheirequationsofmotion,whicheliminate allof A except A m 0 ,andfromtheformoftheremaining A y Q 0 A termwendthat alltheremainingcomponentsof A except A 00 dropout(i.e.,arepuregauge).This leavesonlytheterm A y 00 Q 0 A 00 .Thus,allthecomponentsexceptthegroundstate withrespecttothenewoscillatorscanbeeliminatedasauxi liaryorgaugedegreesof freedom.Thenetresultisthatallthenewoscillatorsareel iminatedfromtheelds andoperatorsintheaction(4.2.1),with Q thusreplacedby Q 0 (andsimilarlyfor J 3 ). (Asimilaranalysiscanbeperformeddirectlyontheequatio nsofmotion Q =0, givingthisgeneralresultforthecohomologyof Q evenincaseswhentheactionisnot givenby(4.2.1).)Thiseliminationofnewmodesrequiredth atthecreationoperators in(4.3.3)beleft-invertible: a y 1 a y =1 a y 1 = 1 a y a +1 a = 1 N +1 a N 0 ; (4 : 3 : 4) implyingthatallstatesmustbeexpressibleascreationope ratorsactingonaground state,asin(4.3.2)(theusualboundaryconditionsonharmo nicoscillatorwavefunctions,exceptthathere b and g correspondtoaspaceofindenitemetric).This provestheequivalenceoftheIGL(1)actions,andthus,byth epreviousargument, alsotheOSp(1,1 j 2)actions,withandwithoutextramodes,andthattheextram odes simplyintroducemoregaugeandauxiliarydegreesoffreedo m. Suchextramodes,althoughredundantinfreetheories,mayb eusefulinformulatinglargergaugeinvarianceswhichsimplifytheformofinte ractingtheories(as,e.g., innonlinear models).Theuseinstringtheoriesofsuchextramodescorre sponding totheworld-sheetmetricwillbediscussedinsect.8.3.4.4.Gaugexing Wenowconsidergaugexingofthesegauge-invariantaction susingtheBRST algebrafromthelightcone,andrelatethismethodtothesta ndardsecond-quantized BRSTmethodsdescribedinsects.3.1-3[4.1].Wewillndtha ttherst-quantized BRSTtransformationsoftheeldsintheusualgauge-xedac tionare identical tothe second-quantizedBRSTtransformations,buttherst-quan tizedBRSTformalismhas alargersetofelds,someofwhichdropoutoftheusualgauge -xedaction.(E.g.,see (3.4.19).However,gaugesexistwheretheseeldsalsoappe ar.)EvenintheIGL(1) formalism,althoughallthe\propagating"eldsappear,on lyasubsetoftheBRST

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4.4.Gaugexing 69 \auxiliary"eldsappear,sincethetwosetsareequalinnum berintherst-quantized IGL(1)buttheBRSTauxiliariesarefewerintheusualsecond -quantizedformalism. Wewillalsoconsidergaugexingtothelight-conegauge,an dreobtaintheoriginal light-conetheoriestowhich2+2dimensionswereadded. ForcovariantgaugexingwewillworkprimarilywithintheI GL(1)formalism, butsimilarmethodsapplytoOSp(1,1 j 2).Sincetheentire\hamiltonian" 2 M 2 vanishesundertheconstraint Q =0(actingontheeld),thefreegauge-xedaction oftheeldtheoryconsistsofonlya\gauge-xing"term: S = i Q; Z dxdc 1 2 y O c = Z dxdc 1 2 y [ O ;iQ g = Z dxdc 1 2 y K ; (4 : 4 : 1) forsomeoperator O ,wheretherst Q ,appearinginthecovariantbracket,isunderstoodtobethesecond-quantizedone.Inordertoget 2 M 2 asthekineticoperator forpartof,wechoose O = c; @ @c # K = c ( 2 M 2 ) 2 @ @c M + : (4 : 4 : 2) Whenexpandingtheeldin c 2 M 2 isthekineticoperatorforthepiececontaining allphysicalandghostelds.Explicitly,(3.4.3b),whensu bstitutedintothelagrangian L = 1 2 y K andintegratedover c ,gives @ @c L = 1 2 y ( 2 M 2 ) + y M + ; (4 : 4 : 3) andintheBRSTtransformations = iQ gives = i ( Q + M + ) ; = i h Q + 1 2 ( 2 M 2 ) ) i : (4 : 4 : 4) containspropagatingeldsand containsBRSTauxiliaryelds.Althoughthe propagatingeldsarecompletelygauge-xed,theBRSTauxi liaryeldshavethe gaugeinvariance = ;M + =0 : (4 : 4 : 5) Thesimplestcaseisthescalar= ( x ).Inthiscase,allof canbegaugedaway by(4.4.5),since M + =0.Thelagrangianisjust 1 2 ( 2 m 2 ) .Forthemassless vector(cf.(4.2.4b)), = i E A i + ic E B; (4 : 4 : 6)

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70 4.GENERALGAUGETHEORIES wherewehaveagainused(4.4.5).Bycomparing(4.4.3)with( 4.2.3),weseethatthe 2 termisextendedtoall J 3 ,the 2 termhastheoppositesign,andthecrossterm isdropped.Wethusnd L = 1 2 ( A i ) y 2 A i +2 B 2 = 1 2 A a 2 A a i e C 2 C +2 B 2 ; (4 : 4 : 7) wherewehavewritten A = i ( C; e C ),dueto(3.4.4,13).Thisagreeswiththeresult (3.2.11)inthegauge =1,wherethis B = 1 2 e B .TheBRSTtransformations(4.4.4), using(4.2.5),are A a = i@ a C; C =0 ; e C = (2 B @ A ) ; B = i 1 2 2 C; (4 : 4 : 8) whichagreeswiththelinearizedcaseof(3.2.8). Wenextprovetheequivalenceofthisformofgaugexingwith theusualapproach,describedinsect.3.2[4.1](aswehavejustprovenf orthecaseofthemassless vector).Thestepsare:(1)AddtermstotheoriginalBRSTaux iliaryelds,which vanishonshell,tomakethemBRSTinvariant,astheyareinth eusualBRSTformulationofeldtheory.(InYang-Mills,thisistheredeni tion B e B in(3.2.11).) (2)UsetheBRSTtransformationstoidentifythephysicale lds(whichmayinclude auxiliarycomponents).Wecanthenreobtainthegauge-inva riantactionbydropping allothereldsfromthelagrangian,withthegaugetransfor mationsgivenbyreplacing theghostsintheBRSTtransformationsbygaugeparameters. Inthelagrangian(4.4.3)onlythepartoftheBRSTauxiliary eld whichappears in M + occursintheaction;therestof ispuregaugeanddropsoutoftheaction. Thus,weonlyrequirethattheshifted M + beBRST-invariant: = e + A;M + e =0 : (4 : 4 : 9) UsingtheBRSTtransformations(4.4.4)andtheidentities( from(3.4.3b)) Q 2 =0 [ 2 M 2 ;M + ]=[ 2 M 2 ; Q + ]=[ M + ; Q + ]=0 ; Q +2 = 1 2 ( 2 M 2 ) M + ; (4 : 4 : 10) weobtaintheconditionson A : ( Q + M + A ) Q + =( Q + M + A ) M + =0 : (4 : 4 : 11)

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4.4.Gaugexing 71 Thesolutiontotheseequationsis A = M + 1 Q + : (4 : 4 : 12) Performingtheshift(4.4.9),thegauge-xedlagrangianta kestheform @ @c L = 1 2 y ^ K +2 e y (1 M 3 ; 2 T ) Q + + e y M + e ; (4 : 4 : 13 a ) ^ K = 2 M 2 2 Q + M + 1 Q + ; (4 : 4 : 13 b ) where T isthe\isospin,"asin(4.2.11),for M .TheBRSTtransformationscan nowbewrittenas M 3 ; 2 T = M 3 ; 2 T Q + ; (1 M 3 ; 2 T ) = M + e ; e = M 3 ; 2 T ( Q + e 1 2 ^ K ) : (4 : 4 : 14) TheBRSTtransformationof e ispuregauge,andcanbedropped.(Insomeof themanipulationswehaveusedthefactthat Q Q + ,and 2 M 2 aresymmetric, i.e.,evenunderintegrationbyparts,while M + isantisymmetric,and Q and Q are antihermitianwhile M + and 2 M 2 arehermitian.Inacoordinaterepresentation,particularlyfor c ,allsymmetrygenerators,suchas Q Q + ,and M + ,wouldbe antisymmetric,sincetheeldswouldbereal.) WecannowthrowawaytheBRST-invariantBRSTauxiliaryeld s e ,butwe mustalsoseparatetheghosteldsin fromthephysicalones.Accordingtothe usualBRSTprocedure,thephysicalmodesofatheoryarethos ewhichareboth BRST-invariantandhavevanishingghostnumber(aswellass atisfytheeldequations).Inparticular,physicaleldsmaytransformintogh osts(correspondingto gaugetransformations,sincethegaugepiecesareunphysic al),butnevertransform intoBRSTauxiliaryelds.Therefore,from(4.4.14)wemust requirethatthephysicaleldshave M 3 = 2 T toavoidtransformingintoBRSTauxiliaryelds,butwe alsorequirevanishingghostnumber M 3 =0.Hence,thephysicalelds(locatedin )areselectedbyrequiringthesimpleconditionofvanishin gisospin T =0.Ifwe projectouttheghostswiththeprojectionoperator T 0 andusetheidentity(4.2.8), weobtainalagrangiancontainingonlyphysicalelds: L 1 = 1 2 y ( 2 M 2 + Q 2 ) T 0 : (4 : 4 : 15) ItsgaugeinvarianceisobtainedfromtheBRSTtransformati onsbyreplacingthe ghosts(thepartof appearingontheright-handsideofthetransformationlaw)

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72 4.GENERALGAUGETHEORIES withthegaugeparameter(thereverseoftheusualBRSTquant izationprocedure), andweaddgaugetransformationstogaugeawaythepartof with T 6 =0: = i 1 2 Q + 1 2 M ; (4 : 4 : 16) wherewehaveobtainedthe Q termfromclosureof Q + with M .Theinvariance of(4.4.15)under(4.4.16)canbeveriedusingtheaboveide ntities.Thisactionand invariancearejusttheoriginalonesoftheOSp(1,1 j 2)gauge-invariantformalism(or theIGL(1)one,aftereliminatingtheNLauxiliaryelds).T hegauge-xingfunctions forthetransformationsarealsogivenbytheBRSTtransfor mations:Theyarethe transformationsofghostsintophysicalelds: F GF = Q + T 0 = p a ( M + a T 0 )+ M ( M + m T 0 ) : (4 : 4 : 17) (ThersttermistheusualLorentz-gaugegauge-xingfunct ionformasslesselds, thesecondtermtheusualadditionforStueckelberg/Higgs elds.)Thegauge-xed lagrangian(physicaleldsonly)isthus L GF = L 1 1 4 F GF y M F GF = 1 2 y ( 2 M 2 ) T 0 ; (4 : 4 : 18) inagreementwith(4.4.3). Insummary,weseethatthisrst-quantizedgauge-xingpro cedureisidentical tothesecond-quantizedonewithregardto(1)thephysicalg augeelds,theirgauge transformations,andthegauge-invariantaction,(2)theB RSTtransformationsofthe physicalelds,(3)theclosureoftheBRSTalgebra,(4)theB RSTinvarianceofthe gauge-xedaction,and(5)theinvertibilityofthekinetic operatorafterelimination oftheNLelds.(1)impliesthatthetwotheoriesarephysica llythesame,(2)and (3)implythattheBRSToperatorsarethesame,uptoaddition almodesasinsect. 4.3,(4)impliesthatbotharecorrectlygaugexed(butperh apsindierentgauges), and(5)impliesthatallgaugeinvarianceshavebeenxed,in cludingthoseofghosts. Concerningtheextramodes,fromthe 2 M 2 formofthegauge-xedkineticoperator weseethattheyareexactlytheonesnecessarytogivegoodhi gh-energybehaviorof thepropagator,andthatwehavechosenageneralizedFermiFeynmangauge.Also, notethefactthatthe c =0(or x =0)partoftheeldcontainsexactlytheright setofghostelds,aswasmanifestbytheargumentsofsect.2 .6,whereasintheusual second-quantizedformalismonebeginswithjustthephysic aleldsmanifest,andthe ghostsandtheirghosts,etc.,mustbefoundbyastep-by-ste pprocedure.Thuswe seethattheOSpfromthelightconenotonlygivesastraightf orwardwayforderiving

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4.4.Gaugexing 73 generalfreegauge-invariantactions,butalsogivesameth odforgaugexingwhich isequivalentto,butmoredirectthan,theusualmethods. Wenowconsidergaugexingtothelightcone.Inthisgaugeth egaugetheory reducesbacktotheoriginallight-conetheoryfromwhichit washeuristicallyobtained byadding2+2dimensionsinsects.3.4,4.1.Thisprovesagen eral\no-ghosttheorem,"thattheOSp(1,1 j 2)(andIGL(1))gaugetheoryisequivalenton-shelltothe correspondinglight-conetheory,foranyPoincarerepres entation(includingstringsas aspecialcase). Consideranarbitrarybosonicgaugeeldtheory,withactio n(4.1.6).(Fermions willbeconsideredinthefollowingsection.)Withoutlosso fgenerality,wecanchoose M 2 =0,sincethemassiveactioncanbeobtainedbydimensionalr eduction.The light-conegaugeisthendescribedbythegauge-xedeldeq uations p 2 =0(4 : 4 : 19 a ) subjecttothegaugeconditions,intheLorentzframe p a = a + p + M =0 ; Q = M p + =0 ; (4 : 4 : 19 b ) withtheresidualpartofthegaugeinvariance = i 1 2 Q M ; (4 : 4 : 19 c ) where nowrefertotheusual\longitudinal"Lorentzindices.(The light-conegauge isthusafurthergauge-xingoftheLandaugauge,whichuses only(4.4.19ab).) (4.4.19bc)implythattheonlysurvivingeldsaresinglets ofthenewOSp(1,1 j 2) algebrageneratedby M M M + (withlongitudinalLorentzindices ):i.e., thosewhichsatisfy M AB =0andcan'tbegaugedawayby = M BA AB WethereforeneedtoconsiderthesubgroupSO(D 2) n OSp(1,1 j 2)ofOSp(D 1,1 j 2) (thespingroupobtainedbyadding2+2dimensionstotheorig inalSO(D 2)),and determinewhichpartsofanirreducibleOSp(D 1,12)representationareOSp(1,1 j 2) singlets.Thisisdonemostsimplybyconsideringthecorres pondingYoungtableaux (whichisalsothemostconvenientmethodforadding2+2dime nsionstotheoriginal representationofthelight-coneSO(D 2)).Thismeansconsideringtensorproductsofthevector(dening)representationwithvariousgr adedsymmetrizationsand antisymmetrizations,and(graded)tracelessnessconditi onsonalltheindices.The obviousOSp(1,1 j 2)singletisgivenbyallowingallthevectorindicestotake only

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74 4.GENERALGAUGETHEORIES SO(D 2)values.However,theresultingSO(D 2)representationisreducible,since itisnotSO(D 2)-traceless.TheOSp(D 1,1 j 2)-tracelessnessconditionequatesits SO(D 2)-tracestoOSp(1,1 j 2)-tracesofrepresentationswhichdieronlybyreplacing thetracedSO(D 2)indiceswithtracedOSp(1,1 j 2)ones.However,OSp(1,1 j 2)(or OSp(2N j 2N)moregenerally)hastheunusualpropertythatitstraces arenottrue singlets.Thesimplestexample[4.11](andonewhichwe'lls howissucienttotreat thegeneralcase)isthegravitonof(4.1.15).Consideringj usttheOSp(1,1 j 2)values oftheindices,thereare2stateswhicharesingletsunderth ebosonicsubgroupGL(2) generatedby M M + ,namely (+ E ) E j ij i .However,ofthesetwostates,one linearcombinationispuregaugeandoneispureauxiliary: 1 E = A E A E M AB 1 E =0 ;but 1 E = 1 2 M ( E ) E ; 2 E = (+ E ) E E E 2 E 6 = M AB BA E ;but M 2 E = 2 ( E ) E 6 =0 : (4 : 4 : 20) Thisresultisduebasicallytothefactthatthegradedtrace can'tbeseparatedout intheusualwaywiththemetricbecauseoftheidentity AB BA = A A =( 1) A A A =2 2=0 : (4 : 4 : 21) Similarly,thereducibleOSp(1,1 j 2)representationwhichconsistsoftheunsymmetrized directproductofanarbitrarynumberofvectorrepresentat ionswillcontainnosinglets,sinceanyonetracereducestothecasejustconsidere d,andthustherepresentationswhichresultfromgradedsymmetrizationsandantis ymmetrizationswillalso containnone.Thus,noSO(D 2)-tracesoftheoriginalOSp(D 1,1 j 2)representation needbeconsidered,sincetheyareequatedtoOSp(1,1 j 2)-nonsinglettracesbythe OSp(D 1,1 j 2)-tracelessnesscondition.Hence,theonlysurvivingSO( D 2)representationistheoriginalirreduciblelight-coneone,obta inedbyrestrictingallvector indicestotheirSO(D 2)valuesandimposingSO(D 2)-tracelessness. Thesemethodsapplydirectlytoopenstrings.Themodicati onforclosedstrings willbediscussedinsect.11.1.

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4.5.Fermions 75 4.5.Fermions TheresultsofthissectionarebasedontheOSp(1,1 j 2)algebraofsect.3.5. Theactionandinvariancesareagaingivenby(4.1.1),witht hemodied J AB of sect.3.5,and(4.1.3)isunchangedexceptfor M c M .WealsoallowfortheinclusionofamatrixfactorintheHilbert-spacemetrictomainta inthe(pseudo)hermiticity ofthespinoperators(e.g., r 0 foraDiracspinor,so y = y r 0 ).Underthe actionofthe functions,wecanmakethereplacement p + 2 J 2 3 4 ( p 2 + M 2 )+( M b p b + M m M ) 2 1 2 ~ r ( M b p b + M m M ) h ~ r +~ r + 1 2 ( p 2 + M 2 ) i ; (4 : 5 : 1) where p 2 p a p a .Undertheactionofthesame functions,thegaugetransformation generatedby J isreplacedwith = J n ( M b p b + M m M ) 1 2 ~ r h ~ r +~ r + 1 2 ( p 2 + M 2 ) io : (4 : 5 : 2) Choosing =~ r ,the~ r partofthisgaugetransformationcanbeusedtochoose thepartialgauge ~ r + =0 : (4 : 5 : 3) Theactionthenbecomes S = Z d D x ( c M 2 ) i ~ r ( M b p b + M m M ) ; (4 : 5 : 4) wherewehavereducedtothehalfrepresentedby byusing(4.5.3).(The~ r can berepresentedas2 2matrices.)Theremainingpartof(4.5.2),togetherwithth e J transformation,canbewrittenintermsof~ r -independentparametersas =( M b p b + M m M )( A + 1 2 ~ r ~ r A )+ h 1 4 ( p 2 + M 2 )+( M b p b + M m M ) 2 i B + 1 2 c M : (4 : 5 : 5) Onewaytogetgeneralirreduciblespinorrepresentationso forthogonalgroups (exceptforchiralityconditions)istotakethedirectprod uctofaDiracspinorwithan irreducibletensorrepresentation,andthenconstrainitb ysettingtozerotheresultof multiplyingbya r matrixandcontractingvectorindices.SincetheOSprepres entationsusedhereareobtainedbydimensionalcontinuation,t hismeansweusethesame constraints,withthevectorindices i runningoverallcommutingandanticommuting

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76 4.GENERALGAUGETHEORIES values(including m ,ifwechoosetodene M im bydimensionalreductionfromone extradimension).TheOSpspinoperatorscanthenbewritten as M ij = M ij + 1 4 [ r i ;r j g ; (4 : 5 : 6) where M arethespinoperatorsforsometensorrepresentationand r i aretheOSp Diracmatrices,satisfyingtheOSpCliordalgebra f r i ;r j ]=2 ij : (4 : 5 : 7) Wechoosesimilarrelationsbetween r 'sand~ r 's: f r i ; ~ r B ]=0 : (4 : 5 : 8) Then,notingthat 1 2 ( r + i ~ r )and 1 2 ( r i ~ r )satisfythesamecommutationrelations ascreationandannihilationoperators,respectively,wed ene r = a + a y ; ~ r = i a a y ; h a ;a y i = : (4 : 5 : 9) Wealsond c M = M + a y ( a ) : (4 : 5 : 10) Thismeansthatanarbitraryrepresentation ( ) ofthepartoftheSp(2)generated by M thatisalsoasingletunderthefullSp(2)generatedby c M canbewritten as ( ) = a y a y ;a =0 : (4 : 5 : 11) Inparticular,foraDiracspinor M =0,sotheaction(4.5.4)becomessimply(see also(4.1.40)) S = Z d D x ( =p + =M ) ; (4 : 5 : 12) where =p r a p a =M r m M ( r m islike r 5 ),alldependenceon r and~ r has beeneliminated,andthegaugetransformation(4.5.5)vani shes.(Thetransformation e ir m = 4 takes =p + =M =p + iM .) Inthecaseofthegravitino,westartwith = j i i i ,where j i i isabasisfor therepresentationof M (only). mustsatisfynotonly c M =0butalsothe irreducibilitycondition r i i =0 = 1 2 r r a a ;a a =0 : (4 : 5 : 13)

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4.5.Fermions 77 ( a y or i ~ r couldbeusedinplaceof r inthesolutionfor .)Thenusing(4.1.8) for M on ,straightforwardalgebragivestheaction S = Z d D x a ( ab =p p ( a r b ) + r a =pr b ) b = Z d D x a r abc p b c ; (4 : 5 : 14) where r abc =(1 = 3!) r [ a r b r c ] ,givingtheusualgravitinoactionforarbitrary D .The gaugeinvarianceremainingin(4.5.5)afterusing(4.5.1314),andmakingsuitable redenitions,reducestotheusual a = @ a : (4 : 5 : 15) Wenowderiveanalternativeformofthefermionicactionwhi chcorrespondsto actionsgivenintheliteratureforfermionicstrings.Inst eadofexplicitlysolvingthe constraint c M =0asin(4.5.11),weusethe c M gaugeinvarianceof(4.5.5)to \rotate"the a y 's.Forexample,writing a =( a + ;a ),wecanrotatethemsothey allpointinthe\+"direction:Thenweneedconsideronly 'softheform ( a + y ) j 0 i (The+valueof shouldnotbeconfusedwiththe+indexon p + .)The ( c M 2 )then picksoutthepieceoftheform(4.5.11).(It\smears"overdi rectionsinSp(2).This useof a issimilartothe\harmoniccoordinates"ofharmonicsupers pace[4.12].)We canalsopick = e ia + y a y ( a + y ) j 0 i ; (4 : 5 : 16 a ) sincetheexponential(after ( c M )projection)justredenessomecomponentsby shiftingbycomponentsoflower M -spin.Inthisgauge,writing r =( s;u ),~ r = (~ s; ~ u ),wecanrewrite as ~0 E = e ia + y a y j 0 i$ ~ s ~0 E = u ~0 E =0 ; = ( s ) ~0 E $ ~ s =0 : (4 : 5 : 16 b ) Byusinganappropriate(indenite-metric)coherent-stat eformalism,wecanchoose s tobeacoordinate(and u = 2 i@=@s ).Wenextmakethereplacement ( c M 2 ) Z ds ( s ) (2 M + + s 2 ) M 3 + s @ @s (4 : 5 : 17) afterpushingittotherightin(4.5.4)soithitsthe ,wherewehavejustreplaced projectiononto c M 2 =0with c M + = c M 3 =0(whichimplies c M =0).Therst functionfactorisaDirac function,whilethesecondisaKronecker ( s )isan

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78 4.GENERALGAUGETHEORIES appropriatemeasurefactor;insteadofdeterminingitbyan explicituseofcoherent states,wexitbycomparisonwiththesimplestcaseoftheDi racspinor:Using Z ds ( s ) ( s 2 + r 2 )( a + bs )= b; (4 : 5 : 18 a ) wend ( s ) ( s ) : (4 : 5 : 18 b ) Thenonlythe~ u termof(4.5.4)contributesinthisgauge,andweobtain(the ~ u itself havingalreadybeenabsorbedintothemeasure(4.5.18b)) S = 2 Z d D xds ( s ) Q + (2 M + + s 2 ) M 3 + s @ @s : (4 : 5 : 19) (Alldependenceon r and~ r hasbeenreducedto beingafunctionofjust s .This actionwasrstproposedforthestring[4.13].)Sincether st functioncanbeused toreplaceany s 2 witha 2 M + ,wecanperformallsuchreplacements,orequivalently choosethegauge ( s )= 0 + s 1 : (4 : 5 : 20) (Anequivalentprocedurewasperformedforthestringin[4. 14].)FortheDiracspinor, afterintegrationover s (includingthatin Q + = 1 2 s ( =p + =M )),theDiracactioniseasily found( 1 dropsout).Forgeneralspinorrepresentations, Q + hasanadditional Q + term,and s integrationgivesthelagrangian L = 0 ( =p + =M ) ( M 3 ) 0 +2 h 1 Q + ( M 3 ) 0 0 Q + ( M 3 +1) 1 i : (4 : 5 : 21) However, r -matrixtraceconstraints(suchas(4.5.13))muststillbes olvedtorelate thecomponents. TheexplicitformoftheOSp(1,1 j 2)operatorsforthefermionicstringtousewith theseresultsfollowsfromthelight-conePoincaregenera torswhichwillbederivedin sect.7.2.The s dependenceof Q + isthenslightlymorecomplicated(italsohasa @=@s term).(Theresultingactionrstappearedin[4.14,15].) Theproofofequivalencetothelightconeissimilartothatf orbosonsinthe previoussection.Againconsideringthemasslesscase,the basicdierenceisthatwe nowhavetouse,from(3.5.3b), c M = M + S ; b Q = M p + + S ; (4 : 5 : 22) andothercorrespondinggenerators,asgeneratingthenewO Sp(1,1 j 2).Thisisjust thediagonalOSp(1,1 j 2)obtainedfrom S AB andtheoneusedinthebosoniccase.

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Exercises 79 Inanalogytothebosoniccase,weconsiderreducibleOSp(D 1,1 j 2)representations correspondingtodirectproductsofarbitrarynumbersofve ctorrepresentationswith onespinorrepresentation(representedbygraded r -matrices).Wethentakethedirect productofthiswiththe S AB representation,whichisanOSp(1,1 j 2)spinorbutan SO(D 2)scalar.Sincethedirectproductof2OSp(1,1 j 2)spinorsgivesthedirect sumofall(graded)antisymmetrictensorrepresentations, eachonce(bytheusual r -matrixdecomposition),fromthebosonicresultweseethat theonlywaytogetan OSp(1,1 j 2)singletisifallvectorindicesagaintakeonlytheirSO(D 2)values.The OSp(D 1,1 j 2)spinoristhedirectproductofanSO(D 2)spinorwithanOSp(1,1 j 2) spinor,sothenetresultistheoriginallight-coneone.Int hebosoniccasetracesin OSp(1,1 j 2)vectorindicesdidnotgivesingletsbecauseof(4.4.21); asimilarresult holdsfor r -matrixtracesbecauseof r A r A = A A =0 : (4 : 5 : 23) Moregeneralrepresentationsfor S AB couldbeconsidered,e.g.,asinsect.3.6.The actioncanthenberewrittenas(4.1.6),butwith M and Q replacedby c M and b Q of(3.5.3b).Inanalogyto(4.5.2,3),the S partofthe J transformationcanbeused tochoosethegauge S + =0.Then,dependingonwhethertherepresentationallows applicationofthe\lowering"operators S 0,1,or2times,onlythetermsofzeroth, rst,orsecondorderin S ,respectively,cancontributeinthekineticoperator. Sincethesetermsarerespectivelysecond,rst,andzeroth orderinderivatives,they canbeusedtodescribebosons,fermions,andauxiliaryeld s. Theargumentforequivalencetothelightconedirectlygene ralizestotheU(1)type4+4-extendedOSp(1,1 j 2)ofsect.3.6.Then c M AB = M AB + S AB hasasinglet onlywhen M AB and S AB arebothsinglets(forbosons)orbothDiracspinors(for fermions).Exercises (1)Derive(4.1.5).(2)Derive(4.1.6).(3)Findthegauge-invarianttheoryresultingfromtheligh t-conetheoryofatotally symmetric,tracelesstensorofarbitraryrank. (4)Findtheexplicitinnitesimalgaugetransformationso f e a m e m a ,e 1 g mn g mn and abc from(4.1.20-22).Linearize,andshowthegauge e [ am ] =0canbeob-

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80 4.GENERALGAUGETHEORIES tainedwith ab .Findthetransformationforacovariantvector A a (fromasimilaritytransformation,like(4.1.22)). (5)Write c abc explicitlyintermsof e a m .Find T abc and R abcd intermsof c abc and abc .Derive(4.1.31).Linearizetoget(4.1.17). (6)Findanexpressionfor abc when(4.1.24)isnotimposed,intermsof T abc and the of(4.1.27). (7)DeriveglobalPoincaretransformationsbyndingthes ubgroupof(4.1.20)which leaves(4.1.28)invariant. (8)Findtheeldequationfor from(4.1.36),andshowthat(4.1.38)satisesit. (9)Derivethegauge-covariantactionforgravityintheGL( 1)-type4+4-extensionof OSp(1,1 j 2),andcomparewiththeU(1)result,(4.1.47). (10)FindtheBRSTtransformationsfortheIGL(1)formalism ofsect.4.2(BRST1, derivedfromthelightcone)forfreegravity.Findthosefor theusualIGL(1)formalismofsect.3.2(BRST2,derivedfromsecond-quantizing thegauge-invariant eldtheory).AftersuitableredenitionsoftheBRST1eld s(includingauxiliariesandghosts),showthatasubsetoftheseeldsthatc orrespondstothe completesetofeldsintheBRST2formalismhasidenticalBR STtransformations. (11)Formulate 3 theoryasin(4.2.19),usingthebracketof(3.4.7). (12)DerivethegaugetransformationsforinteractingYang -Millsbythecovariant second-quantizedoperatormethodof(4.2.21),inboththeI GL(1)andOSp(1,1 j 2) formalisms. (13)Findthefreegauge-invariantactionforgravityinthe IGL(1)formalism,and comparewiththeOSp(1,1 j 2)result(4.1.17).Findthegauge-xedactionby (4.4.1-5). (14)PerformIGL(1)gauge-xing,asinsect.4.4,forasecon d-rankantisymmetric tensorgaugeeld.Performtheanalogousgaugexingbythem ethodofsect. 3.2,andcompare.Notethattherearescalarcommutingghost swhichcanbe interpretedastheghostsforthegaugeinvariancesoftheve ctorghosts(\ghosts forghosts"). (15)Derive(4.5.14,15).

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5.1.Bosonic 81 5.PARTICLE5.1.Bosonic Ifcoordinatesareconsideredaselds,andtheirarguments asthecoordinatesof smallspacetimes,thenthemechanicsofparticlesandstrin gscanbeconsideredas 1-and2-dimensionaleldtheories,respectively(seesect .1.1).(However,toavoid confusion,wewillavoidreferringtomechanicstheoriesas eldtheories.)Thus,the particleisausefulanalogofthestringinonelower\dimens ion",andweherereview itspropertiesthatwillbefoundusefulbelowforthestring Asdescribedinsect.3.1,themechanicsactionforanyrelat ivisticparticleis completelydeterminedbytheconstraintsitsatises,whic hareequivalenttothefree equationsofmotionofthecorrespondingeldtheory.Ther st-order(hamiltonian) form((3.1.10))ismoreconvenientthanthesecond-orderon ebecause(1)itmakes canonicalconjugatesexplicit,(2)theinversepropagator (and,inmoregeneralcases, allotheroperatorequationsofmotion)canbedirectlyread oasthehamiltonian,(3) path-integralquantizationiseasier,and(4)treatmentof thesupersymmetriccaseis clearer.Thesimplestexampleisamassless,spinlessparti cle,whoseonlyconstraint istheKlein-Gordonequation p 2 =0.From(3.1.10),theaction[5.1]canthusbe writteninrst-orderformas S = Z d L ; L = x p g 1 2 p 2 ; (5 : 1 : 1) where isthepropertime,ofwhichtheposition x ,momentum p ,and1-dimensional metric g arefunctions,and = @=@ .TheactionisinvariantunderPoincaretransformationsinthehigher-dimensionalspacetimedescribed by x ,aswellas1Dgeneral coordinatetransformations( -reparametrizations).Thelattercanbeobtainedfrom (3.1.11): x = p;p =0 ;g = : (5 : 1 : 2) Thesedierfromtheusualtransformationsbytermswhichva nishonshell:Ingeneral, anyactionwithmorethanoneeldisinvariantunder i = ij S= j ,where ij is

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82 5.PARTICLE antisymmetric.Suchinvariancesmaybenecessaryforo-sh ellclosureoftheabove algebra,butareirrelevantforobtainingtheeldtheoryfr omtheclassicalmechanics. (Infact,inthecomponentformalismforsupergravity,gaug einvarianceismoreeasily provenusingarst-orderformalismwiththetypeoftransfo rmationsin(5.1.2)rather thantheusualtransformationswhichfollowfromthesecond -orderformalism[5.2].) Inthiscase,ifweaddthetransformations 0 x = S p ; 0 p = S x (5 : 1 : 3) to(5.1.2),andchoose = g 1 ,weobtaintheusualgeneralcoordinatetransformations(seesect.4.1) 00 x = x; 00 p = p; 00 g = ( g ) : (5 : 1 : 4) Thesecond-orderformisobtainedbyeliminating p : L = g 1 1 2 x 2 : (5 : 1 : 5) Thetransformations(5.1.4)for x and g alsofollowdirectlyfrom(5.1.2)uponeliminating p byitsequationofmotion.Themassivecaseisobtainedbyrep lacing p 2 with p 2 + m 2 in(5.1.1).Whentheadditionaltermiscarriedoverto(5.1. 5),weget L = 1 2 g 1 x 2 1 2 gm 2 : (5 : 1 : 6) g cannowalsobeeliminatedbyitsequationofmotion,produci ng S = m Z d q x 2 = m Z p dx 2 ; (5 : 1 : 7) whichisthelengthoftheworldline. Besidesthe1Dinvarianceof(5.1.1)underreparametrizati onof ,italsohasthe discreteinvarianceof reversal.Ifwechoose x ( ) x ( )underthisreversal, then p ( ) p ( ),andthusthisproper-timereversalcanbeidentiedasthe classicalmechanicalanalogofcharge(orcomplex)conjuga tionineldtheory[5.3], where ( x ) ( x )implies ( p ) ( p )forthefouriertransform.(Also,the electromagneticcoupling q R d x A ( x )whenaddedto(5.1.1)requiresthecharge q q .) Therearetwostandardgaugesforquantizing(5.1.1).Inthe light-coneformalism thegaugeiscompletelyxed(for p + 6 =0,uptoglobaltransformations,whichare eliminatedbyboundaryconditions)by x + = p + : (5 : 1 : 8)

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5.1.Bosonic 83 Wetheneliminate p asalagrangemultiplierwitheldequation g =1.Thelagrangianthensimpliesto L = x p + + x i p i 1 2 ( p i 2 + m 2 ) ; (5 : 1 : 9) with(retarded)propagator i ( ) e i 1 2 ( p i 2 + m 2 ) (5 : 1 : 10 a ) (where( )=1for > 0and0otherwise)or,fouriertransformingwithrespectto 1 i @ @ + 1 2 ( p i 2 + m 2 )+ i = 1 p + p + 1 2 ( p i 2 + m 2 )+ i = 1 1 2 ( p 2 + m 2 )+ i : (5 : 1 : 10 b ) Forinteractingparticles,it'spreferabletochoose x + = ; (5 : 1 : 11) sothatthesame coordinatecanbeusedforallparticles.Then g =1 =p + ,so thehamiltonian 1 2 ( p i 2 + m 2 )getsreplacedwith( p i 2 + m 2 ) = 2 p + ,whichmoreclosely resemblesthenonrelativisticcase.Ifwealsousetheremai ning(global)invarianceof reparametrizations(generatedby p 2 ),wecanchoosethegauge x + =0,whichis thesameaschoosingtheSchrodingerpicture. Alternatively,inthecovariantformalismonechoosestheg auge g = constant; (5 : 1 : 12) where g can'tbecompletelygauge-xedbecauseoftheinvarianceof the1Dvolume T = R dg .Thefunctionalintegralover g isthusreplacedbyanordinaryintegral over T [5.4],andthepropagatoris[5.3,5] i Z 1 0 d T ( T ) e i T 1 2 ( p 2 + m 2 ) = 1 1 2 ( p 2 + m 2 )+ i : (5 : 1 : 13) Theuseofthemechanicsapproachtotheparticleissomewhat pointlessforthe freetheory,sinceitcontainsnoinformationexceptthecon straints(fromwhichit wasderived),anditrequirestreatmentoftheirrelevant\o -shell"behaviorinthe \coordinate" .However,theproper-timeisusefulininteractingtheorie sforstudying certainclassicallimitsandvariouspropertiesofperturb ationtheory.Inparticular, theformofthepropagatorgivenin(5.1.13)(withWick-rota ted :seesect.2.5)is themostconvenientfordoingloopintegralsusingdimensio nalregularization:The

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84 5.PARTICLE momentumintegrationsbecomesimpleGaussianintegrals,w hichcanbetrivially evaluatedinarbitrarydimensionsbyanalyticcontinuatio nfromintegerones: Z d D pe ap 2 = Z d 2 pe ap 2 D= 2 = a D= 2 : (5 : 1 : 14) (Theformerintegralfactorsinto1-dimensionalones;thel atteriseasilyperformedin polarcoordinates.)TheSchwingerparameters arethenconvertedintotheusual Feynmanparameters byinsertingunityas R 1 0 d ( P ),rescaling i = i andintegratingout ,whichnowappearsinstandard-functionintegrals,toget theusualFeynmandenominators.Anidenticalprocedureisa ppliedinstringtheory, butwritingtheparametersas x = e w = e .(See(9.1.10).)Bynotconverting the 'sinto 's,thehigh-energybehaviorofscatteringamplitudescanb eanalyzed moreeasily[5.6].Also,thesingularitiesinanamplitudec orrespondtoclassicalpaths oftheparticles,andthisidenticationcanbeseentobesim plytheidentication ofthe parameterswiththeclassicalproper-time[5.7].1-loopca lculationscanbe performedbyintroducingexternalelds(seealsosect.9.1 )andtreatingthepath oftheparticleinspacetimeasclosed[5.5,8].Suchcalcula tionscantreatarbitrary numbersofexternallines(ornonperturbativeexternalel ds)forcertainexternal eldcongurations(suchasconstantgauge-eldstrengths ).Finally,theintroduction ofsuchexpressionsforpropagatorsinexternaleldsallow sthestudyofclassical limitsofquantumeldtheoriesinwhichsomequantumelds( representedbythe externaleld)becomeclassicalelds,asintheusualclass icallimit,whileotherelds (representedbytheparticlesdescribedbythemechanicsac tion)becomeclassical particles[5.9]. Thisclassicalmechanicsanalysiswillbeappliedtothestr inginchapt.6. 5.2.BRST Inthissectionwe'llapplythemethodsofsect.3.2-3tostud yBRSTquantization ofparticlemechanics,andndresultsequivalenttothoseo btainedbymoregeneral methodsinsect.3.4. Inthecaseofparticlemechanics(accordingtosect.3.1),f ortheactionofthe previoussectionwehave G = i 1 2 ( p 2 + m 2 ),andthus[4.4],forthe\temporal"gauge g =1,from(3.2.6) Q = ic 1 2 ( p 2 + m 2 ) ; (5 : 2 : 1) whichagreeswiththegeneralresult(3.4.3b).Wecouldwrit e c = @=@ C sothat intheclassicaleldtheorywhichfollowsfromthequantumm echanicstheeld

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5.2.BRST 85 ( x; C )couldbereal(seesect.3.4).Thisalsofollowsfromthefac tthatthe( reparametrization)gauge-parametercorrespondingto c carriesa(proper-)timeindex (it'sa1Dvector),andthuschangessignunder -reversal(mechanics'equivalentof eldtheory'scomplexconjugation),andso c isamomentum( ( x; C )= ( x; C ), ( p;c )= ( p; c )). WenowconsiderextendingIGL(1)toOSp(1,1 j 2)[3.7].By(3.3.2), Q = ix 1 2 ( p 2 + m 2 ) b@ : (5 : 2 : 2) Inordertocomparewithsect.3.4,wemaketheredenitions( see(3.6.8)) b = i @ @g ;g = 1 2 p + 2 ; (5 : 2 : 3 a ) (where g istheworld-linemetric)andtheunitarytransformation lnU = ( lnp + ) 1 2 [ x ;@ ] ; (5 : 2 : 3 b ) nding UQ U 1 = ix 1 2 p + p 2 + m 2 + p p ix p ; (5 : 2 : 4) whichagreeswiththeexpressiongivenin(3.4.2)forthegen erators J forthecase ofthespinlessparticle,asdoestherestoftheOSp(1,1 j 2)obtainedfrom(3.3.7). Inalagrangianformalism,fortheaction(5.1.6)withinvar iance(5.1.4),(3.3.2) givestheBRSTtransformationlaws Q x a = x x a ; Q g = ( x g ) ; Q x = 1 2 x ( x ) C b; Q b = 1 2 ( x b b x ) 1 4 ( x 2 x + x 2 .. x ) : (5 : 2 : 5) Werstmaketheredenition ~ b = b 1 2 ( x 2 )(5 : 2 : 6) tosimplifythetransformationlawof x andthus b : Q x = x x C ~ b; Q ~ b = x ~ b: (5 : 2 : 7) Wethenmakefurtherredenitions x g 1 x ; ~ b g 1 b 2 ( g 1 x 2 ) ; (5 : 2 : 8)

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86 5.PARTICLE whichsimplifythe g transformation,allowingafurthersimplicationfor b : Q x a = x g 1 x a = x p a ; Q g = x ; Q x = C b; Q b =0 : (5 : 2 : 9) TogetjustaBRSToperator(asfortheIGL(1)formalism),wec anrestricttheSp(2) indicesin(5.2.9)tojustonevalue.Then x fortheothervalueof (theantighost) and b canbedropped.(TheyformanindependentIGL(1)multiplet, asdescribedin sect.3.2.) TogettheOSp(1,1 j 2)formalism,wechoosea\Lorentz"gauge.Wethenquantize withtheISp(2)-invariantgauge-xingterm L 1 = Q 2 f ( g )= f 00 ( g )( gb x 2 )(5 : 2 : 10) forsomearbitraryfunction f suchthat f 00 6 =0.Canonicallyquantizing(where f 0 ( g ) isconjugateto b ),andusingtheequationsofmotion,wend Q fromitsNoether current(whichin D =1isalsothecharge)tobegivenby(5.2.2).ForanIGL(1) formalism,wecanusethetemporalgauge(writing x =( c; ~ c )) L 1 = iQ [~ cf ( g )]= bf ( g ) if 0 ( g )~ c c: (5 : 2 : 11) (Comparethediscussionsofgaugechoicesinsect.3.2-3.) AlthoughLorentz-gaugequantizationgavearesultequival enttothatobtained fromthelightconeinsect.3.4,we'llndinsect.8.3forthe stringaresultequivalent tothatobtainedfromthelightconeinsect.3.6.5.3.Spinning Themechanicsofarelativisticspin-1/2particle[5.1]iso btainedbysymmetrizing theparticleactionforaspinlessparticlewithrespectto one-dimensional (local) supersymmetry.Wethusgeneralize x ( ) X ( ; ),etc.,where isasingle,real, anticommutingcoordinate.Werstdeneglobalsupersymme trybythegenerators q = @ @ i@;i@ i @ @ = q 2 ; (5 : 3 : 1 a )

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5.3.Spinning 87 whichleaveinvariantthederivatives d = @ @ + i@;@ = i d 2 : (5 : 3 : 1 b ) Thelocalinvariancesarethengeneratedby(expandingcova riantly) K = i d + ki@; (5 : 3 : 2 a ) whichactcovariantly(i.e.,as() 0 = e iK () e iK )onthederivatives D = G d + G @; i D 2 : (5 : 3 : 2 b ) Thisgivestheinnitesimaltransformation D = i [ K; D ]= i ( KG i D ) d + i ( K G i D k ) @ +2 iG@: (5 : 3 : 2 c ) Wenextuse bythelastpartofthistransformationtochoosethegauge G =0 = i 1 2 d k: (5 : 3 : 3) Theaction(5.1.1)becomes S = Z ddG 1 h ( i D 2 X ) P 1 2 P D P i = Z dd h iG ( d X ) ( d P ) 1 2 P d P i : (5 : 3 : 4) Whenexpandedincomponentsby R d d ,anddening X = x; D X = ir ; P = ; D P = p ; G = g 1 = 2 ; d G = ig 1 ;(5 : 3 : 5) whenevaluatedat =0(inanalogytosect.3.2),wend S = Z d x p + i r p ir g 1 2 p 2 + 1 2 i : (5 : 3 : 6) The( g;x;p )sectorworksasforthebosoniccase.Inthe( ;r; )sectorweseethat thequantity i ( r 1 2 )iscanonicallyconjugateto ,andthus r = @ @ + 1 2 ; (5 : 3 : 7 a )

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88 5.PARTICLE whichhas r -matrixtypecommutationrelations.Itanticommuteswith ^ r = r = @ @ 1 2 ; (5 : 3 : 7 b ) whichisanindependentsetof r -matrices.However,itis r whichappearsinthe Diracequation,obtainedbyvarying Inalight-coneformalism,weagaineliminateallauxiliary \ "componentsby theirequationsofmotion,andusethegaugeinvariance(5.3 .2-3)toxthe\+" components X + = p + x + = p + ;r + =0 : (5 : 3 : 8) Wethennd G =1,and(5.3.6)reducesto L = x p + + x i p i 1 2 p i 2 i ( r ) + ir i i + 1 2 i i i : (5 : 3 : 9) Inordertoobtaintheusualspinoreld,it'snecessarytoad dalagrangemultiplier termtotheactionconstraining^ r =0.Thisconstraintcaneitherbesolvedclassically (butonlyforevenspacetimedimension D )bydetermininghalfofthe 'stobe thecanonicalconjugatesoftheotherhalf(consider 1 + i 2 vs. 1 i 2 ,etc.),or byimposingitquantummechanicallyontheeldGupta-Bleul erstyle.Theformer approachsacricesmanifestLorentzinvarianceinthecoor dinateapproach;however, ifthe r 'sareconsideredsimplyasoperators(withoutreferenceto theircoordinate representation),thentheeldistheusualspinorrepresen tation,andbothcanbe representedintheusualmatrixnotation.Thisconstrained actionisequivalenttothe second-orderaction S = Z dd 1 2 G 1 ( D 2 X ) ( D X ) = Z dd 1 2 ( G d X ) d ( G d X ) ; = Z d ( 1 2 g 1 x 2 + ig 1 r x 1 2 ir r ) ; (5 : 3 : 10 a ) or,inrst-orderformfor x only, S = Z d ( x p g 1 2 p 2 + 1 2 i r r + i r p ) : (5 : 3 : 10 b ) Theconstraint r p =0(theDiracequation)isjustafactorizedformoftheconst raint (2.2.8)forthisparticularrepresentationoftheLorentzg roup. AfurtherconstraintisnecessarytogetanirreduciblePoin carerepresentationin even D .Sinceanyfunctionofananticommutingcoordinatecontain sbosonicand

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5.3.Spinning 89 fermionictermsasthecoecientsofevenandoddpowersofth atcoordinate,we needtheconstraint r D = 1ontheeld(where r D meansjusttheproductofallthe r 's)topickoutaeldofjustonestatistics(inthiscase,aWe ylspinor:noticethat D iseveninorderforthepreviousconstrainttobeapplied).I ntheOSpapproach thisWeylchiralityconditioncanalsobeobtainedbyanexte nsionofthealgebra [4.10]:OSp(1,1 j 2) n U(1),wheretheU(1)ischiraltransformations,resultsina nextra Kronecker whichisjusttheusualchiralityprojector.ThisU(1)gener ator(forat leastthespecialcaseofaDiracspinororRamondstring)can alsobederivedasa constraintfromrst-quantization:Theclassicalmechani csactionforaDiracspinor, undertheglobaltransformation r a abc d r b r c r d ,variesbyaboundaryterm R d @ @ r D ,whereasusual r D abc d r a r b r c r d .Byaddingalagrangemultiplier termfor r D 1,thissymmetrybecomesalocalone,gaugedbythelagrangem ultiplier (asfortheotherequationsofmotion).By1Dsupersymmetriz ation,thereisalsoa lagrangemultiplierfor abc d p a r b r c r d .TheactionthendescribesaWeylspinor. Manysupersymmetricgaugesarepossiblefor g and .Thesimplestsetsboth toconstants(\temporal"gauge G =1),butthisgaugedoesn'tallowanOSp(1,1 j 2) algebra.Thenextsimplestgauge, d G =0,doesthesameto butsetsthe derivativeof g tovanish,makingitanextracoordinateintheeldtheory(r elated to x ,or p + ),givingthegeneratorsof(3.4.2).However,thegaugewhic halsokeeps asacoordinate(andasapartnerto g )is G =0.Inordertogetthemaximal coordinates(oratleastzero-modes,forthestring)wechoo seanOSp(1,1 j 2)which keeps (relatedto~ r ,andthecorrespondingextraghost,relatedto~ r ).Thisgives themodiedBRSTalgebraof(3.5.1). An \isospinning"particle [5.10]canbedescribedsimilarly.Bydroppingthe termin(5.3.6,10b)it'spossibletohaveadierentsymmetr yontheindicesof( r; ) thanonthoseof( x;p ).Infact,eventherangeoftheindicesandthemetriccan bedierent.Thus,spinseparatesfromorbitalangularmome ntumandbecomes isospin.Thereisnolongerananticommutinggaugeinvarian ce,butwithapositive denitemetricontheisospinorindicesit'snolongerneces sarytohaveonetomaintain unitarity.Ifweusetheconstraint^ r =0wegetanisospinor,butifwedon'twegeta matrix,withthe r 'sactingononesideandthe^ r 'sontheother.Notingthat reversal switches r with ^ r ,weseethatthematrixgetstransposed.Therefore,thecomp lex conjugationthatisthequantum-mechanicalanalogof reversalisactually hermitian conjugation,particularlyonaeldwhichisamatrixrepres entationofsomegroup. (When^ r isconstrainedtovanish, reversalisnotaninvariance.)Bycombining

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90 5.PARTICLE theseanticommutingvariableswiththepreviousonesweget anisospinningspinning particle. Atthispointwetakeaslightdiversiontodiscusspropertie sofspinorsinarbitrary dimensionswitharbitraryspacetimesignature.Thiswillc ompleteourdiscussionof spinorsinthissection,andwillbeusefulinthefollowings ection,whererepresentationsofsupersymmetry,whichisitselfdescribedbyaspino rgenerator,willbefound todependqualitativelyonthedimension.Theanalysisofsp inorsinEuclideanspace (i.e,theusualspinorrepresentationsofSO(D))canbeobta inedbytheusualgroup theoreticalmethods(see,e.g.,[5.11]),usingeitherDynk indiagramsoranexplicit representationofthe r -matrices.ThepropertiesofspinorsinSO( D + D )canthen beobtainedbyWickrotationof D spacedirectionsintotimeones.(OfparticularinterestaretheD-dimensionalLorentzgroupSO(D 1,1)andtheD-dimensional conformalgroupSO(D,2).)Thisaectsthespinorswithresp ecttoonlycomplex conjugationproperties.Ausefulnotationtoclassifyspin orsandtheirproperties is:Denoteafundamentalspinor(\spin1/2")as ,anditshermitianconjugate as .Denoteanotherspinor suchthatthecontraction isinvariantunderthegroup,anditshermitianconjugate .Therepresentationofthegroupon thesevariousspinorsisthenrelatedbytakingcomplexconj ugatesandinversesof thematricesrepresentingthegrouponthefundamentalone. ForSO( D + D )there arealwayssomeoftheserepresentationsthatareequivalen t,sinceSO(2N)hasonly 2inequivalentspinorrepresentationsandSO(2N+1)just1. (In r -matrixlanguage, theDiracspinorcanbereducedto2inequivalentWeylspinor sbyprojectionwith 1 2 (1 r D )inevendimensions.)Incaseswherethereisanotherfundam entalspinor representationnotincludedinthisset,wealsointroducea 0 andthecorresponding 3otherspinors.(However,inthatcaseall4ineachsetwillb eequivalent,sincethere areatmost2inequivalentaltogether.)Manypropertiesoft hespinorrepresentations canbedescribedbyclassifyingtheindexstructureof:(1)t heinequivalentspinors, (2)thebispinorinvarianttensors,or\metrics,"whichare justthematricesrelating theequivalentspinorsinthesetsof4,and(3)the -matrices( r -matricesfor D odd, butineven D thematriceshalfasbigwhichremainafterWeylprojection) ,whichare simplytheClebsch-Gordancoecientsforrelatingspinor n spinortovector.Inthe latter2cases,wealsoclassifythesymmetryinthe2spinori ndices,whereappropriate. Themetricsareof3types(alongwiththeircomplexconjugat esandinverses):(1) M ,whichgiveschargeconjugationfor(pseudo)realrepresen tations,andisrelated tocomplexconjugationpropertiesof r -matrices,(2) M ,whichisthematrixwhich

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5.3.Spinning 91 relatestheDiracspinorsand ,ifitcommuteswithWeylprojection,andisrelated tohermitianconjugationpropertiesof r -matrices,and(3) M ,whichistheClebschGordancoecientsforspinor n same-representationspinortoscalar,andisrelated totranspositionpropertiesof r -matrices.Foralloftheseit'simportanttoknow whetherthemetricissymmetricorantisymmetric;inpartic ular,forthersttype wegeteitherrealorpseudorealrepresentations,respecti vely.In r -matrixlanguage, thischargeconjugationmatrixisstraightforwardlyconst ructedintherepresentation wherethe r -matricesareexpressedintermsofdirectproductsofthePa ulimatrices forthe2-dimensionalsubspaces.UponWickrotationof1dir ectioneachinany numberofpairscorrespondingtothese2-dimensionalsubsp aces,thecorresponding Paulimatrixfactorinthechargeconjugationmatrixmustbe dropped(withperhaps somechangeinthechoiceofPaulimatrixfactorsfortheothe rsubspaces).Itthen followsthat(pseudo)realityisthesameinSO( D + +1, D +1)asinSO( D + D ),so allcasesfollowfromtheEuclideancase.Forthesecondtype ofmetric, = y in theEuclideancase,so M isjusttheidentitymatrix(i.e.,thespinorrepresentatio ns areunitary).AfterWickrotation,thismatrixbecomesthep roductofallthe r matricesintheWickrotateddirections,sincethose r -matricesgotfactorsof i inthe Wickrotation,andthusneedthisextrafactortopreserveth erealityofthetensors r r .Thesymmetrypropertiesofthismetricthenfollowfromth oseofthe r -matrices.Also,becauseofthesignatureofthe r -matrices,itfollowsthatthis metric,exceptintheEuclideancase,hashalfitseigenvalu es+1andhalf 1.The lasttypeofmetrichasonlyundottedindicesandthushasnot hingtodowithcomplex conjugation,soitspropertiesareunchangedbyWickrotati on.It'sidenticaltothe rsttypeinEuclideanspace(sincethesecondtypeistheide ntitythere;ingeneral, if2ofthemetricsexist,thethirdisjusttheirproduct),wh ichthusdeterminesitin thegeneralcase.Varioustypesofgroupsaredenedbythese metricsalone(real, unitary,orthogonal,symplectic,etc.),withtheSO( D + D )groupasasubgroup. (Infact,thesemetrics completely determinetheSO( D + D )group,uptoabelian factors,in D D + + D 6,andallowallvectorindicestobereplacedbypairs ofspinorindices.Theyalsodeterminethegroupin D =8for D even,dueto \triality,"thediscretesymmetrywhichpermutesthevecto rrepresentationwiththe 2spinors.)Wealsoclassifythe -matricesbytheirsymmetrypropertiesonlywhenits 2spinorindicesareforequivalentrepresentations,sothe yareunrelatedtocomplex conjugation(bothindicesundotted),andthustheirsymmet ryisdeterminedbythe Euclideancase.

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92 5.PARTICLE Wenowsummarizetheresultsobtainedbythemethodssketche daboveforspinors ,metrics (symmetric)andn(antisymmetric),and -matrices,intermsof D mod 8and D mod4: D 0 1 2 3 D Euclidean Lorentz conformal 0 0 0 . n n 0 0 1 . . n n n n ( ) ( ) ( ) ( ) 2 . n n ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 3 n n . n n n n n n . ( ) ( ) ( ) ( ) 0 0 4 n n . n n n n 0 0 5 n n . n n . n n n n [ ] [ ] [ ] [ ] 6 n n . [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] 7 . n n n n . [ ] [ ] [ ] [ ] (Wehaveomittedthevectorindicesonthe -matrices.Wehavealsoomittedmetrics whicharecomplexconjugatesorinversesofthoseshown,ora rethesamebutwithall

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5.3.Spinning 93 indicesprimed,whererelevant.)Also,notindicatedinthe tableisthefactthat ispositivedenitefortheEuclideancaseandhalf-positiv e,half-negativeotherwise. Finally,thedimensionofthespinorsis2 ( D 1) = 2 for D oddand2 ( D 2) = 2 (Weylspinor) for D even.These N N metricsdeneclassicalgroupsassubgroupsof GL ( N;C ): SO ( N;C ) n Sp ( N;C ) GL ( N;R ) n GL ( N )( U ( N )) U ( N )( orU ( N 2 ; N 2 )) n U ( N 2 ; N 2 ) . SO ( N )( orSO ( N 2 ; N 2 )) n n SO ( N ) n n Sp ( N ) n n . USp ( N )( orUSp ( N 2 ; N 2 )) (Whenthematrixhasatrace,thegroupcanbefactoredintoth ecorresponding \ S "-grouptimesanabelianfactor U (1)or GL (1 ;R ).) The -matricessatisfytheobviousrelationanalogoustothe r -matrixanticommutationrelations:Contractapairofspinorson2 -matricesandsymmetrizein thevectorindicesandyouget(twice)themetricforthevect orrepresentation(the SO( D + D )metric)timesaKronecker intheremainingspinorindices: ( a b ) = ( a 0 b ) 0 = ( a [ r ] b )[ r ] = ( a ( r ) b )( r ) =2 ab ; (5 : 3 : 11) andsimilarlyforexpressionswithdottedandundotted(orp rimedandunprimed) indicesswitched.(Wehaveraisedindiceswithspinormetri cswhennecessary.) Although(irreducible)spinorsthushavemanydierencesi ndierentdimensions, therearesomepropertieswhicharedimension-independent ,anditwillproveuseful tochangenotationtoemphasizethosesimilarities.Wether eforedenespinorswhich arerealinalldimensions(orwouldberealafteracomplexsi milaritytransformation, andthereforesatisfyageneralizedMajoranacondition).F orthosekindsofspinors intheabovetablewhicharecomplexorpseudoreal,thismean smakingabigger

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94 5.PARTICLE spinorwhichcontainstherealandimaginarycomponentsoft hepreviousoneas independentcomponents.Iftheoriginalspinorwascomplex ( D + D twiceodd),the newspinorisreducibletoanirreduciblespinoranditsineq uivalentcomplexconjugate representation,whichtransformoppositelywithrespectt oaninternalU(1)generator (\ r 5 ").Iftheoriginalspinorwaspseudoreal( D + D =3 ; 4 ; 5 mod 8),thenew spinorreducesto2equivalentirreduciblespinorrepresen tations,whichtransformas adoubletwithrespecttoaninternalSU(2). Thenetresultfortheserealspinorsisthatwehavethefollo winganalogofthe abovetableforthosepropertieswhichholdforallvaluesof D + : D 0 1 2 3 Euclidean Lorentz conformal 0 0 n r 0 r ( ) r ( ) r 0 r [ ] r [ ] These r -matricessatisfythesamerelationsasthe -matricesin(5.3.11).(Infact,they areidenticalfor D + = D mod 8.)Theiradditional, D + -dependentpropertiescan bedescribedbyadditionalmetrics:(1)theinternalsymmet rygeneratorsmentioned above;and(2)for D odd,ametric M or M 0 whichrelatesthe2typesofspinors (sincethereare2independentirreduciblespinorrepresen tationsonlyfor D even). Similarmethodsofrst-quantizationwillbeappliedinsec t.7.2tothespinning string,whichhasspin-0andspin-1/2groundstates.Classi calmechanicsactionsfor particleswithotherspins(orstringswithgroundstateswi thotherspins),i.e.,gauge elds,arenotknown.(Forthesuperstring,however,anonma nifestlysupersymmetric formalismcanbeobtainedbyatruncationofthespinningstr ing,eliminatingsome ofthegroundstates.)Ontheotherhand,theBRSTapproachof chap.3allows thetreatmentofthequantummechanicsofarbitrarygaugee lds.Furthermore,the superparticle,describedinthefollowingsection,isdesc ribedclassical-mechanically byaspin-0orspin-1/2supereld,whichincludescomponent gaugeelds,justasthe stringhascomponentgaugeeldsinitsexcitedmodes.

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5.4.Supersymmetric 95 5.4.Supersymmetric Thesuperparticleisobtainedfromthespinlessparticleby symmetrizingwith respecttothesupersymmetryofthehigher-dimensionalspa ceinwhichtheonedimensionalworldlineoftheparticleisimbedded.(Forrev iewsofsupersymmetry, see[1.17].)Asforthespinlessparticle,afullunderstand ingofthisactionconsistsof justunderstandingthealgebrasofthecovariantderivativ esandequationsofmotion. Inordertodescribearbitrary D ,weworkwiththegeneralrealspinorsoftheprevioussection.Thecovariantderivativesare p a (momentum)and d (anticommuting spinor),with f d ;d g =2 r a p a (5 : 4 : 1) (theothergradedcommutatorsvanish),wherethe r matricesaresymmetricintheir spinorindicesandsatisfy r ( a r r b ) r =2 ab ; (5 : 4 : 2) asdescribedintheprevioussection.Thisalgebraisrepres entedintermsofcoordinates x a (spacetime)and (anticommuting),andtheirpartialderivatives @ a and @ ,as p a = i@ a ;d = @ + ir a @ a : (5 : 4 : 3) Thesecovariantderivativesareinvariantundersupersymm etrytransformationsgeneratedby p a and q ,whichformthealgebra f q ;q g = 2 r a p a : (5 : 4 : 4) p a isgivenabove,and q isrepresentedintermsofthesamecoordinatesas q = @ ir a @ a : (5 : 4 : 5) (See(5.3.1)for D =1.)AlltheseobjectsalsotransformcovariantlyunderLor entz transformationsgeneratedby J ab = ix [ a p b ] + 1 4 r [ ar r b ] r @ + M ab ; (5 : 4 : 6) wherewehaveincludedthe(coordinate-independent)spint erm M ab .(Incomparison with(2.2.4),thespinoperatorheregivesjustthespinofth esupereld,whichis afunctionof x and ,whereasthespinoperatorthereincludesthe @ term,and thusgivesthespinofthecomponenteldsresultingastheco ecientsofaTaylor expansionofthesupereldinpowersof .)

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96 5.PARTICLE Asdescribedintheprevioussection,evenfor\simple"supe rsymmetry(thesmallestsupersymmetryforthatdimension),thesespinorsarere ducibleiftheirreducible spinorrepresentationisn'treal,andreducetothedirects umofanirreduciblespinor anditscomplexconjugate.However,wecanfurthergenerali zebylettingthespinor representmorethanoneofsuchrealspinors(andsomeofeach ofthe2typesthat areindependentwhen D istwiceodd),andstillusethesamenotation,withasingle indexrepresentingallspinorcomponents.(5.4.1-6)areth enunchanged(exceptfor therangeofthespinorindex).However,thenatureofthesup ersymmetryrepresentationswilldependon D ,andonthenumberofminimalsupersymmetries.In theremainderofthissectionwe'llsticktothisnotationto manifestthoseproperties whichareindependentofdimension,andincludesuchthings asinternal-symmetry generatorswhenrequiredfordimension-dependentpropert ies. Foramassless,real,scalareld, p 2 =0istheonlyequationofmotion,butfora massless,real,scalarsupereld,theadditionalequation =pd =0(where d isthespinor derivative)isnecessarytoimposethatthesupereldisaun itaryrepresentationof (on-shell)supersymmetry[5.12]:Sincethehermitiansupe rsymmetrygenerators q satisfy f q;q g p ,wehavethat f =pq;=pq g p 2 =p =0,butonunitaryrepresentations anyhermitianoperatorwhosesquarevanishesmustalsovani sh,so0= =pq = =pd upto atermproportionalto p 2 =0.Thismeansthatonlyhalfthe q 'sarenonvanishing.We canfurtherdividetheseremaining q 'sin(complex)halvesascreationandannihilation operators.Amassless,irreduciblerepresentationofsupe rsymmetryisthenspecied inthisnonmanifestlyLorentzcovariantapproachbyxingt he\Cliord"vacuumof thesecreationandannihilationoperatorstobeanirreduci blerepresentationofthe Poincaregroup. Unfortunately,the p 2 and =pd equationsarenotsucienttodetermineanirreduciblerepresentationofsupersymmetry,evenforascalar supereld(withcertain exceptionsin D 4)since,althoughtheykilltheunphysicalhalfofthe q 's,they don'trestricttheCliordvacuum.Thelatterrestrictionr equiresextraconstraints inamanifestlyLorentzcovariantformalism.Thereareseve ralwaystondthese additionalconstraints:Oneistoconsidercouplingtoexte rnalelds.Thesimplest caseisexternalsuper-Yang-Mills(whichwillbeparticula rlyrelevantforstrings).The generalizationofthecovariantderivativesis d !r = d + ; p a !r a = p a + a : (5 : 4 : 7)

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5.4.Supersymmetric 97 Wethushaveagradedcovariantderivative r A A =( a; ).Withoutlossofgenerality, weconsidercaseswheretheonlyphysicaleldsinthesuperYang-Millsmultipletare avectorandaspinor.Theothercases(containingscalars)c anbeobtainedeasilyby dimensionalreduction.Thenthecommutationrelationsoft hecovariantderivatives become[5.13] fr ; r g =2 r a r a ; [ r ; r a ]=2 r a W ; [ r a ; r b ]= F ab ; (5 : 4 : 8) where W isthesuper-Yang-Millseldstrength(at =0,thephysicalspinoreld), andconsistencyoftheJacobi(Bianchi)identitiesrequire s r a ( r a r ) =0 : (5 : 4 : 9) Thiscondition(whenmaximalLorentzinvarianceisassumed ,i.e.,SO(D-1,1)for a taking D values)impliesspacetimedimensions D =3 ; 4 ; 6 ; 10,and\antispacetime" dimensions(thenumberofvaluesoftheindex ) D 0 =2( D 2).(Thelatteridentity followsfrommultiplying(5.4.9)by r b andusing(5.4.2).)Thegeneralizationofthe equationsofmotionis[5.14] =pd r a r a r ; 1 2 p 2 1 2 r a r a + W r ; (5 : 4 : 10) butclosureofthisalgebraalsorequiresnewequationsofmo tionwhicharecertain Lorentzpiecesof r [ r ] .Specically,in D =3thereisonlyoneLorentzpiece(a scalar),anditgivestheusualeldequationsforascalarmu ltiplet[5.15];in D =4the scalarpieceagaingivestheusualequationsforachiralsca larmultiplet,butthe(axial) vectorpiecegivesthechiralitycondition(afterappropri atenormalordering);in D =6 onlytheself-dualthird-rankantisymmetrictensorpiecea ppearsinthealgebra,and itgivesequationssatisedbyscalarmultiplets(butnotby tensormultiplets,which arealsodescribedbyscalarsupereldstrengths,butcan't coupleminimallytoYangMills)[5.16];butin D =10nomultipletisdescribedbecausetheonlyonepossible wouldbeYang-Millsitself,butitseldstrength W carriesaLorentzindex,andthe equationsdescribedabove(whichapplyonlytoscalars)nee dextratermscontaining Lorentzgenerators. Anotherwaytoderivethemodicationsistousesuperconfor maltransformations. Thesuperconformalgroups[5.17]areactuallyeasiertoder ivethanthesupersymmetrygroupsbecausetheyarejustgradedversionsofclassica lgroups.Specically,the

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98 5.PARTICLE classicalsupergroups(see[5.18]forareview)havedenin grepresentationsdenedin termsofametric M A B ,whichmakesthemunitary(orpseudounitary,ifthemetric isn'tpositivedenite),andsometimesalsoagraded-symme tricmetric M AB ,andthus M A B bycombiningthem(andtheirinverses).Thegeneratorswhic hhavebosonicbosonicorfermionic-fermionicindicesarebosonic,andth osewithbosonic-fermionic arefermionic.(Thechoiceofwhichpartsofthe A indexarebosonicandwhichare fermioniccanbereversed,butthisdoesn'taectthestatis ticsofthegroupgenerators.)Sincethebosonicsubgroupofthesupergroupswithju stthe M A B metricis thedirectproductof2unitarygroups,thosesupergroupsar ecalled(S)SU(M j N)for MvaluesoftheindexofonestatisticsandNoftheother,wher etheSisbecausea (graded)traceconditionisimposed,andthereisasecondSf orM=Nbecausethen asecondtracecanberemoved(soeachofthe2unitarysubgrou psbecomesSU(N)). Thesupergroupswhichalsohavethegraded-symmetricmetri c M AB haveabosonic subgroupwhichisorthogonalinthesectorwherethemetrici ssymmetricandsymplecticinthesectorwhereitisantisymmetric.Inthiscase wechoosethemetric M A B alsotohavegradedsymmetry,insuchawaythatthemetric M A B obtainedfrom theirproductistotallysymmetric,sothedeningrepresen tationisreal,ortotally antisymmetric,sotherepresentationispseudoreal.Thefo rmerisgenerallycalled OSp(M j 2N),andwecallthelatterOSp (M j 2N). Wenextassumethattheanticommutinggeneratorsofthesesu pergroupsare tobeidentiedwiththeconformalgeneralizationofthesup ersymmetrygenerators. Thus,oneindexistobeidentiedwithaninternalsymmetry, andtheotherwitha conformalspinorindex.Theconformalspinorreducesto2Lo rentzspinors,oneof whichistheusualsupersymmetry,the\squareroot"oftrans lations,andtheother ofwhichis\S-supersymmetry,"thesquarerootofconformal boosts.Thechoiceof supergroupthenfollowsimmediatelyfromthegradedgenera lizationoftheconformal spinormetricsappearinginthetableoftheprevioussectio n[5.19]: Dmod8 superconformal bosonicsubgroup dim-0/dilatations 0,4 (S)SU(N j ) SU( ) n SU(N)( n U(1)) SL( ,C) n (S)U(N) 1,3 OSp(N j 2 ) Sp(2 ) n SO(N) SL( ,R) n SO(N) 2 OSp(N j 2 ) " or (") + n (") (") + n (") (") + n (") 5,7 OSp (2 j 2N) SO (2 ) n USp(2N) SL ( ) n USp(2N) 6 OSp (2 j 2N) " or (") + n (") (") + n (") (") + n (")

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5.4.Supersymmetric 99 where\dim-0"arethegeneratorswhichcommutewithdilatat ions(seesect.2.2),so thelastcolumngivesLorentz n internalsymmetry(atleast). isthedimensionof the(irreducible)Lorentzspinor(1/2thatoftheconformal spinor),Nisthenumber ofminimalsupersymmetries,and D ( > 2)isthedimensionof(Minkowski)spacetime (withconformalgroupSO(D,2),so2lessthanthe D intheprevioustable).The2 choicesfortwice-odd D dependonwhetherwechoosetorepresentthesuperconformal grouponbothprimedandunprimedspinors.Ifso,therecanbe aseparateNand N 0 .Again,wehaveused toindicategroupswhichareWickrotationssuchthat thedeningrepresentationispseudorealinsteadofreal.( SL issometimesdenoted SU .) Unfortunately,asdiscussedintheprevioussection,thebo sonicsubgroupacting ontheconformalspinorpartofthedeningrepresentation, asdenedbythespinor metrics(plusthetracecondition,whenrelevant)givesagr oupbiggerthantheconformalgroupunless D 6.However,wecanstilluse D 6,andperhapssomeofthe qualitativefeaturesof D> 6,forouranalysisofmasslesseldequations.Wethen generalizeouranalysisofsect.2.2fromconformaltosuper conformal.It'ssucient toapplyjusttheS-supersymmetrygeneratorstojusttheKle in-Gordonoperator.We thennd[2.6,5.19]: 1 2 p 2 =pq = =pd ( 1 2 f p b ;J ab g + 1 2 f p a ; g = p b M ab + p a d D 2 2 q [ q ] + = d [ d ] + ; (5 : 4 : 11) wherethelastexpressionmeanscertainLorentzpiecesof dd pluscertaintermscontainingLorentzandinternalsymmetrygenerators.Inparti cular, 1 16 r abc d d + 1 2 p [ a M bc ] isthesupersymmetricanalogofthePauli-Lubanskyvector[ 5.20].Thevectorequationis(2.2.8)again,derivedinessentiallythesa meway. Fortheconstraintswethereforechoose[5.21] A = 1 2 p 2 ; B = r a p a d ; C abc = 1 16 r abc d d + 1 2 p [ a M bc ] ; D a = M a b p b + kp a ;(5 : 4 : 12 a ) or,inmatrixnotation, A = 1 2 p 2 ; B = =pd;

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100 5.PARTICLE C abc = 1 16 dr abc d + 1 2 p [ a M bc ] ; D a = M a b p b + kp a ;(5 : 4 : 12 b ) whereoutof(5.4.11)wehavechosen A and D asfornonsupersymmetrictheories (sect.2.2), B forunitarity(asexplainedabove),andjustthePauli-Luba nskypart oftherest(whichisallofitforD=10),thesignicanceofwh ichwillbeexplained below. Theseconstraintssatisfythealgebra fB ; Bg =4 =p A ; [ D a ; D b ]= 2 M ab A p [ a D b ] ; [ C abc ; B ]= 8 r abc d A ; [ C abc ; D d ]= d [ a p b D c ] ; [ C abc ; C def ]= 1 4 [ [ d [ a p e C bc ] f ] ( abc $ def )] 1 128 d (4 r abc def 2 [ d [ a e b r c ] f ] ) B ; rest =0 ; (5 : 4 : 13) withsomeambiguityinhowtheright-handsideisexpressedd uetotherelations p D =2 k A ; =p B =2 d A ; d B =2( trI ) A ; 1 6 p [ a C bcd ] = 1 16 dr abcd B ; p c C cab =2 M ab A + p [ a D b ] + 1 16 dr ab B : (5 : 4 : 14) ( r ab = 1 2 r [ a r b ] ,etc.) Inthecaseofsupersymmetrywithaninternalsymmetrygroup (extendedsupersymmetry,orevensimplesupersymmetryinD=5,6,7),the reisanadditionalconstraintanalogousto C abc forsuperisospin: C a;int = 1 8 dr a int d + p a M int : (5 : 4 : 15) int arethematrixgeneratorsoftheinternalsymmetrygroup,in therepresentation towhich d belongs,and M int arethosewhichactontheexternalindicesofthe supereld.

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5.4.Supersymmetric 101 Unfortunately,therearefewsuperspin-0multipletsthata recontainedwithin spin-0,isospin-0superelds(i.e.,thatthemselvesconta inspin-0,isospin-0componentelds).Infact,theonlysuchmultipletsofphysicalin terestinD > 4areN=1 Yang-MillsinD=9andN=2nonchiralsupergravityinD=10.(F oraconvenientlisting ofmultiplets,see[5.22].)However,bythemethoddescribe dintheprevioussection, spinorrepresentationsfortheLorentzgroupcanbeintrodu ced.Byincluding\ r matrices"forinternalsymmetry,wecanalsointroduceden ingrepresentationsforthe internalsymmetrygroupsforwhichtheyareequivalenttoth espinorrepresentations oforthogonalgroups(i.e.,SU(2)=USp(2)=SO(3),USp(4)=S O(5),SU(4)=SO(6), SO(4)-vector=SO(3)-spinor n SO(3) 0 -spinor,SO(8)-vector=SO(8) 0 -spinor).Furthermore,arbitraryU(1)representationscanbedescribedbyad dingextratermswithout introducingadditionalcoordinates.Thisallowsthedescr iptionofmostsuperspin0multiplets,butwithsomenotableexceptions(e.g.,11Dsu pergravity).However, theseequationsarenoteasilygeneralizedtononzerosuper spins,since,althoughthe superspinoperatoriseasytoidentifyinthelight-conefor malism(seebelow),thecorrespondingoperatorwouldbenonlocalinacovariantdescri ption(orappearalways withanadditionalfactorofmomentum). Wenextconsidertheconstructionofmechanicsactions.The seequationsdescribe onlymultipletsofsuperspin0,i.e.,thesmallestrepresen tationsofagivensupersymmetryalgebra,forreasonstobedescribedbelow.(Thisisno restrictioninD=3, wheresuperspindoesn'texist,andinD=4arbitrarysupersp incanbetreatedbya minormodication,sincetheresuperspinisabelian.)Asde scribedintheprevious section,onlyspin-0andspin-1/2supereldscanbedescrib edbyclassicalmechanics, andwebeginwithspin-0,droppingspintermsin(5.4.12),an dthegenerator D .The actionisthengivenby(3.1.10),where[5.23] z M =( x m ; ) ; A =( p a ;id ) ; z M e M A ( z )=( x i r; ) ; i G i ( )=( A ; B ; C abc ) : (5 : 4 : 16) Uponquantization,thecovariantderivativesbecome A = ie A M @ M = i ( @ a ;@ + ir a @ a ) ; (5 : 4 : 17) whichareinvariantunderthesupersymmetrytransformatio ns x = ir; = : (5 : 4 : 18)

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102 5.PARTICLE Thetransformationlawsthenfollowdirectlyfrom(3.1.11) ,withtheaidof(5.4.13) forthe transformations. Theclassicalmechanicsactioncanbequantizedcovariantl ybyBRSTmethods. Inparticular,thetransformationsgeneratedby B [5.24](withparameter )closeon thosegeneratedby A (withparameter ): x = p + i ( rd + =pr ) ; = =p; p =0 ;d =2 p 2 ; g = +4 i=p ; = : (5 : 4 : 19) Becauseofthesecondlineof(5.4.14),theghostshaveagaug einvariancesimilarto theoriginal invariance,andthentheghostsofthoseghostsagainhavesu chan invariance,etc.,adinnitum.Thisisaconsequenceofthef actthatonlyhalfof canbegaugedaway,butthereisgenerallynoLorentzreprese ntationwithhalfthe componentsofaspinor,sothespinorgaugeparametermustit selfbehalfgauge,etc. Althoughsomewhatawkward,theinnitesetofghostsisstra ightforwardtond. Furthermore,ifderivedfromthelightcone,theOSp(1,1 j 2)generatorsautomatically containthisinnitenumberofspinors:There, isrst-quantizedinthesamewayas theDiracspinorwassecond-quantizedinsect.3.5,and obtainsaninnitenumber ofcomponents(asaninnitenumberofordinaryspinors)asa resultofbeinga representationofagradedCliordalgebra(specically,t heHeisenbergalgebrasof r and~ r ).Thisanalysiswillbemadeinthenextsection. Ontheotherhand,theanalysisoftheconstraintsissimples tinthelight-cone formalism.The A B ,and D equationscanbesolveddirectly,becausetheyareallof theform p f = p + f + : A =0 p = 1 2 p + p i 2 ; B =0 r d = 1 2 p + r i p i r r + d; D =0 M i = 1 p + ( M i j p j + kp i ) ;M + = k; (5 : 4 : 20 a ) wherewehavechosenthecorrespondinggauges x + =0 ; r + =0 ; M + i =0 : (5 : 4 : 20 b )

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5.4.Supersymmetric 103 Thesesolutionsrestrictthe x 's, 's,andLorentzindices,respectively,tothoseofthe lightcone.(Eectively,Disreducedby2,exceptthat p + remains.) However,asupereldwhichisafunctionofalight-cone isnotanirreducible representationofsupersymmetry(exceptsometimesinD 4),althoughitisaunitary one.Infact, C isjustthesuperspinoperatorwhichseparatestherepresen tations:Due totheotherconstraints,allitscomponentsarelinearlyre latedto C + ij = p + M ij + 1 16 dr + r ij d: (5 : 4 : 21) Uptoafactorof p + ,thisisthelight-conesuperspin:Onanirreduciblerepres entation ofsupersymmetry,itactsasanirreduciblerepresentation ofSO(D-2).InD=4this canbeseeneasilybynotingthattheirreduciblerepresenta tionscanberepresented intermsofchiralsuperelds( d =0)withdierentnumbersof d 'sactingonthem, andthe d d in C justcountsthenumbersof d 's.Ingeneral,ifwenotethatthefull light-coneLorentzgeneratorcanbewrittenas J ij = ix [ i p j ] + 1 2 r ij @ @ + M ij = ix [ i p j ] 1 16 p + qr + r ij q + 1 16 p + dr + r ij d + M ij = ^ J ij + 1 p + C + ij ; (5 : 4 : 22) then,byexpressinganystateintermsof q 'sactingontheCliordvacuum,weseethat ^ J givesthecorrecttransformationforthose q 'sandthe x -dependenceoftheCliord vacuum,so C =p + givesthespinoftheCliordvacuumlessthecontributionof the qq termonit,i.e.,thesuperspin.Unfortunately,themechani csactioncan'thandle spinoperatorsforirreduciblerepresentations(eitherfo r M ij orthesuperspin),sowe mustrestrictourselvesnotonlytospin0(referringtothee xternalindicesonthe supereld),butalsosuperspin0(atleastattheclassicalm echanicslevel).Thus,the remainingconstraint C =0istheonlypossible(rst-class)constraintwhichcanma ke thesupersymmetryrepresentationirreducible.Theconstr aints(5.4.12)aretherefore necessaryandsucientforderivingthemechanicsaction.H owever,ifweallowthe trivialkindofsecond-classconstraintsthatcanbesolved intermsofmatrices,wecan generalizetospin-1/2.Inprinciple,wecouldalsodosuper spin-1/2,butthisleadsto covarianteldswhicharejustthoseforsuperspin-0withan extraspinorindextagged on,whichdiersbyfactorsofmomentum(withappropriatein dexcontractions)from thedesiredexpressions.(Thus,thesuperspinoperatorwou ldbenonlocalonthe

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104 5.PARTICLE latter.)Isospin-1/2canbetreatedsimilarly(andsuperis ospin-1/2,butagainasa tagged-onindex). ThesimplestnontrivialexampleisD=4.(InD=3, C + ij vanishes,sincethetransverseindex i takesonlyonevalue.)Therelight-conespinorshaveonly1( complex) component,andsodoes C + ij .Forthiscase,wecan(andmust,foranoddnumberN ofsupersymmetries)modify C abc : C abc = 1 16 dr abc d + 1 2 p [ a M bc ] + iH abcd p d ; (5 : 4 : 23) where H isthe\superhelicity."Wethennd C + ij = p + ij M + iH i 1 4 p + [ d a ; d a ] ; (5 : 4 : 24) where M ij = ij M and f d a ; d b g = p + a b ,and a isanSU(N)index.The\helicity" h isgivenby M = ih ,andwethenndbyexpandingtheeldoverchiralelds [5.25-27]( d =0) ( d ) n H = h + 1 4 (2 n N ) : (5 : 4 : 25) Specifyingboththespinandsuperhelicityoftheoriginals upereldxesboth h and H ,andthusdetermines n .Notethatthisrequires H tobequarter-(odd-)integralfor oddN.Ingeneral,theSU(N)representationof alsoneedstobespecied,andthe relevantpartof(5.4.15)is C + a b = p + M a b i 1 4 p + [ d a ; d b ] 1 4 a b [ d c ; d c ] # ; (5 : 4 : 26) andthevanishingofthisquantityforces tobeanSU(N)-singlet.(Moregeneral casescanbeobtainedsimplybytackingextraindicesontoth eoriginalsupereld, andthusonto .) Wenextconsider10DsuperYang-Mills.Theappropriatesupe reldisaWeyl orMajoranaspinor,soweincludetermsasintheprevioussec tioninthemechanics action.Tosolvetheremainingconstraint,werstdecompos eSO(9,1)covariant spinorsand r -matricestoSO(8)light-coneonesas d =2 1 = 4 d + d ; =p = p 2 p + =p T =p T y p 2 p 0 0 ;=p = 0 0 p 2 p =p T =p T y p 2 p + ;

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5.4.Supersymmetric 105 =a T =b T y + =b T =a T y = =a T y =b T + =b T y =a T =2 a T b T ; =p T = =p T ; = y = ; 2 = I: (5 : 4 : 27) (WecouldchoosetheMajoranarepresentation= I ,butotherrepresentationscan bemoreconvenient.)Theindependentsupersymmetry-covar iantderivativesarethen d + = @ @ + + p + + ;p T ;p + : (5 : 4 : 28) Inordertointroducechirallight-conesuperelds,wefurt herreduceSO(8)to SO(6) n SO(2)=U(4)notation: d + = p 2 d a d a ;=p T = p ab a b p L b a p L p ab ; = 0 I I 0 ( p ab = 1 2 abcd p cd ) : (5 : 4 : 29) Intermsofthis\euphoric"notation,theconstraints C + ij arewrittenontheSO(6)spinorsupereldas 1 p + C + a b c c = i 1 2 c b a 1 4 a b c a c b + 1 4 a b c i 1 4 p + [ d a ; d b ] 1 4 a b [ d d ; d d ] c c ; 1 p + C + c c = i 1 2 c c i 1 4 p + [ d a ; d a ] c c ; 1 p + C + ab c c = i 1 2 abcd d 0 i 1 2 p + d a d b c c ; (5 : 4 : 30) andthecomplexconjugateequationfor C + ab .(Notethatitiscrucialthattheoriginal SO(10)supereld wasaspinorofchiralityoppositetothatof d inordertoobtain solubleequations.)Thesolutiontotherst2equationsgiv es intermsofachiral supereld a = d a ; (5 : 4 : 31 a ) andthattothethirdequationimposestheself-dualitycond ition[5.25-27] 1 24 abcd d a d b d c d d = p + 2 : (5 : 4 : 31 b ) Thiscanalsobewrittenas n Y h 1 2 p + 1 = 2 d io = Z de a a p + = 2 ( a )=[ ( a )] : (5 : 4 : 32) Thiscorrespondstothefactthatinthemechanicsaction reversalon includes multiplicationbythechargeconjugationmatrix,whichswi tches a with a ,which equals (2 =p + ) @=@ a bythechiralitycondition d a =0.

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106 5.PARTICLE Theseresultsareequivalenttothoseobtainedfromrst-qu antizationofamechanicsactionwith d =0asasecond-classconstraint[5.28].(Thisistheanalogo f theconstraint^ r =0oftheprevioussection.)Thisiseectivelythesameasdr oppingthe d termsfromtheaction,whichcanthenbewritteninsecond-or derformby eliminating p byitsequationofmotion.Thiscanbesolvedeitherbyusinga chiral supereld[5.27]asasolutiontothisconstraintinaGuptaBleulerformalism, d a =0 d a =0 Z d + =0 ; (5 : 4 : 33) orbyusingasupereldwitha real 4-component [5.26]asasolutiontothisconstraintbeforequantization(butaftergoingtoalight-con egauge),determininghalf theremainingcomponentsof r tobethecanonicalconjugatesoftheotherhalf. However,whereaseitherofthesemethodswithsecond-class constraintsrequiresthe breakingofmanifestLorentzcovariancejustfortheformul ationofthe(eld)theory, themethodwehavedescribedabovehasconstraintsontheel dswhicharemanifestlyLorentzcovariant((5.4.12)).Furthermore,thisse cond-classapproachrequires that(5.4.32)beimposedinaddition,whereasintherst-cl assapproachitandthe chiralityconditionautomaticallyfollowedtogetherfrom (5.4.21)(andtheordinary realityoftheoriginalSO(9,1)spinorsupereld). Ontheotherhand,theformalismwithsecond-classconstrai ntscanbederived fromtherst-classformalism without M ab terms(andthuswithout D in(5.4.12)) [5.29]:Justas A and B weresolvedattheclassicalleveltoobtain(5.4.20), C =0can alsobesolvedclassically.Tobespecic,weagainconsider D =10.Then C + ij =0 isequivalentto d [ d ] =0(where isan8-valuedlight-conespinorindex).(They arejustdierentlinearcombinationsofthesame28antisym metricquadraticsin d ,whicharetheonlynonvanishing d productsclassically.)Thisconstraintimplies thecomponentsof d areallproportionaltothesameanticommutingscalar,time s dierentcommutingfactors: d d =0 d = c ; (5 : 4 : 34 a ) where c isanticommutingand commuting.Furthermore, C + ij arejustSO(8)generatorson d ,andthustheirgaugetransformationcanbeusedtorotateit inany direction,thuseliminatingallbut1component[5.30].(Th isisclearfromtriality, sincethespinor8representationislikethevector8.)Spec ically,wechoosethe gaugeparametersofthe C transformationstodependon insuchawayastorotate inanyonedirection,andthenredene c toabsorbtheremaining factor: C gauge : d = m 1 c: (5 : 4 : 34 b )

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5.4.Supersymmetric 107 Inthisgauge,the C constraintitselfistrivial,sinceitisantisymmetricin d 's.Finally, wequantizethisoneremainingcomponent c of d tond quantization : c 2 = p + c = p p + : (5 : 4 : 34 c ) c hasbeendeterminedonlyuptoasign,butthereisaresidual C gaugeinvariance, sincethe C rotationcanalsobeusedtorotate intheoppositedirection,changingits sign.Afterusingthegaugeinvariancetomakeallbutonecom ponentof vanish,this signchangeistheonlypartofthegaugetransformationwhic hsurvives.Itcanthen beusedtochoosethesignin(5.4.34c).Thus,allthe d 'saredetermined(although1 componentisnonvanishing),andweobtainthesamesetofcoo rdinates( x and q ,no d )asinthesecond-classformalism.The C constraintcanalsobesolvedcompletely atthequantummechanicallevelbyGupta-Bleulermethods[5 .29].TheSO(D-2) generatorsrepresentedby C arethendividedupintotheCartansubalgebra,raising operators,andloweringoperators.Theraisingoperatorsa reimposedasconstraints (ontheket,andtheloweringoperatorsonthebra),implying onlythehighest-weight statesurvives,andthegeneratorsoftheCartansubalgebra areimposedonlyupto \normal-ordering"constants,whicharejusttheweightsof thatstate. Thecomponentsofthischiralsupereldcanbeidentiedwit htheusualvector +spinor[5.26,27]: ( x; a )= ( p + ) 1 A L ( x )+ a ( p + ) 1 a ( x )+ 2 ab A ab ( x )+ 3 a a ( x )+ 4 ( p + ) A L ( x ) ; (5 : 4 : 35) andthe =0componentsof r + =( a ; a )canbeidentiedwiththespinor.Alternatively,thevector+spinorcontentcanbeobtaineddirect lyfromthevanishingof C + ij of(5.4.21),withoutusingeuphoricnotation:Werstnotet hatthe-matricesof M ij arerepresentedby2spinors,correspondingtothe2dieren tchiralitiesofspinors inSO(8).SO(8)hasthepropertyof\triality,"whichisthep ermutationsymmetry ofthese2spinorswiththevectorrepresentation.(Allare8 -componentrepresentations.)Sincetheanticommutationrelationsof d arejustatrialitytransformationof thoseofthe's(modulo p + 's),theyarerepresentedbytheother2representations: aspinoroftheotherchiralityandavector.Thesameholdsfo rtherepresentation of q .Thus,thedirectproductoftherepresentationsofand d includesasinglet (superspin0),pickedoutby C + ij =0,sothetotalSO(8)representation(generated by J ij of(5.4.22))isjustthatof q ,aspinor(ofoppositechirality)andavector. Thereisanotheron-shellmethodofanalysisof(super)conf ormaltheoriesthatis manifestlycovariantandmakesessentialuseofspinors.Th ismethodexpressesthe

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108 5.PARTICLE eldsintermsofthe\spinor"representationofthesuperco nformalgroup.(TheordinaryconformalgroupisthecaseN=0.)Thespinorisdenedin termsofgeneralized r -matrices(\twistors"[5.31]or\supertwistors"[5.32]): f r A ;r B ]= A B ; (5 : 4 : 36) wheretheindexhasbeenloweredby M A B ,andthegradingissuchthattheconformal spinorparthasbeenchosen commuting andtheinternalpart anticommuting ,justas ordinary r -matriceshavebosonic(vector)indicesbutareanticommut ing.Theanticommuting r 'sarethencloselyanalogoustothelight-conesupersymmet rygenerators r q .Thegeneratorsarethenrepresentedas G A B r A r B ; (5 : 4 : 37) withgraded(anti)symmetrizationortracessubtracted,as appropriate.Thecase ofOSp(1,1 j 2)hasbeentreatedin(3.5.1).Arepresentationintermsoft heusual superspacecoordinatescanthenbegeneratedbycoset-spac emethods,asdescribedin sect.2.2.Webeginbyidentifyingthesubgroupofthesuperg roupwhichcorresponds tosupersymmetry(bypicking1ofthe2Lorentzspinorsinthe conformalspinor generator)andtranslations(byclosureofsupersymmetry) .Wethenequatetheir representationin(5.4.37)(analogoustothe ^ J 'sofsect.2.2)withtheirrepresentation in(5.4.3,5)(analogoustothe J 'sofsect.2.2).(Theconstant r -matricesof(5.4.3,5) shouldnotbeconfusedwiththeoperatorsof(5.4.36).)This resultsinanexpression analogousto(2.2.6),where(0)isafunctionofhalfof(lin earcombinationsof) the r 'sof(5.4.36)(theotherhalfbeingtheircanonicalconjuga tes).(Forexample, thebosonicpartof r A isaconformalspinorwhichisexpressedasaLorentzspinor anditscanonicalconjugate.(5.4.37)thengives p a = r a ,whichimplies p 2 =0in D =3 ; 4 ; 6 ; 10dueto(5.4.9).)Wethenintegrateoverthese r 'stoobtain afunctionofjusttheusualsuperspacecoordinates x and .Duetothequadratic formofthemomentumgeneratorintermsofthe r 's,itdescribesonlypositiveenergy. Negativeenergies,forantiparticles,canbeintroducedby addingtotheeldaterm forthecomplexconjugaterepresentation.Atleastin D =3,4,6,10thissupereld satises p 2 =0asaconsequenceoftheexplicitformofthegenerators(5. 4.37),and asaconsequencealltheequationswhichfollowfromsuperco nformaltransformations. Theseequationsformasuperconformaltensorwhichcanbewr ittencovariantlyasan expressionquadraticin G A B .In D =4anadditionalU(1)actingonthetwistorspace canbeidentiedasthe(littlegroup)helicity,andin D =6asimilarSU(2)( n SU(2) 0 if

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5.4.Supersymmetric 109 theprimedsupergroupisalsointroduced)appears.((5.4.3 4a)isalsoasupertwistor typeofrelation.) Thecases D =3,4,6,10[5.33]areespeciallyinterestingnotonlyforth eabove reasonand(5.4.9)butalsobecausetheirvariousspacetime groupsformaninteresting patternifweconsiderthesegroupstobethesameforthesedi erentdimensions exceptthattheyareoverdierentgeneralizednumbersyste ms A called\division algebras."Thesearegeneralizationsofcomplexnumberswh ichcanbewrittenas z = z 0 + P n1 z i e i f e i ;e j g = 2 ij ,where n =0,1,3,or7.Choosingforthedierent dimensionsthedivisionalgebras D A 3 real 4 complex 6 quaternion 10 octonion wehavethecorrespondence SL 1 (1 ; A )= SO ( D 2) SU (2 ; A )= SO ( D 1) SU (1 ; 1 ; A )= SO ( D 2 ; 1) SL (2 ; A )= SO ( D 1 ; 1) SU 0 (4 ; A )= SO ( D; 2) SU ( N j 4 ; A )= superconformal 9>>>>>>>>>>>=>>>>>>>>>>>; =SO ( D 3) where SL 1 meansonlytherealpart( z 0 )ofthetraceofthedeningrepresentation vanishes,by SU 0 wemeantracelessandhavingthemetricn (vs. for SU ), andthegraded SU hasmetric M A B =( a b ; n )(inthatorder).The refers togeneralizedconjugation e i e i (andthe e i areinvariantundertransposition, althoughtheirorderinginsidethematriceschanges).The\ =SO ( D 3)"refersto thefactthattogetthedesiredgroupswemustincluderotati onsofthe D 3 e i 's amongthemselves.Theonlypossibleexceptionisforthe D =10superconformal groups,whichdon'tcorrespondtothe OSp ( N j 32)above,andhaven'tbeenshown toexist[5.34].Thelight-coneformoftheidentity(5.4.9) ((7.3.17))isequivalentto thedivisionalgebraidentity j xy j = j x jj y j ( j x j 2 = xx = x 0 2 + P x i 2 ),whereboth thevectorandspinorindicesonthelight-cone r -matricescorrespondtotheindexfor ( z 0 ;z i )(allrangingover D 2values)[5.35].

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110 5.PARTICLE Asimilarrst-quantizationanalysiswillbemadeforthesu perstringinsect.7.3. 5.5.SuperBRST Insteadofusingthecovariantquantizationwhichwouldfol lowdirectlyfromthe constraintanalysisof(5.4.12),wewillderiveheretheBRS Talgebrawhichfollows fromthelight-conebythemethodofsect.3.6,whichtreatsb osonsandfermionssymmetrically[3.16].Webeginwithany(reducible)light-con ePoincarerepresentation whichisalsoasupersymmetryrepresentation,andextendal sothelight-conesupersymmetrygeneratorsto4+4extradimensions.Theresulting OSp(D+1,3 j 4)spinor doesnotcommutewiththeBRSTOSp(1,1 j 2)generators,andthusmixesphysicaland unphysicalstates.Fortunately,thisextendedsupersymme tryoperator q caneasily beprojecteddowntoitsOSp(1,1 j 2)singletpiece q 0 .Webeginwiththefactthatthe light-conesupersymmetrygeneratorisatensoroperatorin aspinorrepresentationof theLorentzgroup: [ J ab ;q ]= 1 2 r ab q; (5 : 5 : 1) where r ab = 1 2 r [ a r b ] .(All r 'sarenowDirac r -matrices,notthegeneralized r 'sof (5.4.1,2).)Asaresult,itsextensionto4+4extradimensio nstransformswithrespect totheU(1)-typeOSp(1,1 j 2)as [ J AB ;q g = 1 2 ( r AB + r A 0 B 0 ) q: (5 : 5 : 2) Itwillbeusefultocombine r A and r A 0 intocreationandannihilationoperatorsasin (4.5.9): r A = a A + a y A ;r A 0 = i ( a A a y A ); f a A ;a y B ]= AB (5 : 5 : 3) 1 2 ( r AB + r A 0 B 0 )= a y [ A a B ) : (5 : 5 : 4) ( a y A a B ,withoutsymmetrization,arearepresentationof U (1 ; 1 j 1 ; 1).)Wechoose boundaryconditionssuchthatall\states"canbecreatedby thecreationoperators a y froma\vacuum"annihilatedbytheannihilationoperators a .(Thischoice,eliminatingstatesobtainedfromasecondvacuumannihilatedby a y ,isatypeofWeyl projection.)Thisvacuumisafermionicspinor(actedonby r a )whosestatisticsare changedby a y (butnotby a y ).If q isarealspinor,wecanpreservethisrealityby choosingarepresentationwhere r A isrealand r A 0 isimaginary.(Inthesameway,for theordinaryharmonicoscillatorthegroundstatecanbecho sentobearealfunction

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5.5.SuperBRST 111 of x ,andthecreationoperator x @=@x preservesthereality.)Thecorresponding charge-conjugationmatrixis C = ir 5 0 ,where r 5 0 = 1 2 [ r + 0 ;r 0 ] e 1 2 f r c 0 ;r ~ c 0 g ; (5 : 5 : 5) with r 0 =( r c 0 ;r ~ c 0 ).( e ir 5 0 = 4 convertstotherepresentationwhereboth r A and r A 0 arereal.) WenowprojecttotheOSp(1,1 j 2)singlet [ J AB ;q 0 g =0 q 0 = ( a y A a A ) q; (5 : 5 : 6) wheretheKronecker projectsdowntogroundstateswithrespecttothesecreatio n operators.Itsatises ( a y a ) a y = a ( a y a )=0 : (5 : 5 : 7) Thisprojectorcanberewritteninvariousforms: ( a y A a A )= ( a y a ) a + a a y + a y = Z du 2 e iua y A a A : (5 : 5 : 8) Wenextcheckthatthissymmetryofthephysicalstatesisthe usualsupersymmetry.Westartwiththelight-conecommutationrelations f q; q g =2 P =p; (5 : 5 : 9) where P isaWeylprojector,whennecessary,and,asusual, q = q y ,with the hermitianspinormetricsatisfying r y = r .( 'sexplicitformwillchangeupon addingdimensionsbecauseofthechangeinsignatureoftheL orentzmetric.)We thennd f q 0 ; q 0 g = ( a y a )2 P =p ( a y a )=2 P ( a y a ) r a p a ; (5 : 5 : 10) wherethe r A and r A 0 termshavebeenkilledbythe ( a y a )'sontheleftandright. Thefactorsotherthan2 r a p a projecttothephysicalsubspace(i.e.,restricttherange oftheextendedspinorindextothatofanordinaryLorentzsp inor).Theanalogous constructionfortheGL(1)-typeOSp(1,1 j 2)fails,sinceinthatcasethecorresponding projector ( BA r A r B 0 )= ( BA r A 0 r B ) y (whereas ( a y a )ishermitian),andthe2 's thenkillalltermsin =p except BA r A 0 p B Asaspecialcase,weconsiderarbitrarymasslessrepresent ationsofsupersymmetry.Thelight-conerepresentationofthesupersymmetryge neratorsis(cf.(5.4.20a)) q = q + 1 2 p + r + r i p i q + ; (5 : 5 : 11 a )

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112 5.PARTICLE where q + isaself-conjugatelight-conespinor: r q + =0 ; f q + ; q + g =2 P r p + : (5 : 5 : 11 b ) Thus, q + hasonlyhalfasmanynonvanishingcomponentsasaLorentzsp inor,and onlyhalfofthoseareindependent,theotherhalfbeingthei rconjugates.ThePoincare algebraisthenspeciedby M ij = 1 16 p + qr + r ij q + M ij ; (5 : 5 : 12) where M isanirreduciblerepresentationofSO(D-2),thesuperspin ,specifyingthe spinoftheCliordvacuumof q + .(Cf.(5.4.21,22).Wehavenormalizedthe qq term forMajorana q .) Afteradding4+4dimensions, q + canbeLorentz-covariantlyfurtherdividedusing r 0 : q + = p p + @ @ +2 r ; r @ @ = r + = r + 0 @ @ = r 0 =0 ; ( @ @ ; # = P ( 1 2 r r + )( 1 2 r + 0 r 0 ) : (5 : 5 : 13) Aftersubstitutionof(5.5.11,12)into(5.5.6),wend q 0 = ( a y a ) p p + @ @ + 2 p p + ( r a p a + r p ) # : (5 : 5 : 14) From(5.5.12)weobtainthecorrespondingspinoperators(f orMajorana ) M ab = 1 2 r ab @ @ + M ab ;M + M 0 0 = a y ( a ) @ @ + M + M 0 0 ; M 0 + 0 = 1 2 @ @ + M 0 + 0 ;M + 0 0 = 1 2 r r + 0 r 0 + M + 0 0 ; M a = 1 2 r r a @ @ + M a ;M 0 0 = 1 16 @ @ r + r 0 r 0 @ @ + M 0 0 : (5 : 5 : 15) Finally,weperformtheunitarytransformations(3.6.13)t ond q 0 = ( a y a ) @ @ +2 r a p a : (5 : 5 : 16) The nowprojectsoutjusttheOSp(1,1 j 2)-singletpartof (i.e.,theusualLorentz spinor): q 0 = @ @ 0 + r a p a 0 ;

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5.5.SuperBRST 113 @ @ 0 = ( a y a ) @ @ ; 0 =2 ( a y a ) ; ( @ @ 0 ; 0 ) = P ( a y a ) : (5 : 5 : 17) (5.5.15)canbesubstitutedinto(3.6.14).WethenndtheOS p(1,1 j 2)generators J + = ix p + ;J + = ix p + ;J = ix ( p ) + c M ; J = ix p + 1 p + h ix 1 2 ( p a 2 + p p )+ c M p + b Q i ;(5 : 5 : 18 a ) c M = a y ( a ) @ @ + M + M 0 0 ; b Q = i 1 8 @ @ + ir r + 0 r a p a r + r 0 a y @ @ ir r + 0 r a p a + M 0 0 + M a p a + 1 2 M + 0 0 p a 2 ; (5 : 5 : 18 b ) and J ab = ix [ a p b ] + 1 2 r ab @=@ + M ab fortheLorentzgenerators.Finally,wecan removealldependenceon r and r 0 byextractingthecorresponding r 0 factors contributingtothespinormetric : r 0 = i 1 2 ( r + r )( r + 0 r 0 )= r 0 y ; ( a y a ) r 0 = r 0 ( a y a )= ( a y a ) ; @ @ r 0 @ @ = i 1 2 r r + 0 @ @ ; r 0 = i 1 2 r + r 0 ; ( r + ;r 0 ) @ @ =( r + ;r 0 ) = ( r ;r + 0 )= @ @ ( r ;r + 0 )=0 ; ( @ @ ; # = P ( 1 2 r + r )( 1 2 r 0 r + 0 );(5 : 5 : 19 a ) andthenconverttotheharmonicoscillatorbasiswithrespe cttothese r 's: @ @ e a y + a y @ @ ; 1 2 e a y + a y ; 1 2 e a + a ; @ @ @ @ e a + a ; a @ @ = a = a y = @ @ a y =0; ( @ @ ; # = P ( a + a a y + a y ) : (5 : 5 : 19 b ) All a 'sand a y 'scanthenbeeliminated,andthecorrespondingprojection operators (thefactormultiplying P in(5.5.19b))dropped.Theonlypartof(5.5.18)(orthe Lorentzgenerators)whichgetsmodiedis b Q = 1 2 q a y d + M 0 0 + M a p a + 1 2 M + 0 0 p a 2 ;

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114 5.PARTICLE q = @ @ + r a p a ; d = @ @ r a p a ; q 0 = ( a y a ) q ;d 0 = ( a y a ) d : (5 : 5 : 20) Ifweexpandthersttermin b Q levelbylevelin a y 's,wenda q ateachlevel multiplyinga d ofthepreviouslevel.Inparticular,therst-levelghost q 1 multiplies thephysical d 0 .Thismeansthat d 0 =0iseectivelyimposedforonlyhalfofits spinorcomponents,sincethecomponentsof q arenotallindependent. AninterestingcharacteristicofthistypeofBRST(aswella smoreconventional BRSTobtainedbyrst-quantization)isthatspinorsobtain innitetowersofghosts. Infact,thisisnecessarytoallowthemostgeneralpossible gauges.Thesimplest explicitexampleisBRSTquantizationoftheactionofsect. 4.5fortheDiracspinor quantizedinagaugewherethegauge-xedkineticoperators areall p 2 insteadof =p .However,theseghostsarenotallnecessaryforthegaugein varianttheory,orfor certaintypesofgauges.Forexample,forthetypeofgaugein variantactionsfor spinorsdescribedinsect.4.5,theonlypartsoftheinnite -dimensionalOSp(D-1,1 j 2) spinorswhicharenotpuregaugearetheusualLorentzspinor s.(E.g.,theOSp(D1,1 j 2)DiracspinorreducestoanordinarySO(D-1,1)Diracspino r.)Forgauge-xed, 4DN=1supersymmetrictheories,supergraphsusechainsofg hostsupereldswhich alwaysterminatewithchiralsuperelds.Chiralsupereld scanbeirreducibleo-shell representationsofsupersymmetrysincetheyeectivelyde pendononlyhalfofthe componentsof .(Ananalogalsoexistsin6D,withorwithouttheuseofharmo nic superspacecoordinates[4.12].)However,nochiraldivisi onof existsin10D( isa realrepresentationofSO(9,1)),soaninnitetowerofghos tsupereldsisnecessary forcovariantbackground-eldgauges.(Forcovariantnonbackground-eldgauges, allbuttheusualniteFaddeev-Popovghostsdecouple.)Thu s,theinnitetoweris notjustapropertyofthetypeofrst-quantizationused,bu tisaninherentproperty ofthesecond-quantizedtheory.However,eveninbackgroun d-eldgaugestheinnite tower(exceptfortheFaddeev-Popovs)contributeonlyaton elooptotheeective action,sotheirevaluationisstraightforward,andtheonl yexpectedproblemwould betheirsummation. Thebasicreasonforthetowerof 'sisthefactthatonly1/4(or,inthemassive case,1/2)ofthemappearinthegauge-invarianttheoryon-s hell,butif isanirreducibleLorentzrepresentationit'simpossibletocancel3 /4(or1/2)ofitcovariantly. Wethuseectivelyobtainthesums 1 1+1 1+ = 1 2 ; (5 : 5 : 21 a )

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5.5.SuperBRST 115 1 2+3 4+ = 1 4 : (5 : 5 : 21 b ) (Thelatterseriesisthe\square"oftheformer.)Thepositi vecontributionsrepresentthephysicalspinor(or )andfermionicghostsatevenlevels,thenegative contributionsrepresentbosonicghostsatoddlevels(cont ributinginloopswiththe oppositesign),andthe 1 2 or 1 4 representsthedesiredcontribution(asobtaineddirectly inlight-conegauges).Addingconsecutivetermsinthesumg ivesanonconvergent (butnondivergentincase(5.5.21a))resultwhichoscillat esaboutthedesiredresult. However,thereareunambiguouswaystoregularizethesesum s.Forexample,if werepresentthelevelsintermsofharmonicoscillators(on ecreationoperatorfor (5.5.21a),andthe2 a y 'sfor(5.5.21b)),thesesumscanberepresentedasintegral s overcoherentstates(see(9.1.12)).For(5.5.21a),wehave : str (1)= tr h ( 1) N i = Z d 2 z e j z j 2 D z ( 1) a y a z E = Z d 2 z e j z j 2 h z j z i = Z d 2 z e 2 j z j 2 = 1 2 Z d 2 z e j z j 2 = 1 2 ; (5 : 5 : 22) where str isthesupertrace;for(5.5.21b),thesupertraceoverthedi rectproduct correspondingto2setsofoscillatorsfactorsintothesqua reof(5.5.22).(Thecorrespondingpartitionfunctionis str ( x N )=1 = (1+ x )=1 x + x 2 ,andfor2sets ofoscillators1 = (1+ x ) 2 =1 2 x +3 x 2 .) Aninterestingconsequenceof(5.5.21)isthepreservation oftheidentity D 0 =2 ( D k ) = 2 ; D 0 = str S (1) ;D = str V (1);(5 : 5 : 23) uponadding(2 ; 2 j 4)dimensions,where D 0 and D arethe\superdimensions"ofa spinorandvector,denedintermsofsupertracesoftheiden tityforthatrepresentation,and k isanintegerwhichdependsonwhetherthedimensioniseveno roddand whetherthespinorisWeyland/orMajorana(seesect.5.3). k isunchangedbyadding (2 ; 2 j 4)dimensions, D changesbyadditionof4 4=0,and,becauseof(5.5.21), D 0 changesbyafactorof2 2 ( 1 2 ) 2 =1.Thisidentityisimportantforsuper-Yang-Mills andsuperstrings. Beforeconsideringtheactionforarbitrarysupersymmetri ctheories,we'llrst studytheequationsofmotion,sincethenaivekineticopera torsmayrequireextra factorstowriteasuitablelagrangian.WithintheOSp(1,1 j 2)formalism,thegaugexedeldequationsare(cf.(4.4.19)) ( p 2 + M 2 ) =0(5 : 5 : 24 a )

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116 5.PARTICLE whensubjecttothe(Landau)gaugeconditions[2.3] b Q = c M =0 : (5 : 5 : 24 b ) Applyingthesegaugeconditionstothegaugetransformatio ns,wendtheresidual gaugeinvariance = i 1 2 b Q ; h 3 2 ( p 2 + M 2 )+ b Q 2 i = c M r + C r ( ) =0 : (5 : 5 : 24 c ) (IntheIGL(1)formalism,sect.4.2,thecorrespondingequa tionsinvolvejustthe b Q + componentof Q andthe c M + and c M 3 componentsof c M ,butareequivalent,since c M + = c M 3 =0 c M =0,and b Q + = c M =0 b Q =0.) Forsimplicity,weconsiderthemasslesscase,and M ij =0.Wecanthenchoose thereferenceframewhere p a = a + p + ,andsolvetheseequationsinlight-conenotation.(The+'sand 'snowrefertotheusualLorentzcomponents;theunphysical x p + ,and r 0 havealreadybeeneliminated.)Thegaugeconditions(5.5.2 4b)eliminate auxiliarydegreesoffreedom(as @ A = p + A =0eliminates A inYang-Mills), and(5.5.24c)eliminatesremaininggaugedegreesoffreedo m(as A + inlight-conegaugeYang-Mills).Wedividethespinors d q @=@ ,and intohalvesusing r ,and thenfurtherdividethoseintocomplexconjugatehalvesasc reationandannihilation operators,asin(5.4.27,29): d r + d ;r d d a ; d a ;@ a ; @ a ; q r + q ;r q q a ; q a ;@ a ; @ a ; (5 : 5 : 25) wherethe\ "partsof d and q arebothjustpartialderivativesbecausethemomentumdependencedropsoutinthisframe,and d; d and q; q havegradedharmonic oscillatorcommutators(uptofactorsof p + ).(5.5.20)thenbecomes b Q q a a y @ a + q a a y @ a + @ a a y d a + @ a a y d a : (5 : 5 : 26) Since b Q consistsoftermsoftheform AB ,either A or B canbechosenastheconstraintin(5.5.24b),andtheotherwillgenerategaugetran sformationsin(5.5.24c). Wecanthuschooseeither @ a ,or d a and q a ,andsimilarlyforthecomplexconjugates, exceptfortheSp(2)singlets,wherethechoiceisbetween @ a andjust d a (andsimilarlyforthecomplexconjugates).However,choosingboth d and d (orboth q and q ) forconstraintscausestheeldtovanish,andchoosingthem bothforgaugegeneratorsallowstheeldtobecompletelygaugedaway.Asaresult ,theonlyconsistent constraintsandgaugetransformationsare d a =( a q a ) = @ a =0 ; = @ a a ; (5 : 5 : 27 a )

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5.5.SuperBRST 117 subjecttotherestrictionthattheresidualgaugetransfor mationspreservethegauge choice(explicitly,(5.5.24c),althoughit'smoreconveni enttore-solvefortheresidual invarianceinlight-conenotation).(Thereisalsoacomple xconjugatetermin ifit satisesarealitycondition.Foreachvalueoftheindex a ,thechoiceofwhichoscillator iscreationandwhichisannihilationisarbitrary,andcorr espondingcomponentsof d and d or q and q canbeswitchedbychanginggauges.)Choosingthegauge a =0 ; (5 : 5 : 27 b ) fortheresidualgaugetransformationgeneratedby @ a = @=@ a ,theeldbecomes ( + ; + ; ; )= ( a + ) ( ) ( 0 + ; 0 + ) ; d 0 =0 : (5 : 5 : 28) istheusualchirallight-conesupereld(asinsect.5.4),a functionofonly1/4of theusualLorentzspinor 0 .Thisagreeswiththegeneralresultofequivalencetothe lightconeforU(1)-type4+4-extendedBRSTgivenattheendo fsect.4.5. Sincethephysicalstatesagainappearinthemiddleofthe expansion(including ghost 's),wecanagainuse(4.1.1)astheaction:Inthelight-cone gauge,from (5.5.28),integratingoverthe -functions, S = Z dxd 0 + d 0 + 2 '; (5 : 5 : 29) whichisthestandardlight-conesuperspaceaction[5.25]. Asusualfortheexpansion ofsupereldsintolight-conesuperelds,thephysicallig ht-conesupereldappearsin themiddleofthenon-light-coneexpansionofthegaugesupereld,withauxiliary light-conesupereldsappearingathigherordersandpureg augeonesatlowerorders. Becausesomeoftheghost 'sarecommuting,wethereforeexpectaninnitenumberofauxiliaryeldsinthegauge-covariantaction,asint heharmonicsuperspace formalism[4.12].Thismaybenecessaryingeneral,because thistreatmentincludes self-dualmultiplets,suchas10Dsuper-Yang-Mills.(This multipletissuperspin0, andthusdoesnotrequirethesuperspin M ij tobeself-dual,soitcanbetreatedinthe OSp(1,1 j 2)formalism.However,anadditionalself-conjugacycondi tiononthelightconesupereldisrequired,(5.4.31b),andacovariantOSp( 1,1 j 2)statementofthis conditionwouldbenecessary.)However,insomecases(such as4DN=1supersymmetry)itshouldbepossibletotruncateoutallbutanitenu mberoftheseauxiliary superelds.Thiswouldrequirean(innite)extensionofth egroupOSp(1,1 j 2)(perhapsinvolvingpartoftheunphysicalsupersymmetries a q ),inthesamewaythat extendingIGL(1)toOSp(1,1 j 2)eliminatesNakanishi-Lautrupauxiliaryelds.

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118 5.PARTICLE Theunusualformofthe OSp (1 ; 1 j 2)operatorsforsupersymmetricparticlesmay requirenewmechanicsactionsforthem.Itmaybepossibleto derivetheseactions byinvertingthequantizationprocedure,rstusingtheBRS Talgebratoderivethe hamiltonianandthenndingthegauge-invariantclassical mechanicslagrangian. Exercises (1)Derive(5.1.2)from(5.1.1)and(3.1.11).(2)Showthat,undertheusualgaugetransformation A A + @ exp [ iq R f i d x A ( x )]transformswithafactor exp f iq [ ( x ( f )) ( x ( i ))] g .(InaFeynman pathintegral,thiscorrespondstoagaugetransformationo ftheendsofthepropagator.) (3)Fouriertransform(5.1.13),using(5.1.14).Explicitl yevaluatetheproper-time integralinthemasslesscasetondthecoordinate-spaceGr eenfunctionsatisfying 2 G ( x;x 0 )= D ( x x 0 )forarbitrary D> 2.Do D =2bydierentiatingwith respectto x 2 ,thendoingtheproper-timeintegral,andnallyintegrati ngback withrespectto x 2 .(Thereisaninniteconstantofintegrationwhichmustbe renormalized.)Forcomparison,do D =2bytakingthelimitfrom D> 2. (4)Usethemethoddescribedin(5.1.13,14)toevaluatethe1 -looppropagatorcorrectionin 3 theory.Comparethecorrespondingcalculationwiththecov ariantized light-conemethodofsect.2.6.(Seeexercise(7)ofthatcha pter.) (5)Derive(5.2.1),andnd J 3 .Derive(5.2.2)andtherestoftheOSp(1,1 j 2)algebra. Showtheseresultsagreewiththoseofsect.3.4. (6)Quantize(5.3.10)inthe3supersymmetricgaugesdescri bedinthatsection,and ndthecorrespondingIGL(1)(andOSp(1,1 j 2),whenpossible)algebrasineach case,usingthemethodsofsects.3.2-3.Notethatthemethod sofsect.3.3require somegeneralization,sincecommutingantighostscanbecon jugatetothecorrespondingghostsandstillpreserveSp(2):[ C ;C ] C .Showequivalenceto theappropriatealgebrasofsects.3.4-5. (7)Derivethetablesinsect.5.3.(Reviewthegrouptheoryo fSO(N)spinors,if necessary.)Usethetablestoderivethegroups,equivalent toSO( D + D )for D 6,forwhichthesespinorsarethedeningrepresentation. (8)Expressthereal-spinor r -matricesofsect.5.3intermsofthe -matricestherefor arbitrary D + and D .UsetheMajoranarepresentationwherethespinorisnot

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Exercises 119 necessarilyexplicitlyreal,butequivalenttoarealone,s uchthat:(1)forcomplex representations,thebottomhalfofthespinoristhecomple xconjugateofthetop half(eachbeingirreducible);(2)forpseudorealrepresen tations,thebottomhalf isthecomplexconjugateagainbutwiththeindexconvertedw ithametricto makeitexplicitlythesamerepresentationasthetop;(3)fo rrealrepresentations, thespinorisjustthereal,irreducibleone.Findthematric esrepresentingthe internalsymmetry(U(1)forcomplexandSU(2)forpseudorea l). (9)ChecktheJacobiidentitiesforthecovariantderivativ eswhosealgebraisgiven in(5.4.8).Checkclosureofthealgebra(5.4.10)in D =10,includingtheextra generatordescribedinthetext. (10)Derive(5.4.13).(11)Writetheexplicitactionandtransformationlawsfor( 5.4.16). (12)Writetheexplicitequationsofmotion(5.4.12),modi edby(5.4.23),forascalar supereldforN=1supersymmetryinD=4.Showthatthisgives theusualcovariantconstraintsandeldequations(uptoconstantsofi ntegration)forthe chiralscalarsupereld(scalarmultiplet).Dothesamefor aspinorsupereld, andobtaintheequationsforthevector-multipleteldstre ngth. (13)Derivetheexplicitformofthetwistoreldsfor D =3,4,6.FindanexplicitexpressionforthesupersymmetrizedPauli-Lubanskyvectori n D =4intermsof supertwistors,andshowthatitautomaticallygivesanexpl icitexpressionforthe superhelicity H of(5.4.23)asanoperatorinsupertwistorspace.Showthesu pertwistorPauli-Lubanskyvectorautomaticallyvanishes in D =3,andderivean expressionin D =6.

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120 6.CLASSICALMECHANICS 6.CLASSICALMECHANICS6.1.Gaugecovariant Inthischapterwe'llconsiderthemechanicsactionforthes tringanditsgauge xing,asadirectgeneralizationofthetreatmentofthepar ticleintheprevious chapter. Therst-orderactionforstringmechanicsisobtainedbyge neralizingthe1dimensionalparticlemechanicsworld-lineof(5.1.1)toa2 -dimensionalworldsheet [6.1]: S = 1 0 Z d 2 2 h ( @ m X ) P m + g mn 1 2 P m P n i ; (6 : 1 : 1) where X ( m )isthepositioninthehigher-dimensionalspaceinwhichth eworld sheetisimbeddedofthepointwhoselocationintheworldshe etitselfisgivenby m =( 0 ; 1 )=( ; ), d 2 = d 0 d 1 = dd @ m =( @ 0 ;@ 1 )=( @=@;@=@ ),and g mn =( g ) 1 = 2 g mn istheunit-determinantpartofthe2Dmetric.(Actually,it has determinant 1.)1 = 2 0 isboththestringtensionandtherest-massperunitlength. (Theirratio,thesquareofthevelocityofwavepropagation inthestring,isunityin unitsofthespeedoflight:Thestringisrelativistic.)Thi sactionisinvariantunder 2Dgeneralcoordinatetransformations(generalizing(5.1 .4);seesect.4.1): X = m @ m X; P m = @ n ( n P m ) P n @ n m ; g mn = @ p ( p g mn ) g p ( m @ p n ) : (6 : 1 : 2) Otherformsofthisactionwhichresultfromeliminatingvar iouscombinationsof theauxiliaryelds P m and g mn are S = 1 0 Z d 2 2 X P 0 1 g 11 1 2 ( P 0 2 + X 0 2 ) g 01 g 11 P 0 X 0 # (6 : 1 : 3 a ) = 1 0 Z d 2 2 h X P 0 + 1 4 ( P 0 + X 0 ) 2 1 4 ( P 0 X 0 ) 2 i (6 : 1 : 3 b )

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6.1.Gaugecovariant 121 = 1 0 Z d 2 2 g mn 1 2 ( @ m X ) ( @ n X )(6 : 1 : 3 c ) = 1 0 Z d 2 2 h ( @ m X ) P m + p detP m P n i (6 : 1 : 3 d ) = 1 0 Z d 2 2 q det ( @ m X ) ( @ n X )= 1 0 Z q ( dX a ^ dX b ) 2 ; (6 : 1 : 3 e ) where 1 0 P 0 isthemomentum( -)density(themomentum p = 1 0 R d 2 P 0 ), 0 = @=@ andtoobtain(6.1.3d)wehaveusedthedeterminantofthe g equationsofmotion P m P n 1 2 g mn g pq P p P q =0 : (6 : 1 : 4) Asaconsequenceofthisequationandtheequationofmotionf or P ,the2Dmetricis proportionaltothe\induced"metric @ m X @ n X (asappearsin(6.1.3e)),whichresults frommeasuringdistancesintheusualMinkowskiwayintheDdimensionalspacein whichthe2Dsurfaceisimbedded(using dX = d m @ m X ).(Theequationsofmotion don'tdeterminetheproportionalityfactor,sinceonlythe unit-determinantpartof themetricappearsintheaction.)Inanalogytotheparticle ,(6.1.4)alsorepresents thegeneratorsof2Dgeneral-coordinatetransformations. (6.1.3a)isthehamiltonian form,(6.1.3b)isarewritingofthehamiltonianformtorese mbletheexample(3.1.14) (butwiththe indenite-metricsum ofsquaresofbothleftand right-handedmodes constrained),(6.1.3c)isthesecond-orderform,and(6.1. 3e)istheareasweptoutby theworldsheet[6.2].Ifthetheoryisderivedfromtheform( 6.1.3b),thereare2sets oftransformationlawsoftheform(3.1.15)(withappropria tesigndierences),and (6.1.2,3c)canthenbeobtainedas(3.1.16,17). Fortheopenstring,the X equationsofmotionalsoimplycertainboundary conditionsin .(Bydenition,theclosedstringhasnoboundaryin .)Varyingthe ( @X ) P termandintegratingbypartstopulloutthe X factor,besidestheequation ofmotionterm @P wealsogetasurfaceterm n m P m ,where n m isthenormaltothe boundary.Ifweassume2Dcoordinatessuchthattheposition ofthe boundaries areconstantin ,thenwehavetheboundarycondition P 1 =0 : (6 : 1 : 5) It'sconvenienttodenethequantities b P ( ) = 1 p 2 0 ( P 0 X 0 )(6 : 1 : 6 a )

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122 6.CLASSICALMECHANICS becausethehamiltonianconstraintsappearingin(6.1.3b) (equivalentto(6.1.4))can beexpressedverysimplyintermsofthemas b P ( )2 =0 ; (6 : 1 : 6 b ) andbecausetheyhavesimplePoissonbracketswitheachothe r.Fortheopenstring, it'sfurtherusefultoextend :Ifwechoosecoordinatessuchthat =0forall at oneendofthestring,andsuchthat(6.1.5)implies X 0 =0atthatend,thenwecan dene X ( )= X ( ) ; b P ( )= 1 p 2 0 ( P 0 + X 0 )= b P ( ) ( ) for > 0 ; (6 : 1 : 7 a ) sotheconstraint(6.1.6b)simpliesto b P 2 =0 : (6 : 1 : 7 b ) 6.2.Conformalgauge Theconformalgaugeisgivenbythegaugeconditions(on(6.1 .1,3ac)) g mn = mn ; (6 : 2 : 1) where isthe2Drat(Minkowski)spacemetric.(Since g isunit-determinant,ithas only2independentcomponents,sothe2gaugeparametersof( 6.1.2)aresucient todetermineitcompletely.)Asfortheparticle,thisgauge can'tbeobtainedeverywhere,soit'simposedeverywhereexcepttheboundaryin .Thenvariationof g at initialornal implies(6.1.4)there,andtheremainingeldequationsthe nimplyit everywhere.Inthisgaugethoseequationsare P m = @ m X;@ m P m =0 @ 2 X =0 : (6 : 2 : 2) Itisnoweasytoseethattheendpointsofthestringtravelat thespeedoflight.(With slightgeneralization,thiscanbeshowninarbitrarygauge s.)From(6.1.4,5)and (6.2.1,2),wendthat dX = d X ,andthus dX dX = d 2 X 2 = d 2 ( X 2 + X 0 2 )=0. (6.2.2)ismosteasilysolvedbytheuseof2Dlight-conecoor dinates = 1 p 2 ( 1 0 ) mn = 0110 = mn ; (6 : 2 : 3) where2Dindicesnowtakethevalues .Wethenhave @ + @ X =0 X = 1 2 [ ^ X (+) ( + )+ ^ X ( ) ( )] : (6 : 2 : 4 a )

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6.2.Conformalgauge 123 Fortheopenstring,theboundaryconditionatoneboundary, chosentobe =0,is ( @ + + @ ) X =0 ^ X (+) ( )= ^ X ( ) ( ) ^ X (+) ( )= ^ X ( ) ( ) ; (6 : 2 : 4 b ) withoutlossofgenerality,sincetheconstantpartsof ^ X ( ) appearin X onlyastheir sum.Thus,themodesoftheopenstringcorrespondtothemode sofonehandedness oftheclosedstring.Theboundaryconditionattheotherbou ndaryoftheopen string,takenas = ,andthe\boundary"conditionoftheclosedstring,whichis simplythatthe\ends"at = arethesamepoint(andthustheclosedstring X isperiodicin withperiod2 ,orequivalently X and X 0 havethesamevaluesat = asat = )bothtaketheform ^ X ( ) ( +2 )= ^ X ( ) ( ) ^ X ( ) ( +2 )= ^ X ( ) ( )+4 0 p ( ) ; (6 : 2 : 4 c ) p (+) = p ( ) : (6 : 2 : 4 d ) Theconstraints(6.1.4)alsosimplify: P 2 ( ; )= 1 2 ^ X ( ) 0 2 ( )=0 : (6 : 2 : 5) TheseconstraintswillbeusedtobuildtheBRSTalgebrainch apt.8. Thefactthatthemodesoftheopenstringcorrespondtohalft hemodesof theclosedstring(exceptthatbothhave1zero-mode)meanst hattheopenstring canbeformulatedasaclosedstringwithmodesofonehandedn ess(clockwiseor counterclockwise).Thisisaccomplishedbyaddingtotheac tion(6.1.1)fortheclosed stringtheterm S 1 = Z d 2 2 1 2 u a u b mn 1 2 ( g mp mp ) 1 2 ( g nq nq )( @ p X a )( @ q X b ) ; (6 : 2 : 6) where u isaconstant,timelikeorlightlike(butnotspacelike)vec tor( u 2 0),and + = 1. isalagrangemultiplierwhichconstrains( u @ X ) 2 =0,andthus u @ X =0,inthegauge(6.2.1).Togetherwith(6.2.5),thisimplie stheLorentz covariantconstraint @ X a =0,so X dependsonlyon ,asin(3.1.14-17).Thus, theformulationusing(6.2.6)isLorentzcovarianteventho ugh S 1 isnotmanifestly so(becauseoftheconstantvector u ).Wecanthenidentifythenew X withthe ^ X of (6.2.4).Since itselfappearsmultipliedby @ X intheequationsofmotion,itthus dropsout,implyingthatit'sagaugedegreeoffreedomwhich ,like g ,canbegauged awayexceptatinnity.

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124 6.CLASSICALMECHANICS Similarmethodscanbeappliedtotheone-handedmodesofthe heteroticstring [1.13].(Thenin(6.2.6)onlyspacelike X 'sappear,soinsteadof u a u b anypositivedenitemetriccanbeused,eectivelysummingovertheonehanded X 's.)Various propertiesoftheactions(3.1.17,6.2.6)havebeendiscuss edintheliterature[6.3],particularlyinrelationtoanomaliesinthegaugesymmetryoft helagrangemultipliers uponnaivelagrangianquantization.Onesimplewaytoavoid theseanomalieswhile keepingamanifestlycovariant2Dlagrangianistoaddscala rs withthesquaresof both @ and @ + appearinginlagrange-multiplierterms[6.4].Alternativ ely(oradditionally),onecanaddWeyl-Majorana2Dspinors(i.e.,re al,1-component,1-handed spinors)whosenonvanishingenergy-momentumtensorcompo nentcouplestotheappropriatelagrangemultiplier.(E.g.,aspinorwithkineti cterm @ + appearsalso intheterm [( @ ) 2 + @ ].)Thesenonpropagatingeldsappeartogetherwith scalarswithonlyoneortheotherhandednessorneithercons trained,andunconstrainedfermionswhichareWeyland/orMajoranaorneither .Thereare(atmost) 2lagrangemultipliers,oneforeachhandedness. Intheconformalgaugethereisstillaresidueofthegaugein variance,whichoriginallyincludednotonly2Dgeneralcoordinatetransformati onsbutalsolocalrescalings ofthe2Dmetric(sinceonlyitsunit-determinantpartappea redintheaction).By denition,thesubsetofthesetransformationswhichleave (6.2.1)invariantistheconformalgroup.Unlikeinhigherdimensions,the2Dconformal grouphasaninnite numberofgenerators.Itcaneasilybeshownthatthesetrans formationsconsistof thecoordinatetransformations(restrictedbyappropriat eboundaryconditions) 0 = ( ) ; (6 : 2 : 7 a ) with 'snotmixing(correspondingto21Dgeneralcoordinatetran sformations),since thesecoordinatetransformationshaveaneectonthemetri cwhichcanbecanceled byalocalscaling: d 2 0 =2 d + 0 d 0 = + 0 0 2 d + d : (6 : 2 : 7 b ) Onshell,thesetransformationsaresucienttogaugeawayo neLorentzcomponent of X ,anotherbeingkilledbytheconstraint(6.2.5).These2Lor entzcomponentscan beeliminatedmoredirectlybyoriginallychoosingstronge rgaugeconditions,asin thelight-conegauge. Theconformalgaugeisatemporalgauge,sinceitisequivale nttosettingthe timecomponentsofthegaugeeldtoconstants: g m0 = m0 .Whengeneralizedto D> 2,itisthechoiceofGaussiannormalcoordinates.Wecanins teadchoosea

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6.3.Lightcone 125 Lorentzgauge, @ m g mn =0.ThisistheDeDondergauge,orharmoniccoordinates, whichisstandardlyusedin D> 2.We'lldiscussthisgaugeinmoredetailinsect. 8.3.6.3.Lightcone Inalight-coneformalism[6.5]notonlyaremoregaugedegre esoffreedomeliminatedthanincovariantgauges,butalsomore(Lorentz)aux iliaryelds.Wedo thelatterrstbyvaryingtheaction(6.1.1)withrespectto alleldscarryinga\ Lorentzindex( X P m ): X @ m P m + =0;(6 : 3 : 1 a ) P m g mn =( A r B r ) 1 ( pm A p qn A q B m B n ) ; A m = P m + ;B m = @ m X + : (6 : 3 : 1 b ) Wenexteliminatealleldswitha\+"indexbygaugeconditio ns: : X + = k : P 0 + = k; (6 : 3 : 2 a ) where k isanarbitraryconstant.(Thesameprocedureisappliedinl ight-coneYangMills,where A iseliminatedasanauxiliaryeldand A + asagaugedegreeof freedom:seesect.2.1.)Thelatterconditiondetermines tobeproportionaltothe amountof+-momentumbetween =0andthepointatthatvalueof (sothestring lengthisproportionalto R dP 0 + ,whichisaconstant,since @ m P m + =0).Thus, isdetermineduptoafunctionof (correspondingtothechoiceofwhere =0). However, P 1 + isalsodetermineduptoafunctionof (sincenow @ 1 P 1 + =0),so iscompletelydetermined,uptoglobaltranslations + constant ,bythefurther condition P 1 + =0 : (6 : 3 : 2 b ) (6.3.2)implies(6.2.1).Fortheopenstring,bytheconvers eoftheargumentleading totheboundarycondition(6.1.5),thisdeterminesthevalu esof attheboundaries uptoconstants,sotheremainingglobalinvarianceisusedt ochoose =0atone boundary.Fortheclosedstring,theglobalinvariancerema ins,andiscustomarily dealtwithinthequantumtheorybyimposingaconstraintofi nvarianceunderthis transformationontheeldorrst-quantizedwavefunction

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126 6.CLASSICALMECHANICS Thelengthofthestringisthengivenbyintegrating(6.3.2a ): p + = 1 0 R d 2 P 0 + = (1 = 2 0 ) k length .Thetwomostconvenientchoicesare k =1 length =2 0 p + length = (2 ) k =2 0 p + ( 0 p + ) foropen ( closed ) : (6 : 3 : 3) Forthefreestringthelatterchoiceismoreconvenientfort hepurposeofmode expansions.Inthecaseofinteractions k mustbeconstanteventhroughinteractions, andthereforecan'tbeidentiedwiththevalueof p + ofeachstring,sotheformer choiceismade.Notethatfor p + < 0thestringthenhasnegativelength.Itisthen interpretedasan anti string(oroutgoingstring,asopposedtoincomingstring). The useofnegativelengthsisparticularlyusefulforinteract ions,sincethenthevertices are(cyclically)symmetricinallstrings:e.g.,astringof length1breakinginto2 stringsoflength 1 2 isequivalenttoastringoflength 1 2 breakingintostringsoflength 1and 1 2 Theactionnowbecomes S = Z d ( x p + + 1 0 Z d 2 h ( @ m X i ) P m i + mn 1 2 P m i P n i i ) ; (6 : 3 : 4 a ) or,inhamiltonianform, S = Z d ( x p + + 1 0 Z d 2 h X i P 0 i 1 2 ( P 0 i 2 + X 0 i 2 ) i ) : (6 : 3 : 4 b ) X i isfoundasin(6.2.4),but X + isgivenby(6.3.2),and X isgivenbyvaryingthe originalactionwithrespecttotheauxiliaryelds g mn and P m + ,conjugatetothose variedin(6.3.1): g mn P m = P m i P 0 i 1 2 0 m np P n i P p i P m + X = x Z d 2 P 1 + constant; (6 : 3 : 5) wheretheconstantischosentocanceltheintegralwheninte gratedover ,sothat 1 0 R d 2 X P 0 + = x p + in(6.3.4).(Thistermhasbeenrestoredin(6.3.4)inorder toavoidusingequationsofmotiontoeliminateanycoordina teswhoseequationsof motioninvolvetimederivatives,andarethusnotauxiliary .In(6.3.1a)allbutthe zero-modepartofthe X equationcanbeusedtosolveforallbuta -independent partof P 1 + ,withoutinvertingtimederivatives.)

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Exercises 127 Afterthecontinuous2Dsymmetrieshavebeeneliminatedbyc oordinatechoices, certaindiscretesymmetriesremain: and reversal.Asintheparticlecase, reversalcorrespondstoaformofchargeconjugation.However ,theopenstringhas agrouptheoryfactorassociatedwitheachendwhichistheco mplex-conjugaterepresentationofthatattheotherend(soforunitaryrepresen tationstheycancancel forsplittingorjoiningstrings),soforchargeconjugatio nthe2endsshouldswitch, whichrequires reversal.Furthermore,theclosedstringhasclockwiseand counterclockwisemodeswhicharedistinguishable(especially fortheheteroticstring),so againwerequire reversaltoaccompany reversaltokeep frommixing.We thereforedenechargeconjugationtobethesimultaneousr eversalof and (or 2 0 p + topreservethepositionsoftheboundariesoftheopenstrin g).On theotherhand,somestringsare nonoriented (asopposedtothe oriented onesabove) inthatsolutionswith reversedarenotdistinguished(correspondingtoopenstri ngs withrealrepresentationsforthegrouptheoryfactors,orc losedstringswithclockwisemodesnotseparatedfromcounterclockwise).Forsuchs tringswealsoneedto denea reversalwhich,becauseofitsactiononthe2Dsurface,isca lleda\twist". Inthequantumtheorytheseinvariancesareimposedasconst raintsontheeldsor rst-quantizedwavefunctions(seechapt.10).Exercises (1)Derive(6.1.3)from(6.1.1).Derive(6.1.5).(2)Derive(6.3.1).

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128 7.LIGHT-CONEQUANTUMMECHANICS 7.LIGHT-CONEQUANTUM MECHANICS 7.1.Bosonic Inthissectionwewillquantizethelight-conegaugebosoni cstringdescribedin sect.6.3andderivethePoincarealgebra,whichisaspecia lcaseofthatdescribedin sect.2.3. Thequantummechanicsofthefreebosonicstringisdescribe dinthelight-cone formalism[6.5]intermsoftheindependentcoordinates X i ( )and x ,andtheir canonicalconjugates P 0 i ( )and p + ,with\time"( x + =2 0 p + )dependencegiven bythehamiltonian(see(6.3.4b)) H = 1 0 Z d 2 1 2 ( P 0 i 2 + X i 0 2 ) : (7 : 1 : 1) Functionally, 1 0 P 0 i ( )= i X i ( ) ; X i ( 1 ) ;X j ( 2 ) # = ij 2 ( 2 1 ) : (7 : 1 : 2) (Noteourunconventionalnormalizationforthefunctional derivative.) Fortheopenstring,it'sconvenienttoextend from[0 ; ]to[ ; ]asin(6.1.7) bydening X ( )= X ( ) ; b P ( )= 1 p 2 0 0 i X + X 0 = 1 p 0 P ( ) for > 0 ; (7 : 1 : 3 a ) sothat b P isperiodic(and P + or P ,asin(6.2.5),for > or < 0), or toexpress theopenstringasaclosedstringwithmodeswhichpropagate onlyclockwise(oronly counterclockwise)intermsof ^ X ofsect.6.2: b P ( )= 1 p 2 0 ^ X 0 ( ) ; (7 : 1 : 3 b )

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7.1.Bosonic 129 whichresultsinthesame b P (sameboundaryconditionsandcommutationrelations). Thelatterinterpretationwillproveusefulforgraphcalcu lations.(However,theform oftheinteractionswillstilllookdierentfromatrueclos edstring.)Eitherway,the newboundaryconditions(periodicity)allow(Fourier)exp ansionofalloperatorsin termsofexponentialsinsteadofcosinesorsines.Furtherm ore, b P containsallof X and =X (except x ,whichisconjugateto p R d b P ;i.e.,allthetranslationally invariantpart).Inparticular, H = Z d 2 1 2 b P i 2 : (7 : 1 : 4) Thecommutationrelationsare [ b P i ( 1 ) ; b P j ( 2 )]=2 i 0 ( 2 1 ) ij : (7 : 1 : 5) Fortheclosedstring,wedene2 b P 'sby b P ( ) ( )= 1 p 2 0 0 i X ( ) X 0 ( ) # = 1 p 2 0 ^ X ( ) 0 ( ) : (7 : 1 : 6) Thenoperatorswhichareexpressedintermsofintegralsove r alsoincludesums over :e.g., H = H (+) + H ( ) ,with H ( ) givenintermsof b P ( ) by(7.1.4). Inordertocomparewithparticles,we'llneedtoexpandallo peratorsinFourier modes.Inpracticalcalculations,functionaltechniquesa reeasier,andmodeexpansionsshouldbeusedonlyasthenalstep:externallinefact orsingraphs,orexpansion oftheeectiveaction.(Similarremarksapplyingenerale ldtheoriestoexpansionof supereldsincomponentsandexpansionofeldsaboutvacuu mexpectationvalues.) For b P fortheopenstring(suppressingLorentzindices), b P ( )= 1 X n = 1 n e in ; 0 = p 2 0 p; n =( n ) y = i p na n y ; [ p;x ]= i; [ m ; n ]= n m + n; 0 ; [ a m ;a n y ]= mn : (7 : 1 : 7 a ) Wealsohave ^ X ( )= ^ X y ( )= ^ X ( )=( x +2 0 p )+ p 2 0 1 X 1 1 p n ( a n y e in + a n e in ) ; X ( )= 1 2 h ^ X ( )+ ^ X ( ) i ;P 0 ( )= 0 i X ( ) = 1 2 h ^ X 0 ( )+ ^ X 0 ( ) i : (7 : 1 : 7 b )

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130 7.LIGHT-CONEQUANTUMMECHANICS (The y and havetheusualmatrixinterpretationiftheoperatorsareco nsidered asmatricesactingontheHilbertspace:The y istheusualoperatorialhermitian conjugate,whereasthe istheusualcomplexconjugateasforfunctions,which changesthesignofmomenta.Combinedtheygivetheoperator ialtranspose,which correspondstointegrationbyparts,andthusalsochangest hesignofmomenta,which arederivatives.) H isnowdenedwithnormalordering: H = 1 2 : 1 X 1 n n :+ constant = 0 p i 2 + 1 X 1 na n y a n + constant = 0 p i 2 + N + constant: (7 : 1 : 8 a ) Inanalogytoordinaryeldtheory, i@=@ + H = 0 p 2 + N + constant Theconstantin H hasbeenintroducedasaniterenormalizationafterthein niterenormalizationdonebythenormalordering:Asinordi naryeldtheory,whereverinniterenormalizationisnecessarytoremoveinnit ies,niterenormalization shouldalsobeconsideredtoallowfortheambiguitiesinren ormalizationprescriptions.However,alsoasinordinaryeldtheories,therenor malizationisrequired torespectallsymmetriesoftheclassicaltheorypossible( otherwisethesymmetryis anomalous,i.e.,notasymmetryofthequantumtheory).Inth ecaseofanylight-cone theory,theonesymmetrywhichisnevermanifest(i.e.,anau tomaticconsequenceof thenotation)isLorentzinvariance.(Thissacricewasmad einorderthatunitarity wouldbemanifestbychoosingaghost-freegaugewithonlyph ysical,propagating degreesoffreedom.)Thus,inordertoproveLorentzinvaria nceisn'tviolated,the commutatorsoftheLorentzgenerators J ab = 1 0 R d 2 X [ a P 0 b ] mustbechecked.All aretrivialexcept[ J i ;J j ]=0because X and P 0 arequadraticin X i and P 0 i by (6.3.5).Theproofisleftasan(important)exerciseforthe readerthatthedesired resultisobtainedonlyif D =26andtherenormalizationconstantin H and P 0 ( H = 1 0 R d 2 P 0 )isgivenby H = 0 p i 2 + N 1 0 ( p i 2 + M 2 ) : (7 : 1 : 8 b ) Theconstantin H alsofollowsfromnotingthattherstexcitedlevel( a 1 i y j 0 i )is justatransversevector,whichmustbemassless.Themasssp ectrumoftheopen string,givenbytheoperator M 2 ,isthenaharmonicoscillatorspectrum:Thepossible valuesofthemasssquaredare 1,0,1,2,...times1 = 0 ,withspinsateachmass levelrunningashighas N = 0 M 2 +1.(Thehighest-spinstateiscreatedbythe symmetrictracelesspartof Na 1 y 's.)

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7.1.Bosonic 131 Grouptheoryindicesareassociatedwiththeendsoftheopen string,sothe stringactslikeamatrixinthatspace.Ifthegroupisorthog onalorsymplectic, theusual(anti)symmetryoftheadjointrepresentation(as amatrixactingonthe vectorrepresentation)isimposednotjustbyswitchingthe indices,butbyripping thewholestring(switchingtheindicesand $ ).Asaresult, N oddgives matriceswiththesymmetryoftheadjointrepresentation(b yincludinganextra\ signintheconditiontoputthemasslessvectorinthatrepre sentation),while N even givesmatriceswiththeoppositesymmetry.(Aparticularcu riosityisthe\SO(1)" openstring,whichhas only N even,andnomasslessparticles.)Suchstrings,because oftheirsymmetry,arethus\nonoriented".Ontheotherhand ,ifthegroupisjust unitary,thereisnosymmetrycondition,andthestringis\o riented".(Theendscan belabeledwitharrowspointinginoppositedirections,sin cethehermitianconjugate statecanbethoughtofasthecorresponding\antistring".) Theclosedstringistreatedanalogously.Themodeexpansio nsofthecoordinates andoperatorsaregivenintermsof2setsofhattedoperators ^ X ( ) or b P ( ) expanded over ( ) n : X ( )= 1 2 h ^ X (+) ( )+ ^ X ( ) ( ) i ;P 0 ( )= 1 2 h ^ X (+) 0 ( )+ ^ X ( ) 0 ( ) i ; ^ X ( ) ( )= ^ X ( ) y ( )= ^ X ( ) ( ) ( ) n = ( ) n y = ( ) n : (7 : 1 : 9) However,becauseoftheperiodicitycondition(6.2.4c),an dsincethezero-modes(from (7.1.7))appearonlyastheirsum,theyarenotindependent: x ( ) = x;p ( ) = 1 2 p: (7 : 1 : 10) Operatorswhichhavebeenintegratedover (asin(7.1.4))canbeexpressedassums overtwosets( )ofopen-stringoperators:e.g., H = H (+) + H ( ) = 1 2 0 p i 2 + N 2 ;N = N (+) + N ( ) : (7 : 1 : 11) Rememberingtheconstraintunderglobal translations i d d Z d 2 X 0 i X = N N (+) N ( ) =0 ; (7 : 1 : 12) thespectrumisthengivenbythedirectproductof2openstri ngs,butwiththestates constrainedsothatthe2factors(fromthe2openstrings)gi veequalcontributions tothemass.Themassessquaredarethus 4,0,4,8,...times1 = 0 ,andthehighest spinatanymasslevelis N = 1 2 0 M 2 +2(withthecorrespondingstategivenbythe

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132 7.LIGHT-CONEQUANTUMMECHANICS symmetrictracelessproductof Na 1 y 's,halfofwhicharefromoneopen-stringset andhalffromtheother).Comparedwith 0 M 2 +1fortheopenstring,thismeans thatthe\leadingReggetrajectory" N ( M 2 )(seesec.9.1)hashalftheslopeandtwice theinterceptfortheclosedstringasfortheopenstring.If weapplytheadditional constraintofsymmetryofthestateunderinterchangeofthe 2setsofstringoperators, thestateissymmetricunderinterchangeof $ ,andistherefore\nonoriented"; otherwise,thestringis\oriented",theclockwiseandcoun terclockwisemodesbeing distinguishable(sothestringcancarryanarrowtodisting uishitfromastringthat's rippedover). Fromnowonwechooseunits 0 = 1 2 : (7 : 1 : 13) Inthecaseoftheopenbosonicstring,thefreelight-conePo incaregeneratorscan beobtainedfromthecovariantexpressions(theobviousgen eralizationoftheparticle expressions,becauseof(7.1.2),since X a and P 0 a aredenedtoLorentztransformas vectorsand X totranslatebyaconstant), J ab = i Z d 2 X [ a ( ) P 0 b ] ( ) ;p a = Z d 2 P 0 a ( ) ; (7 : 1 : 14 a ) bysubstitutingthegaugecondition(6.3.2)andfreeeldeq uations(6.1.7)(using (7.1.7)) ^ X + ( )= p + ; b P 2 2=0 b P = b P i 2 2 2 p + J ij = ix [ i p j ] + X a y n [ i a nj ] ; J + i = ix i p + ;J + = ix p + ; J i = i ( x p i x i p )+ X a y n [ a ni ] ; p = 1 2 p + h p i 2 +2 X na y n i a ni 1 i ;a n = 1 p + ( p i a ni + n ) ; n = i 1 p n 1 2 n 1 X m =1 q m ( n m ) a m i a n m;i 1 X m =1 q m ( n + m ) a y m i a n + m;i : (7 : 1 : 14 b ) (Wehaveaddedthenormal-orderingconstantof(7.1.8)toth econstraint(6.1.7).) Asforarbitrarylight-conerepresentations,thePoincar egeneratorscanbeexpressed completelyintermsoftheindependentcoordinates x i and x ,thecorresponding momenta p i and p + ,themassoperator M ,andthegeneratorsofaspinSO(D 1),

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7.1.Bosonic 133 M ij and M im .Fortheopenbosonicstring,wehavetheusualoscillatorre presentation oftheSO(D 2)spingenerators M ij = i Z 0 ^ X i b P j = X n a y n [ i a nj ] ; (7 : 1 : 15 a ) where R 0 meansthezero-modesaredropped.Themassoperator M andtheremaining SO(D 1)operators M im aregivenby M 2 = 2 p + Z 0 b P = Z 0 ( b P i 2 2)=2 X na y n i a ni 1 =2( N 1) ; M im M = ip + Z 0 ^ X i b P = X a y ni n y n a ni : (7 : 1 : 15 b ) Theusuallight-coneformalismfortheclosedstringisnota truelight-coneformalism,inthesensethatnotallconstraintshavebeensolve dexplicitlybyeliminating variables:Theoneconstraintthatremainsisthatthecontr ibutiontothe\energy" p fromtheclockwisemodesisequaltothatfromthecounterclo ckwiseones.Asa result,thenaivePoincarealgebradoesnotclose[4.10]:U singtheexpressions J i = i ( x p i x i p )+ X n; a y ( ) n [ a ( ) ni ] ; p = 1 2 p + h p i 2 +4 N (+) + N ( ) 2 i ; (7 : 1 : 16 a ) wend [ J i ;J j ]= 4 p + 2 N J ij ; (7 : 1 : 16 b ) where J ij isthedierencebetweenthe(+)and( )partsof J ij Weinsteaddene2setsofopen-stringlight-conePoincare generators J ( ) ab and p ( ) a ,builtoutofindependentzero-andnonzero-modes.Theclos ed-stringPoincare generatorsarethen[4.10] J ab = J (+) ab + J ( ) ab ;p a = p (+) a + p ( ) a : (7 : 1 : 17 a ) Sincetheoperators p a = p (+) a p ( ) a (7 : 1 : 17 b ) commutewiththemselvesandtransformasavectorundertheL orentzalgebra,we canconstructaPoincarealgebrafromjust J ab and p a .ThisisthePoincarealgebra whoseextensionisrelevantforstringeldtheory;itclose soshell.(Thisholds ineitherlight-coneorcovariantquantization.)However, asdescribedabove,this

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134 7.LIGHT-CONEQUANTUMMECHANICS resultsinanunphysicaldoublingofzero-modes.Thiscanbe xedbyapplyingthe constraints(see(7.1.10)) p a =0 : (7 : 1 : 18) Inthelight-coneformalism, p + = p i =0(7 : 1 : 19 a ) eliminatesindependentzero-modes,while 0= p = 1 p + ( M 2(+) M 2( ) )= 2 p + N (7 : 1 : 19 b ) isthentheusualremaininglight-coneconstraintequating thenumbersofleft-handed andright-handedmodes. ThegeneratorsoftheLorentzsubgroupagaintaketheform(2 .3.5),asforthe openstring,andtheoperatorsappearingin J ab areexpressedintermsoftheopenstringonesappearingin(7.1.15)as M ij = X M ( ) ij ; M 2 =2 X M 2( ) =4( N 2) ;N = X N ( ) ; N = N (+) N ( ) ; M im M =2 X ( M im M ) ( ) : (7 : 1 : 20) (Since J ab and p a areexpressedassums,soare M ab and M .Thiscausesobjects quadraticintheseoperatorstobeexpressedastwicethesum sinthepresenceofthe constraint p a =0 p ( ) a = 1 2 p a .) ThesePoincarealgebraswillbeusedtoderivetheOSp(1,1 j 2)algebrasusedin ndinggauge-invariantactionsinsects.8.2and11.2.7.2.Spinning Inthissectionwe'lldescribeastringmodelwithfermionso btainedbyintroducing a2Dsupersymmetryintotheworldsheet[7.1,5.1],inanalog ytosect.5.3,andderivethecorrespondingPoincarealgebra.Thedescription ofthesuperstringobtained bythismethodisn'tmanifestlyspacetimesupersymmetric, sowe'llonlygiveabrief discussionofthisformalismbeforegivingthepresentstat usofthemanifestlysupersymmetricformulation.Inbothcases,(free)quantumconsi stencyrequires D =10. Forsimplicity,wegodirectlytothe(rst-)quantizedform alism,sinceasfaras theeldtheoryisconcernedtheonlyrelevantinformationf romthefreetheoryishow

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7.2.Spinning 135 toconstructthecovariantderivativesfromthecoordinate s,andthentheequationsof motionfromthecovariantderivatives.Afterquantization ,when(intheSchrodinger picture)thecoordinatesdependonlyon ,therestillremainsa1Dsupersymmetry in -space[7.2].Thecovariantderivativesaresimplythe1Dsu persymmetrizationof the b P operatorsofsect.7.1(orthoseofsect.6.1attheclassical level,usingPoisson brackets): f ^ D a ( 1 ; 1 ) ; ^ D b ( 2 ; 2 ) g = ab d 2 2 ( 2 1 ) ( 2 1 ) ; (7 : 2 : 1 a ) ^ D ( ; )= ^ ( )+ b P ( ) [ b P; b P ]= asbefore; f ^ a ( 1 ) ; ^ b ( 2 ) g = ab 2 ( 2 1 ) : (7 : 2 : 1 b ) The2Dsuperconformalgenerators(thesupersymmetrizatio nofthegenerators(6.1.7b) of2Dgeneralcoordinatetransformations,orofjusttheres idualconformaltransformationsafterthelagrangemultipliershavebeengaugedawa y)arethen 1 2 ^ D d ^ D =( 1 2 ^ b P )+ ( 1 2 b P 2 + 1 2 i ^ 0 ^ ) : (7 : 2 : 2) Thereare2choicesofboundaryconditions: ^ D ( ; )= ^ D ( +2 ; ) b P ( )= b P ( +2 ) ; ^ ( )= ^ ( +2 ) : (7 : 2 : 3) The+choicegivesfermions(theRamondmodel),whilethe givesbosons(NeveuSchwarz). Expandinginmodes,wenowhave,inadditionto(7.1.7), ^ ( )= 1 X 1 r n e in !f r a m ;r b n g = ab m + n; 0 ;r n = r n y ; (7 : 2 : 4) where m;n areintegralindicesforthefermioncaseandhalf-(odd)int egralforthe bosonic.Theassignmentofstatisticsfollowsfromthefact that,whilethe r n 'sare creationoperators r n = d n y for n> 0,theyare r matrices r 0 = r= p 2for n =0 (asintheparticlecase,butinrelationtotheusual r matricesnowhaveKlein transformationfactorsforboth d n 'sand,intheBRSTcase,theghost b C ,orelsethe d n 'sand b C arerelatedtotheusualbyfactorsof r 11 ).However,asintheparticle case,afunctionalanalysisshowsthatthisassignmentcano nlybemaintainedifthe numberofanticommutingmodesiseven;inotherwords, r 11 ( 1) P d y d =1 : (7 : 2 : 5)

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136 7.LIGHT-CONEQUANTUMMECHANICS (Intermsoftheusual r -matrices,the r -matricesherealsocontaintheKleintransformationfactor( 1) P d y d .However,inpracticeit'smoreconvenienttousethe r -matricesof(7.2.4-5),whichanticommutewithallfermion icoperators,andequate themdirectlytotheusualmatricesafterallfermionicosci llatorshavebeeneliminated.)Asusual,if r 0 isrepresentedexplicitlyasmatrices,thehermiticityin( 7.2.4) meanspseudohermiticitywithrespecttothetimecomponent r 0 0 (theindenitemetricoftheHilbertspaceofaDiracspinor).However,if r 0 isinsteadrepresentedas operators(as,e.g.,creationandannihilationoperators, asfortheusualoperatorrepresentationofSU(N) SO(2N)),noexplicitmetricisnecessary(beingautomatica lly includedinthedenitionofhermitianconjugationfortheo perators). Therestissimilartothebosonicformalism,andisstraight forwardinthe1D supereldformalism.Forexample, H = Z d 2 d 1 2 ^ D d ^ D = Z d 2 ( 1 2 b P 2 + 1 2 i ^ 0 ^ )= 1 2 ( p 2 + M 2 ) ; M 2 =2 X n> 0 n ( a n y a n + d n y d n ) : (7 : 2 : 6 a ) Forthefermionicsectorwealsohave(from(7.2.2)) Z d 2 ^ b P = 1 p 2 ( =p + f M ) ; f M 2 = M 2 : (7 : 2 : 6 b ) (Asdescribedabove,the d 'santicommutewith r ,andthuseectivelyincludean implicitfactorof r 11 f M isthusanalogoustothe =M of(4.5.12).)Thefermionic groundstateismassless(especiallyduetotheabovechiral itycondition),butthe bosonicgroundstateisatachyon.(Thelattercanmosteasil ybeseen,asforthe bosonicstring,bynotingthattherstexcitedlevelconsis tsofonlyamasslessvector.) However,consistentquantuminteractionsrequiretruncat iontothespectrumofthe superstringdescribedinthenextsection.Thismeans,inad ditiontothechirality condition(7.2.5)inthefermionicsector,therestriction inthebosonicsectortoeven M 2 [7.3].(Unlikethefermionicsector,odd M 2 ispossiblebecauseofthehalf-integral modenumbers.)Asforthebosonicstring,besidesdetermini ngtheground-state massesLorentzinvariancealsoxesthedimension,now D =10. Inthelight-coneformulationofthespinningstringwehave insteadof(7.1.14) [7.4] J ab = Z d 2 iX [ a P 0 b ] + 1 2 ^ [ a ^ b ] ; ^ X + = p + ; ^ + =0; b P 2 + i ^ 0 ^ = ^ b P =0

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7.3.Supersymmetric 137 b P = 1 2 p + b P i 2 + i ^ 0 i ^ i ; ^ = 1 p + ^ i b P i : (7 : 2 : 7) TheresultingcomponentexpansionfortheNeveu-Schwarzst ringissimilartothe non-spinningbosonicstring,withextracontributionsfro mthenewoscillators.For thecaseoftheRamondstring,comparingto(2.3.5),wendin placeof(7.1.15) M 2 =2 X n a y n i a ni + d y n i d ni ; M ij = 1 4 r [ i r j ) + X a y n [ i a nj ) + d y n [ i d nj ) ; M im M = 1 2 r i f M + 1 2 r j ji + i j r j + h a y ni n y n a ni + d y ni n y n d ni i ; f M = i i ; ij = i X p 2 n d y ni a nj d ni a y nj ; p = 1 2 p + ( p i 2 + M 2 ) ;r = 1 p + ( r i p i + f M ) ; a n = 1 p + p i a ni i 1 2 q n 2 r i d ni + n ;d n = 1 p + p i d ni + i q n 2 r i a ni + n ; n = i 1 p n ( 1 2 n 1 X m =1 q m ( n m ) a m i a n m;i +( m n 2 ) d m i d m n;i 1 X m =1 q m ( n + m ) a y m i a n + m;i +( m + n 2 ) d y m i d n + m;i ) ; n = i n 1 X m =1 p ma m i d n m;i + 1 X m =1 p n + md y m i a n + m;i p ma y m i d n + m;i # : (7 : 2 : 8) Thisalgebracanbeapplieddirectlytoobtaingauge-invari antactions,aswas describedinsect.4.5.7.3.Supersymmetric Wenowobtainthesuperstring[7.3]asacombinedgeneraliza tionofthebosonic stringandthesuperparticle,whichwasdescribedinsect.5 .4. Althoughthesuperstringcanbeformulatedasatruncationo fthespinning string,amanifestlysupersymmetricformulationisexpect edtohavetheusualadvantagesthatsupereldshaveovercomponentsinordinary eldtheories:simpler constructionsofactions,useofsupersymmetricgauges,ea sierquantumcalculations, no-renormalizationtheoremswhichfollowdirectlyfroman alyzingcounterterms,etc. Asusual,thefreetheorycanbeobtainedcompletelyfromthe covariantderivatives

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138 7.LIGHT-CONEQUANTUMMECHANICS andequationsofmotion[7.5].Thecovariantderivativesar edenedbytheirane Lie,orKac-Moody(or,asappliedtostrings,\Kac-Kradle "),algebraoftheform 1 2 [ G i ( 1 ) ; G j ( 2 ) g = ( 2 1 ) f ij k G k ( 1 )+ i 0 ( 2 1 ) g ij ; (7 : 3 : 1) where f arethealgebra'sstructureconstantsand g its(notnecessarilyCartan)metric (bothconstants).Thezero-modesofthesegeneratorsgivea nordinary(graded)Lie algebrawithstructureconstants f .TheJacobiidentitiesaresatisedifandonlyif f [ ij j l f l j k ) m =0 ; (7 : 3 : 2 a ) f i ( j j l g l j k ] =0 ; (7 : 3 : 2 b ) wheretherstequationistheusualJacobiidentityofaLiea lgebraandthesecond statesthetotal(graded)antisymmetryofthestructurecon stantswithindexlowered bythemetric g .Inthiscase,wewishtogeneralize f d ;d g =2 r a p a forthe superparticleand[ b P a ( 1 ) ; b P b ( 2 )]=2 i 0 ( 2 1 ) ab forthebosonicstring.The simplestgeneralizationconsistentwiththeJacobiidenti tiesis: f D ( 1 ) ;D ( 2 ) g =2 ( 2 1 )2 r a P a ( 1 ) ; [ D ( 1 ) ;P a ( 2 )]=2 ( 2 1 )2 r a n ( 1 ) ; f D ( 1 ) ; n ( 2 ) g =2 i 0 ( 2 1 ) ; [ P a ( 1 ) ;P b ( 2 )]=2 i 0 ( 2 1 ) ab ; [ P; n]= f n ; n g =0 ; (7 : 3 : 3 a ) r a ( r a r ) =0 : (7 : 3 : 3 b ) (7.3.2b)requirestheintroductionoftheoperatorn,and(7 .3.2a)thenimplies(7.3.3b). Thissupersymmetricsetofmodes(as b P forthebosonicstring)describesacomplete openstringorhalfaclosedstring,sotwosuchsetsareneede dfortheclosedsuperstring,whiletheheteroticstringneedsoneoftheseplusap urelybosonicset. Notetheanalogywiththesuper-Yang-Millsalgebra(5.4.8) : ( D ;P a ; n ) $ ( r ; r a ;W ) ; (7 : 3 : 4) andalsothattheconstraint(7.3.3b)occursonthe r -matrices,whichimplies D = 3 ; 4 ; 6 ; or10[7.6]whenthemaximalLorentzinvarianceisassumed(i .e.,allofSO(D 1,1) forthe D -vector P a ).

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7.3.Supersymmetric 139 Thisalgebracanbesolvedintermsof b P a ,aspinorcoordinate ( ),andits derivative = : D = + r a b P a + 1 2 ir a r ar r 0 ; P a = b P a + ir a 0 ; n = i 0 : (7 : 3 : 5) Theseareinvariantundersupersymmetrygeneratedby q = Z d 2 r a b P a 1 6 ir a r ar r 0 ; p a = Z d 2 b P a ; (7 : 3 : 6) where f q ;q g = 2 r a p a Thesmallest(generalizedVirasoro)algebrawhichinclude sgeneralizationsofthe operators 1 2 b P 2 ofthebosonicstringand 1 2 p 2 and =pd ofthesuperparticleisgenerated by A = 1 2 P 2 +n D = 1 2 b P 2 + i 0 ; B = r a P a D ; C = 1 2 D [ D ] ; D a = ir a D D 0 : (7 : 3 : 7) Notethesimilarityof A to(5.4.10),(7.2.6a),and(8.1.10,12).Thealgebragenera ted bytheseoperatorsis(classically) 1 2 [ A (1) ; A (2)]= i 0 (2 1)[ A (1)+ A (2)] ; 1 2 [ A (1) ; B (2)]= i 0 (2 1)[ B (1)+ B (2)] ; 1 2 [ A (1) ; C (2)]= i 0 (2 1)[ C (1)+ C (2)] ; 1 2 [ A (1) ; D a (2)]= i 0 (2 1)[ D a (1)+2 D a (2)] ; 1 2 fB (1) ; B (2) g = i 0 (2 1) 1 2 r ar r a [ C r (1)+ C r (2)]+4 (2 1) [ r a ( P a A + 1 8 D a )+( r ( ) 1 2 r a r a r )n r B ] ; 1 2 [ B (1) ; C r (2)]=4 (2 1)[ [ D r ] A +( [ 1 2 r a r a [ )n C r ] ] ;

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140 7.LIGHT-CONEQUANTUMMECHANICS 1 2 [ B (1) ; D a (2)]= 2 i 0 (2 1)[2 r a D A +(3 r r a r abr r b ) n r C ](1)+2 i (2 1)[4 r a D 0 A +(3 r r a r abr r b )n r C 0 ir ar r br n D b ] ; 1 2 [ C (1) ; C r (2)]= 2 (2 1) P a r a [ r [ C ] ] ; 1 2 [ C (1) ; D a (2)]=2 i 0 (2 1) r a r r br [ P b C ] (1) 4 i (2 1)( r ar [ D 0 ] B r + P b r ba [ C 0 ] ) ; 1 2 [ D a (1) ; D b (2)]= 00 (2 1) r ab D B (1)+(2) + 0 (2 1) h 4 iP ( a D b ) + ab (3 D 0 B D B 0 ) i (1) (2) + (2 1) h 2 i (3 P 0 [ a D b ] P [ a D 0 b ] ) +2 r ab (3 D 0 B 0 D 00 B )+ r abc (3 P 0 c C 0 P 00 c C ) i : (7 : 3 : 8) (Duetoidentitieslike =PDD B D =P C ,thereareotherformsofsomeofthese relations.) BRSTquantizationcanagainbeperformed,andthereareanin nitenumberof ghosts,asintheparticlecase.However,aremainingproble mistondtheappropriategroundstate(andcorrespondingstringeld).Consider ingtheresultsofsect.5.4, thismayrequiremodicationofthegenerators(7.3.7)andB RSToperator,perhaps toincludeLorentzgenerators(actingontheendsofthestri ng?)orseparatecontributionsfromtheBRSTtransformationsofYang-Millseldth eory(thegroundstate oftheopensuperstring,orofasetofmodesofonehandedness ofthecorresponding closedstrings).Ontheotherhand,thegroundstate,rather thanbeingYang-Mills, mightbepurelygaugedegreesoffreedom,withYang-Millsap pearingatsomeexcited level,somodicationwouldbeunnecessary.Thecondition Q 2 =0shouldreproduce theconditions D =10, 0 =1. Thecovariantderivativesandconstraintscanalsobederiv edfroma2Dlagrangian ofthegeneralform(3.1.10),asforthesuperparticle[7.5] .Thisclassicalmechanics LagrangianimposesweakerconstraintsthantheGreen-Schw arzone[7.6](whichsets D =0viaGupta-Bleuler),andthusshouldnotimposestrongerc onditions. Ontheotherhand,quantizationinthelight-coneformalism isunderstood.Spinors areseparatedintohalves,withthecorrespondingseparati onofthe r matricesgiving

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7.3.Supersymmetric 141 thesplittingofvectorsintotransverseandlongitudinalp arts,asin(5.4.27).The light-conegaugeisthenchosenas P + ( )= p + ; =0 : (7 : 3 : 9) Otheroperatorsaretheneliminatedbyauxiliaryeldequat ions: A =0 P = 1 p + 1 2 b P i 2 + i + 0 D + ; B =0 D = 1 p 2 p + b =P T y D + : (7 : 3 : 10) Theremainingcoordinatesare x X i ( ),and + ( ),andtheremainingoperators are D + = + + p + + ;P i = b P i ; n + = i + 0 : (7 : 3 : 11) However,insteadofimposing C = D =0quantummechanically,wecansolvethem classically,inanalogytotheparticlecase.The C + ij ( )arenow local (in )SO(8) generators,andcanbeusedtogaugeawayallbut1Lorentzcom ponentof D + ,by thesamemethodas(5.4.34ab)[5.30,29].Afterthis, D + isjusttheproductofthis onecomponenttimesits -derivative.Furthermore, D + isaVirasoroalgebrafor D + andcanthusbeusedtogaugeawayallbutthezero-mode[5.30, 29](astheusualone A didfor P + in(7.3.9)),afterthisconstraintimplies D + factorsinawayanalogous to(5.4.34a): D + =0 D + = c ( ) : (7 : 3 : 12) (Theproofisidentical,since D + =0isequivalentto D + ( 1 ) D + ( 2 )=0.)We arethusbacktotheparticlecasefor D ,withasinglemoderemaining,satisfying thecommutationrelation c 2 = p + c = p p + ,so D iscompletelydetermined. Alternatively,asfortheparticle,wecouldconsider D =0asasecond-classconstraint [7.6],orimposethecondition D + =0(whicheliminatesallauxiliarystringelds), asaGupta-Bleulerconstraint.Thisrequiresafurtherspli ttingofthespinors,asin (5.4.29),andtheGupta-Bleulerconstraintisagainachira litycondition,asin(5.4.33). C = D =0arethenalsosatisedalaGupta-Bleuler(withappropria te\normal ordering").Thus,ina\chiral"representation(asinordin arysupersymmetry)wehave achiral,\on-shell"stringsupereld,orwavefunction,[ x ;X i ( ) ; a ( )],which satisesalight-coneeldequation i @ @x + + H =0 ;

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142 7.LIGHT-CONEQUANTUMMECHANICS H = p = 1 0 Z d 2 b P = Z d 2 1 2 b P i 2 + i a 0 a : (7 : 3 : 13) Thedimensionofspacetime D =10andtheconstantin H (zero)aredeterminedby considerationssimilartothoseofthebosoniccase(Lorent zinvarianceinthelight-cone formalism,orBRSTinvarianceintheyet-to-be-constructe dcovariantformalism). Similarly,light-coneexpressionsfor q canbeobtainedfromthecovariantones: q + = Z d 2 + p + + Z d 2 Q + ; q = Z d 2 1 p 2 p + b =P T y Q + : (7 : 3 : 14) Ifthesuperstringisformulateddirectlyinthelightcone, (7.3.14)canbeusedasthe startingpoint. Q + and D + canbeconsideredasindependentvariables(insteadof + and = + ),denedbytheirself-conjugatecommutationrelations(a nalogousto thoseof b P ): 1 2 f Q + ( 1 ) ;Q + ( 2 ) g = ( 2 1 )2 p + ; 1 2 f D + ( 1 ) ;D + ( 2 ) g = ( 2 1 )2 p + : (7 : 3 : 15) However,asdescribedaboveandfortheparticle, D + isunnecessaryfordescribing physicalpolarizations,soweneednotintroduceit.Inorde rtomorecloselystudy theclosureofthealgebra(7.3.14),weintroducemorelight -conespinornotation(see sect.5.3):WorkingintheMajoranarepresentation= I ,weintroduce( D 2)dimensionalEuclidean r -matricesas =p T r i 0 p i ; r ( i 0 r j ) 0 =2 ij ;r ( i 0 r j ) 0 =2 ij 0 0 ; (7 : 3 : 16) wherenotonlyvectorindices i butalsospinorindices and 0 canberaisedand loweredbyKronecker 's,andprimedandunprimedspinorindicesarenotnecessari ly related.(However,asforthecovariantindices,theremayb eadditionalrelations satisedbythespinors,irrelevantforthepresentconside rations,thatdierindierent dimensions.)Closureofthesupersymmetryalgebra(onthem omentumintheusual way(5.4.4),butinlight-conenotation,andwiththelightconeexpressionsfor p i p + and p )thenrequirestheidentity(relatedto(7.3.16)by\triali ty") r i ( j 0 r i j ) 0 =2 0 0 : (7 : 3 : 17)

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7.3.Supersymmetric 143 Thisidentityisactually(7.3.3b)inlight-conenotation, andtheequalityofthedimensionsofthespinorandvectorcanbederivedbytracing(7 .3.17)with Returningtoderivingthelight-coneformalismfromthecov ariantone,wecan alsoobtainthelight-coneexpressionsforthePoincarege nerators,whichshouldprove importantforcovariantquantization,viatheOSp(1,1 j 2)method.Asingeneral,they arecompletelyspeciedby M ij M im M ,and M 2 : M ij = Z 0 i ^ X i b P j + M ij ; M im M = Z 0 i ^ X i p + b P + M i j b P j ; M 2 = Z 0 2 p + b P ; (7 : 3 : 18) where p + b P = 1 2 b P i 2 + i 1 8 p + Qr + Q 0 i 1 8 p + D + ; M ij = 1 16 p + Qr + r ij Q + 1 16 p + C + ij ; (7 : 3 : 19) containall D dependence(asopposedto X and Q dependence)onlyintheformof C and D ,whichcanthereforebedropped.(Cf.(5.4.22). r + picksout Q + from Q ,as in(5.4.27).) WecannowconsiderderivingtheBRSTalgebrabythemethodof adding4+4 dimensionstothelight-cone(sects.3.6,5.5).Unfortunat ely,adding4+4dimensions doesn'tpreserve(7.3.17).Infact,fromtheanalysisofsec t.5.3,weseethattopreservethesymmetriesofthe -matricesrequiresincreasingthenumberofcommuting dimensionsbyamultipleof8,andthenumberoftimedimensio nsbyamultipleof4. Thissuggeststhatthisformalismmayneedtobegeneralized toadding8+8dimensionstothelight-cone(4space,4time,8fermionic).Coinc identally,thelight-cone superstringhas8+8physical( -dependent)coordinates,sothiswouldjustdouble thenumberofoscillators.PerformingthereductionfromOS p(4,4 j 8)toOSp(2,2 j 4) toOSp(1,1 j 2),ifonestepischosentobeU(1)-typeandtheotherGL(1)-t ype,itmay bepossibletoobtainanalgebrawhichhasthebenetsofboth formalisms. Asforthebosonicstring,theclosedsuperstringisconstru ctedasthedirect productof2openstrings(1fortheclockwisemodesand1fort hecounterclockwise): Thehamiltonianisthesumof2open-stringones,andtheclos ed-stringgroundstate istheproductof2open-stringones.InthecaseoftypeIorII Bclosedstrings,the 2setsofmodesarethesamekind,andtheformer(thebound-st ateoftypeIopen

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144 7.LIGHT-CONEQUANTUMMECHANICS strings)isnonoriented(inordertobeconsistentwiththe N =1supersymmetryof theopenstring,ratherthanthe N =2supersymmetrygeneratedbythe2setsof modesoforiented,typeIIstrings).TypeIIAclosedstrings have'swiththeopposite chiralitybetweenthetwosetsofmodes(i.e.,onesethasa whiletheotherhasa ).Thegroundstatesoftheseclosedstringsaresupergravit y( N =1supergravity fortypeIand N =2fortypeII).Theheteroticstringisaclosedstringforwh ich onesetofmodesisbosonic(withtheusualtachyonicscalarg roundstate)whilethe otherissupersymmetric(withtheusualsupersymmetricYan g-Millsgroundstate). Thelowest-massphysicalstates,duetothe N =0restriction,aretheproduct ofthemasslesssectorofeachset(sincenow N = H (+) H ( ) =( N (+) 1) N ( ) ).Thedimensionofspacetimeforthe2setsofmodesismadeco nsistentby compacticationofsomeofthe26dimensionsofoneandsome( ornone)ofthe10of theotherontoatorus,leavingthesamenumberofnoncompact ieddimensions(at leastthephysical4)forbothsetsofmodes.Thesecompacti edbosonicmodescan alsobefermionized(seethenextsection),givinganequiva lentformulationinwhich theextradimensionsdon'texplicitlyappear:Forexample, fermionizationof16of thedimensionsproduces32(real)fermioniccoordinates,g ivinganSO(32)internal symmetry(whenthefermionsaregiventhesameboundarycond itions,allperiodicor allantiperiodic).Theresultingspectrumforthemassless sectorofheteroticstrings consistsofsupergravitycoupledtosupersymmetricYang-M illswith N =1(in10D counting)supersymmetry.ThevectorsgaugingtheCartansu balgebraofthefull Yang-Millsgrouparetheobviousonescomingfromthetoroid alcompactication (i.e.,thosethatwouldbeobtainedfromthenoncompactied theorybyjustdropping dependenceonthecompactiedcoordinates),whiletherest correspondto\soliton" modesofthecompactiedcoordinatesforwhichthestringwi ndsaroundthetorus.As forthedimensionandground-statemass,quantumconsisten cyrestrictstheallowed compactications,andinparticularthetoroidalcompacti cationsarerestrictedto thosewhich,inthecaseofcompacticationto D =10,giveYang-MillsgroupSO(32) orE 8 n E 8 .(Thesegroupsgiveanomaly-free10Dtheoriesintheirmass lesssectors. ThereisalsoanSO(16) SO(16)10D-compacticationwhichcanbeconsideredto havebrokenN=1supersymmetry.Thereareother10D-compact icationswhichhave tachyons.) Someaspectsoftheinteractingtheorywillbedescribedinc hapts.9and10.

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Exercises 145 Exercises (1)Use(7.1.7)asthe classical solutionfor ^ X i ,andset a ni =0for n 6 =1and i 6 =1. Find X a ( ; ).Inthecenter-of-massframe,ndtheenergyandspin,andr elate them. (2)Do(1)intheconformalgaugebyusing(7.1.7)for ^ X a for a =0 ; 1(same n ),and applyingtheconstraint(6.2.5).Compareresults. (3)Provethelight-conePoincarealgebraclosesonlyfor D =26,anddeterminesthe constantin(7.1.8a). (4)Findexplicitexpressionsforallthestatesatthe4lowe stmasslevelsoftheopen bosonicstring.Forthemassivelevels,combineSO(D 2)representationsinto SO(D 1)ones.Dothesameforthe4lowest(nontrivial)masslevels ofthe closedstring. (5)Derive(7.1.15),includingtheexpressionsintermsof -integrals.Whathappens tothepartofthisintegralsymmetricin ij for M ij ? (6)Derive(7.2.8).(7)Derive(7.3.17),bothfrom(7.3.3b)andclosureof(7.3. 14).Showthatitimplies D 2=1 ; 2 ; 4 ; 8. (8)Showthat(7.3.5)satises(7.3.3).Showthat(7.3.6)gi vesasupersymmetry algebra,andthattheoperatorsof(7.3.5)areinvariant.Ch eck(7.3.8)tillyou drop. (9)Findallthestatesinthespinningstringatthetachyoni c,massless,andrst massivelevels.Showthat,usingthetruncationofsect.7.2 ,thereareequal numbersofbosonsandfermionsateachlevel.Constructthes amestatesusing the X and Q oscillatorsofsect.7.3.

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146 8.BRSTQUANTUMMECHANICS 8.BRSTQUANTUMMECHANICS8.1.IGL(1) WerstdescribetheformoftheBRSTalgebraobtainedbyrst -quantization ofthebosonicstringbythemethodofsect.3.2,usingthecon straintsfoundinthe conformal(temporal)gaugeinsect.6.2. Theresidualgaugeinvarianceinthecovariantgaugeisconf ormaltransformations (modiedbytheconstraintthattheypreservethepositiono ftheboundaries).After quantizationintheSchrodingerpicture(wherethecoordi nateshaveno dependence), theVirasorooperators[8.1] G ( )= i h 1 2 b P 2 ( ) 1 i ; (8 : 1 : 1) with b P asin(7.1.3)butforallLorentzcomponents,generateonlyt hesetransformations(insteadofthecompletesetof2Dgeneralcoordinat etransformationsthey generatedwhenleftasarbitraryo-shellfunctionsof and intheclassicalmechanics).UsingthehamiltonianformofBRSTquantization,wer stndtheclassical commutationrelations(Poissonbrackets,neglectingthen ormal-orderingconstantin (8.1.1)) [ G ( 1 ) ; G ( 2 )]=2 0 ( 2 1 )[ G ( 1 )+ G ( 2 )] =2 [ ( 2 1 ) G 0 ( 2 )+2 0 ( 2 1 ) G ( 2 )] ; (8 : 1 : 2 a ) orinmodeform i G ( )= X L n e in [ L m ; L n ]=( n m ) L m + n : (8 : 1 : 2 b ) Thesecommutationrelations,rewrittenas Z d 1 2 1 ( 1 ) G ( 1 ) ; Z d 2 2 2 ( 2 ) G ( 2 ) # = Z d 2 [2 ( ) 1] 0 ( ) G ( ) ; (8 : 1 : 3 a )

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8.1.IGL(1) 147 correspondtothe1Dgeneralcoordinatetransformations(i n\zeroth-quantized"notation) 2 ( ) @ @ ; 1 ( ) @ @ # = [2 1] 0 @ @ ; (8 : 1 : 3 b ) givingthestructureconstants f ( 1 ; 2 ; 3 )=2 0 ( 2 1 )[ ( 1 3 )+ ( 2 3 )] ; (8 : 1 : 4 a ) or 1 j 2 i f ij k $ 1 @ @ 2 : (8 : 1 : 4 b ) Moregenerally,wehaveoperatorswhosecommutationrelati ons [ G ( 1 ) ; O ( 2 )]=2 [ ( 2 1 ) O 0 ( 2 )+ w 0 ( 2 1 ) O ( 2 )](8 : 1 : 5 a ) representthetransformationpropertiesofa1Dtensorof(c ovariant)rank w ,ora scalardensityofweight w : Z G ; O = O 0 + w 0 O : (8 : 1 : 5 b ) Equivalently,intermsof2Dconformaltransformations,it hasscaleweight w .(Rememberthatconformaltransformationsin D =2areequivalentto1Dgeneralcoordinatetransformationson :See(6.2.7).)Inparticular,weseefrom(8.1.2a) that G itselfisa2nd-rank-covariant(asopposedtocontravarian t)tensor:Itisthe energy-momentumtensorofthemechanicsaction.(Itwasder ivedbyvaryingthat actionwithrespecttothemetric.)Theniteformofthesetr ansformationsfollows fromexponentiatingtheLiealgebrarepresentedin(8.1.3) :(8.1.5)canthenalsobe rewrittenastheusualcoordinatetransformations @ 0 @ w O 0 ( 0 )= O ( ) ; (8 : 1 : 6 a ) wheretheprimesherestandforthetransformedquantities( not -derivatives)oras ( d 0 ) w O 0 ( 0 )=( d ) w O ( ) ; (8 : 1 : 6 b ) indicatingtheirtensorstructure.Inparticular,acovari antvector( w =1)canbe integratedtogiveaninvariant. b P issuchavector(andthemomentum p thecorrespondingconformalinvariant),whichiswhy G hastwiceitsweight(by(8.1.1)). BeforeperformingtheBRSTquantizationofthisalgebra,we relateittothe light-conequantizationofthepreviouschapter.Theconst raints(8.1.1)canbesolved

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148 8.BRSTQUANTUMMECHANICS inaGupta-Bleulerfashioninlight-conenotation.Thedie rencebetweenthatand actuallight-conequantizationisthatinthelight-conequ antization b P istotally eliminatedattheclassicallevel,whereasinthelight-con enotationforthecovariantgaugequantizationtheconstraintisusedtodeterminethed ependenceonthe oscillatorsintermsofthetransverseoscillators.Oneway todothiswouldbetostart withastateconstructedfromjusttransverseoscillators( asinlight-conequantization) andaddintermsinvolvinglongitudinaloscillatorsuntilt heconstraintsaresatised (oractuallyhalfofthem,alaGupta-Bleuler).Asimplerway istostartattheclassical levelinanarbitraryconformalgaugewithtransverseoscil lators,andthenconformally transformthemtothelight-conegaugetoseewhatatransver seoscillator(inthe physicalsense,notthelight-cone-indexsense)lookslike .Wethuswishtoconsider d 0 b P i 0 ( 0 )= d b P i ( ) ; 0 = 1 p + ^ X + ( )= + oscillator terms: (8 : 1 : 7) (Withoutlossofgenerality,wecanworkat x + =0.Equivalently,wecanexplicitly subtract x + from ^ X + everywherethelatterappearsinthisderivation.)Ifwecon sider thesametransformationon b P + (using @ ^ X=@ b P ),wend b P + 0 = p + ,thelight-cone gauge.(8.1.7)canberewrittenas b P 0 ( 1 )= Z d 2 1 1 p + ^ X + ( 2 ) b P ( 2 ) : (8 : 1 : 8) (8.1.7)followsuponreplacing 1 with 1 0 andintegratingoutthe -function(with theJacobiangivingtheconformalweightfactor).Amorecon venientformforquantizationcomesfromthemodeexpansion:Multiplyingby e in 1 andintegrating, n 0 = Z d 2 e in ^ X + ( ) =p + b P ( ) : (8 : 1 : 9) These(\DDF")operators[8.2](withnormalordering,asusu al,uponquantization, andwithtransverseLorentzindex i ,and n> 0)canbeusedtocreateallphysicalstates.Duetotheirdenitionintermsofaconformaltra nsformationfroman arbitraryconformalgaugetoacompletelyxed(light-cone )gauge,theyareautomaticallyconformallyinvariant:i.e.,theycommutewith G .(Thiscanbeveriedto remaintrueafterquantization.)Consequently,statescon structedfromthemsatisfy theGupta-Bleulerconstraints,sincetheconformalgenera torspushpasttheseoperatorstohitthevacuum.Thus,theseoperatorsallowtheconst ructionofthephysical Hilbertspacewithintheformalismofcovariant-gaugequan tization,andallowadirect comparisonwithlight-conequantization.

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8.1.IGL(1) 149 Ontheotherhand,formostpurposesitismoreconvenienttos olvetheconstraints ascovariantlyaspossible(whichiswewhyweareworkingwit hcovariant-gauge quantizationintherstplace).ThenextstepistheIGL(1)a lgebra[3.4] Q = Z d 2 b C i 1 2 b P 2 + b C 0 b C + i / 0 i Z d 2 b C A ; J 3 = Z d 2 b C b C : (8 : 1 : 10) Expandingintheghostzero-mode c = Z d 2 b C (8 : 1 : 11) wealsond(see(3.4.3b)) p 2 + M 2 =2 Z A ;M + = i Z b C b C 0 : (8 : 1 : 12) / 0 (theinterceptoftheleadingReggetrajectory)isaconstan tintroduced,asin thelight-coneformalism,becauseofimplicitnormalorder ing.Theonlyambiguous constantin J 3 isanoverallone,whichwechoosetoabsorbintothezero-mod etermso thatitappearsas c@=@c ,sothatphysicaleldshavevanishingghostnumber.(This alsomakes J 3 y =1 J 3 .)Inanalogytotheparticle, b C isamomentum(denedto beperiodicon 2 [ ; ]),asfollowsfromconsiderationof reversalintheclassical mechanicsaction,buthere reversalisaccompaniedby reversalinordertoavoid switching+and modes.(Intheclassicalactiontheghostisoddundersucha transformation,sinceitcarriesa2Dvectorindex,asdoest hegaugeparameter,while theantighostiseven,carrying2indices,asdoesthegaugexingfunction g mn .) b C can alsobeseparatedintooddandevenparts,whichisusefulwhe nsimilarlyseparating b P asin(7.1.3a): b C = C + e C;C ( )= C ( ) ; e C ( )= e C ( ) : (8 : 1 : 13) Wenowpayattentiontothequantumeects.Ratherthanexami ningtheBRST algebra,welookattheIGL(1)-invariantVirasorooperator s(from(3.2.13)) ^ G = G + b C 0 b C + b C b C 0 + i ( / 0 1) : (8 : 1 : 14) (The / 0 1justreplacesthe1in(8.1.1)with / 0 .)Correspondingto(8.1.2b), wenowhavetheexactquantummechanicalcommutationrelati ons(afternormal ordering) [ ^ L m ; ^ L n ]=( n m ) ^ L m + n + D 26 12 ( m 3 m )+2( / 0 1) m m; n : (8 : 1 : 15)

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150 8.BRSTQUANTUMMECHANICS Thetermslinearin m intheanomalousterms(thosenotappearingintheclassical result(8.1.2))aretrivial,andcanbearbitrarilymodied byaddingaconstantto ^ L 0 Thattheremainingtermis m 3 followsfrom(1D)dimensionalanalysis:[ ^ G ; ^ G ] 0 ^ G + 000 ,sincethersttermimplies ^ G 1 = 2 dimensionally,soonly 000 1 = 4 canbeused.Thevaluesofthecoecientsinthesetermscanal sobedetermined byevaluatingjustthevacuummatrixelements h 0 j [ ^ L n ; ^ L n ] j 0 i for n =1 ; 2.Further examiningtheseterms,weseethattheghostcontributionsa renecessarytocancel thosefromthephysicalcoordinates(whichhavecoecient D ),anddosoonlyfor D =26.Theremaininganomalycancelsfor / 0 =1.Underthesameconditionsone canshowthat Q 2 =0.Thus,inthecovariantformalism,whereLorentzcovaria nceis manifestandnotunitarity(theoppositeofthelight-conef ormalism), Q 2 =0isthe analogofthelightcone's[ J i ;J j ]=0(andthecalculationisalmostidentical,sothe readerwillhavelittletroublemodifyinghispreviouscalc ulation). b C and = b C canbe expandedinzero-modesandcreationandannihilationopera tors,as b P ((7.1.7a)),but thecreationoperatorsin b C arecanonicallyconjugatetotheannihilationoperatorsin = b C ,andviceversa: b C = c + 1 X 1 1 p n ( c n y e in + c n e in ) ; b C = @ @c + 1 X 1 p n ( i ~ c n y e in + i ~ c n e in ); f c m ; ~ c n y g = i mn ; f ~ c m ;c n y g = i mn : (8 : 1 : 16) (SincetheIGL(1)formalismisdirectlyrelatedtotheOSp(1 ,1 j 2),asinsect.4.2,we havenormalizedtheoscillatorsinawaythatwillmaketheSp (2)symmetrymanifest inthenextsection.)Thephysicalstatesareobtainedbyhit ting j 0 i with a y 'sbutalso requiring Q j i =0;states j i = Q j i arenullstates(puregauge).Thecondition ofbeingannihilatedby Q isequivalenttobeingannihilatedby L n for n 0(i.e., the\nonpositiveenergy"partof G ( ),whichisnownormalorderedandincludesthe / 0 termof(8.1.10,14)),whichisjusttheconstraintinGuptaBleulerquantization. L 0 issimplytheLorentz-covariantizationof H of(7.1.8)(i.e.,alltransverseindices replacedwithLorentzindices). AninterestingfactabouttheVirasoroalgebra(8.1.15)(an ditsgeneralizations, seebelow)isthat,afteranappropriateshiftin L 0 (namely,thechoiceof / 0 =1in thiscase),theanomalydoesnotappearintheSp(2)(=SL(2)= SU(1,1)=SO(2,1)= projectivegroup)subalgebragivenby n =0 ; 1[8.3],independentoftherepresentation(inthiscase, D ).Furthermore,unlikethewholeVirasoroalgebra(evenwhe n

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8.1.IGL(1) 151 theanomalycancels),wecandeneastatewhichisleftinvar iantbythisSp(2). Expanding(8.1.14)inmodes(asin(7.1.7a,8.1.2b)),theon lytermin ^ L 1 containing noannihilationoperatorsis p a 1 y ,sowechoose p =0.Then ^ L 0 =0requiresthe statebeon-shell,whichmeansit'sintheusualmasslesssec tor(Yang-Mills).Further examinationthenshowsthatthisstateisuniquelydetermin edtobethestatecorrespondingtoaconstant( p =0)Yang-Millsghosteld C .Itcanalsobeshownthat thisistheonlygauge-invariant,BRST-invariantstate(i. e.,inthe\cohomology"of Q )ofthatghost-number J 3 [8.4].Sinceithasthesameghostnumberasthegauge parameter(see(4.2.1)),thismeansthatitcanbeidentie dastheonlygaugeinvarianceofthetheorywhichhasnoinhomogeneousterm:Anyg augeparameterof theform= Q notonlyleavesthefreeactioninvariant,butalsotheinter actingone, sinceupongaugexingit'sagaugeinvarianceoftheghosts( whichmeanstheghosts themselvesrequireghosts),whichmustbemaintainedatthe interactinglevelfor consistentquantization.However,anyparametersatisfyi ng Q =0won'tcontribute tothefreegaugetransformationofthephysicalelds,butm aycontributeatthe interactinglevel.Infact,gaugetransformationsintheco homologyof Q arejustthe globalinvariancesofthetheory,oratleastthosewhichpre servethesecond-quantized vacuumaboutwhichthedecompositionintofreeandinteract inghasbeendened. SincetheBRSTtransformation = Q isjustthegaugetransformationwiththe gaugeparameterreplacedbytheghost,thistransformation parameterappearsinthe eldinthesamepositionaswouldthecorrespondingghost.F orthebosonicstring, theonlymasslessphysicaleldisYang-Mills,andthustheo nlyglobalinvarianceis theusualglobalnonabeliansymmetry.Thus,thestateinvar iantunderthisSp(2)directlycorrespondstotheglobalinvarianceofthestringth eory,andtoitsghost.This Sp(2)symmetrycanbemaintainedattheinteractinglevelin treegraphcalculations (seesect.9.2),especiallyforvertices,basicallyduetot hefactthattreegraphshave thesameglobaltopologyasfreestrings.Insuchcalculatio nsit'sthereforesomewhat moreconvenienttoexpandstatesaboutthisSp(2)-invarian t\vacuum"insteadofthe usualone.(Wenowrefertotherst-quantizedvacuumwithre specttowhichfree eldsaredened.It'sredenitionisunrelatedtotheusual vacuumredenitionsof eldtheory,whichareinhomogeneousintheelds.)Thisee ctivelyswitchestherole ofthecorrespondingpairofghostoscillators(justthe n =1mode)betweencreation andannihilationoperators. Theclosedstring[4.5]isquantizedsimilarly,butwith2se tsofmodes( ;except

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152 8.BRSTQUANTUMMECHANICS thattherearestilljustone x and p ),andwecanseparate b C ( )=( C e C )( )(8 : 1 : 17) correspondingto(7.1.6). Since A commuteswithboth M 3 and M + ,itisSp(2)-invariant.Thus,themodiedVirasorooperators L n itgives(inanalogyto(8.1.2b),or,morespecically,the nonnegative-energyones),andinparticulartheirfermion icparts,canbeusedto generate(BRST)Sp(2)-invariantstates,withtheexceptio nofthezerothandrst fermionicVirasorooperators(theprojectivesubgroup),w hichvanishonthevacuum. Wewillnowshowthattheseoperators,togetherwiththeboso nicoscillators,are sucienttogenerate all suchstates,i.e.,thecompletesetofphysicalelds[4.1]. (By physicaleldswemeanalleldsappearinginthegauge-inva riantaction,including StueckelbergeldsandunphysicalLorentzcomponents.)Th isisseenbybosonizing thetwofermioniccoordinatesintoasingleadditionalboso niccoordinate,whosecontributiontotheVirasorooperatorsincludesatermlineari nthenewoscillators,but lackingtherstmode.Thiscorrespondstothefactthat M + containsatermlinear intheannihilationoperatoroftherstmode.Thus,theVira sorooperatorsgenerate excitationsinallbuttherstmodeofthenewcoordinate,an dthecondition M + =0 killsonlyexcitationsintherstmode. J 3 isjustthezeromodeofthenewcoordinate, soitsvanishing(whichthenimplies T =0)completesthederivation. Thebosonizationisessentiallythesameasthestandardpro cedure[8.5],except fordierencesduetotheindenitemetricoftheHilbertspa ceoftheghosts.The fermioniccoordinatescanbeexpressedintermsofabosonic coordinate^ (analogous to ^ X )as b C = e ^ ; b C = e ^ ; (8 : 1 : 18) withourusualimplicitnormalordering(withbothterms^ q +^ p ofthezeromode appearinginthesameexponentialfactor).Notethehermiti cityofthesefermionic coordinates,duetothelackof i 'sintheexponents.(Forphysicalbosonsandfermions, wewoulduse ^ = e i ^ ^ y = e i ^ ,with ^ canonicallyconjugateto ^ y .)^ hasthe modeexpansion ^ =(^ q +^ p )+ 1 X n =1 1 p n (^ a n e in +^ a y n e in ); [^ p; ^ q ]= i; [^ a m ; ^ a y n ]= mn : (8 : 1 : 19)

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8.1.IGL(1) 153 Bycomparisonwith(7.1.7),weseethatthiscoordinatehasa timelikemetric(i.e., it'saghost).Using : e a ^ ( ) :: e b ^ ( 0 ) :=: e a ^ ( )+ b ^ ( 0 ) : h 2 isin 0 2 + i ab ; (8 : 1 : 20) wecanverifythefermionicanticommutationrelations,asw ellas J 3 = i ^ p + 1 2 ;M + = Z d 2 e 2^ ;(8 : 1 : 21 a ) A = 1 2 b P 2 ^ 0 2 ^ 00 9 4 : (8 : 1 : 21 b ) Since J 3 isquantizedinintegralvalues, isdenedtoexistonacircleofimaginary lengthwithanticyclicboundaryconditions.(Theimaginar yeigenvaluesofthishermitianoperatorareduetotheindenitenessinducedbythegho stsintotheHilbert-space metric.)Conversely,choosingsuchvaluesfor i ^ p makes b C periodicin .TheSU(2) whichfollowsfrom J 3 and M + isnottheusualoneconstructedinbosonization[8.6] becauseoftheextrafactorsandinversesof @=@ involved(seethenextsection). Sinceweprojectonto^ p = i 1 2 whenactingon,wendforthepartsof M + and A linearin oscillatorswhenactingon M + = e 2^ q 2^ a 1 + ; L n = 1 2 p n ( n 1)^ a y n + : (8 : 1 : 22) Thisshowshowtheconstraint T =0essentiallyjusteliminatesthezerothandrst oscillatorsof WehaveseensomeexamplesaboveofVirasorooperatorsdene dasexpressions quadraticinfunctionsof (andtheirfunctionalderivatives).Moregenerally,we canconsiderabosonic(periodic)function ^ f ( )witharbitraryweight w .Inorderto obtainthetransformationlaw(8.1.5),wemusthave G ( )= ^ f 0 ^ f w ^ f ^ f 0 ; (8 : 1 : 23 a ) uptoanoverallnormal-orderingconstant(whichwedrop).B ymanipulationslike thoseabove,wend [ L m ; L n ]=( n m ) L m + n + nh ( w 1 2 ) 2 1 12 i m 3 1 6 m o m; n : (8 : 1 : 23 b ) Since ^ f 0 and = ^ f (or ^ f and = ^ f 0 )havethesamecommutationrelationsastwo b P 's, butwitho-diagonalmetric ab =( 01 10 ),for w =0(or w =1)thealgebra(8.1.23) mustgivejusttwicethecontributiontotheanomalyasasing le b P .Thisagreesexactly

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154 8.BRSTQUANTUMMECHANICS with(8.1.15)(the D term).Forfermionic ^ f ,theanomaloustermsin(8.1.23)havethe oppositeoverallsign.Inthatcase, ^ f and = ^ f havetheanticommutationrelations of2physicalfermions(seesect.7.2),againwiththeo-dia gonalmetric,and w = 1 2 givestheVirasorooperatorsfor2physicalfermions(i.e., asintheabovebosonic case, G canberewrittenasthesumof2independent G 's).Theanomalyforasingle physicalfermioninthusgivenbyhalfofthatin(8.1.23b),w ithoppositesign.Another interestingcaseis w = 1(or2),which,forfermions,givestheghostcontribution of(8.1.15)(thenonD terms,for / 0 =0;comparing(8.1.14)with(8.1.23a),wesee b C has w = 1andthus = b C has w =2).Thus,(8.1.23)issucienttogiveallthe Virasoroalgebraswhicharehomogeneousofsecondorderin1 Dfunctions.Bythe methodofbosonization(8.1.18),thefermioniccaseof(8.1 .23a)canberewrittenas G = i h 1 2 b P 2 +( 1 2 w ) b P 0 + 1 8 i ; (8 : 1 : 23 c ) where ^ f = exp ^ intermsofatimelikecoordinate ( b P =^ 0 ).For w = 1 2 ,this givesanindependentdemonstrationthat2physicalfermion sgivethesameanomaly as1physicalboson(modulothenormal-orderingconstant), sincetheyarephysically equivalent(uptotheboundaryconditionsonthezero-modes ).(Therearealsofactors of i thatneedtobeinsertedinvariousplacestodistinguishphy sicalbosonsand fermionsfromghostones,butthesedon'taectthevalueoft heanomaly.) Asbefore,theseVirasorooperatorscorrespondto2Denergy -momentumtensors obtainedbyvaryinganactionwithrespecttothe2Dmetric.U singthevielbein formalismofsect.4.1,werstnotethattheLorentzgroupha sonlyonegenerator, whichactsverysimplyonthelight-conecomponentsofacova rianttensor: M ab = ab M ( + =1) [ M ; ( s ) ]= s ( s ) ; (8 : 1 : 24) wherefortensors s isthenumberof\+"indicesminus\ "indices.However,since the2DLorentzgroupisabelian,thisgeneralizestoarbitra ry\spin,"half-integralas wellasirrational.Thecovariantderivativecanthenbewri ttenas r a = e a + a M ; a = 1 2 cb abc = 1 2 cb c bca = ab e @ m e 1 e bm : (8 : 1 : 25) Wealsohavetheonlynonvanishingcomponentofthecurvatur e R abcd givenby (4.1.31): e 1 R = mn @ n m = @ m h e am @ n (e 1 e a n ) i : (8 : 1 : 26)

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8.1.IGL(1) 155 Thecovariantactioncorrespondingto(8.1.23a)is S Z d 2 x e 1 ( (+) 1 w r (+) w + ( ) 1 w r + ( ) w ) ; [ M ; ( ) w ]= w ( ) w ; (8 : 1 : 27) where ( ) w correspondsto ^ f and ( ) 1 w to = ^ f .Foropen-stringboundaryconditions, ( ) w arecombinedtoform ^ f (as,for w =1, P combinedtoform b P in (7.1.3a));forclosedstrings,the2functionscanbeusedin dependently(astheusual ( )modesforclosedstrings).Wethusseethatthespinisrelat edtotheweight(at leastforthesefree,classicalelds)as s = w for ( ) w .Theactioncorresponding to(8.1.23c)(neglectingthenormal-orderingconstant)is [8.7] S Z d 2 x e 1 h 1 2 2 +( w 1 2 ) R i : (8 : 1 : 28 a ) (Wehavedroppedsomesurfaceterms,asin(4.1.36).)Thefac tthat(8.1.28a)representsaparticlewithspincanbeseenin(atleast)2ways.One wayistoperforma dualitytransformation[8.8]:(8.1.28a)canbewrittenin rst-orderformas S Z d 2 x e 1 h 1 2 ( F a ) 2 + F a r a +( w 1 2 ) R i : (8 : 1 : 28 b ) (Notethat r istheeldstrengthfor undertheglobalinvariance + constant Inthatrespect,thelasttermin(8.1.28b)islikea\Chern-S imons"term,sinceitcan onlybewrittenastheproductof1eldwith1eldstrength,i ntermsoftheelds a and andtheireldstrengths R and r a .)Eliminating F byitsequationof motiongivesback(8.1.28a),whileeliminating gives S Z d 2 x e 1 1 2 ( G a ) 2 ; G a = ab r b ; [ M ; ]= w 1 2 : (8 : 1 : 28 c ) (Actually,since(8.1.28a)and(8.1.28c)areequivalenton shell,wecouldequallywell havestartedwith(8.1.28c)andavoidedthisdiscussionofd ualitytransformations. However,(8.1.28a)isalittlemoreconventional-looking, andtheonethatmorecommonlyappearsintheliterature.)TheunusualLorentztrans formationlawof follows fromthefactthatit'sthelogarithmofatensor: [ M ;e ]=( w 1 2 ) e ; [ M ;e ]=( 1 2 w ) e : (8 : 1 : 29) Thisisanalogousto(8.1.18),buttheweightsthereareincr easedby 1 2 byquantum eects.(Moreexamplesofthiseectwillbediscussedinsec t.9.1.)

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156 8.BRSTQUANTUMMECHANICS Anotherwaytoseethat hasaneectiveLorentzweight w istolookatthe relationshipbetweenLorentzweightsandweightsunder2Dg eneralcoordinatetransformations(or1D,or2Dconformaltransformations),asin( 8.1.5).Thisfollows fromthefactthatconformallyinvarianttheories,whencou pledtogravity,become locallyscaleinvarianttheories(evenwithoutintroducin gthescalecompensator of(4.1.34)).(Conversely,conformaltransformationscan bedenedasthesubgroup ofgeneralcoordinate+localscaletransformationswhichl eavesthevacuuminvariant.)Thismeansthatwecangauge-transformeto1,or,equiv alently,redenethe nongravitationaleldstocancelalldependenceone.Then e a m appearsonlyinthe unit-determinantcombination e a m =e 1 = 2 e a m .Theweights w thenappearinthe scaletransformationwhichleaves(8.1.27)invariant: w 0 = e w w : (8 : 1 : 30) (ThishasthesameformasalocalLorentztransformation,bu twithdierentrelative signsfortheeldsinthe( )termsof(8.1.27).Thisisrelatedtothefactthat,upon applyingtheequationsofmotion,andintheconformalgauge ,the2setsofeldsdependrespectivelyon ,andthereforehaveindependentconformaltransformation s onthese coordinates,exceptasrelatedbyboundaryconditionsfort heopenstring.) Choosing e =e 1 = 2 thenreplacesthe 'switheldswhicharescale-invariant,but transformundergeneralcoordinatetransformationsasden sitiesofweight w (i.e.,as tensorstimese w= 2 ).Intheconformalgauge,thesedensitiessatisfytheusual free (fromgravity)eldequations,sincethevielbeinhasbeene liminated(thedeterminantbyredenition,therestbychoiceofgeneralcoordinat egauge).Similarremarks applyto(8.1.28),butit'snotscaleinvariant.Toisolatet hescalenoninvarianceof thataction,ratherthanmaketheabovescaletransformatio n,wemakeanonlocal redenitionof in(8.1.28a)whichreducestotheabovetypeofscaletransfo rmation intheconformalgauge e a m = a m : ( w 1 2 ) 1 2 R: (8 : 1 : 31) Intheconformalgauge, R = 1 2 2 ln e.(Remember: islikethelogarithmofa tensor.)Underthisredenition,theactionbecomes S Z d 2 x e 1 1 2 2 1 2 ( w 1 2 ) 2 R 1 2 R : (8 : 1 : 32) (Notethat nowsatisestheusualscalareldequation.)Theredened eldisnow scaleinvariant,andthescalenoninvariancecannowbeattr ibutedtothesecondterm,

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8.2.OSp(1,1 j 2) 157 whichisthesamekindoftermresponsiblefortheconformal( Virasoro)anomaliesat thequantumlevel(i.e.,the1-loopcontributiontothe2De ldtheoryinabackground gravitationaleld).Infact,theconformallyinvariantac tion(4.1.39),withafactor proportionalto1 = ( D 2),isthedimensionallyregularizedexpressionresponsib lefor theanomaly:Although(4.1.39)isconformallyinvariantin arbitrary D ,subtraction ofthedivergent(i.e.,withacoecient1 = ( D 2)) R term,whichisconformally invariantonlyin D =2(asfollowsfromconsideringthe D 2limitof(4.1.39) withoutmultiplyingby1 = ( D 2)),leavesarenormalized(nite)actionwhich,inthe limit D 2,isjustthesecondtermof(8.1.32).Thus,thesecondtermi n(8.1.32) contributesclassicallytotheanomalyof(8.1.23b),there mainingcontributionbeing theusualquantumcontributionofthescalar.(Ontheotherh and,inthefermionic theoryfromwhich(8.1.28)canbederivedbyquantummechani calbosonization,all oftheanomalyisquantummechanical.) D< 26canalsobequantized(atleastatthetreelevel),butther eisananomaly in2Dlocalscaleinvariancewhichcauses det ( g mn )toreappearatthequantumlevel [8.9](or,inthelight-coneformalism,anextra\longitudi nal"Lorentzcomponentof X [1.3,4]);however,therearecomplicationsattheone-loop levelwhichhavenotyet beenresolved. Presentlythecovariantformulationofstringinteraction sisunderstoodonly withintheIGL(1)formalism(althoughinprincipleit'sstr aightforwardtoobtain theOSp(1,1 j 2)formalismbyeliminatingtheauxiliaryelds,asinsect. 4.2).These interactionswillbediscussedinsect.12.2.8.2.OSp(1,1 j 2) Wenextusethelight-conePoincarealgebraofthestring,o btainedinsect.7.1,to derivetheOSp(1,1 j 2)formulationasinsect.3.4,whichcanbeusedtondthegau geinvariantaction.Wethenrelatethistotherst-quantized IGL(1)oftheprevious sectionbythemethodsofsect.4.2.ThisOSp(1,1 j 2)formalismcanalsobederived fromrstquantizationsimplybytreatingthezero-modeof g as g fortheparticle (sect.5.1),andtheothermodesofthemetricasintheconfor malgauge(sect.6.2). Alloperatorscomedirectlyfromthelight-coneexpression sofsect.7.1,using thedimensionalextensionofsect.2.6,asdescribedintheg eneralcaseinsect.3.4. Inparticular,toevaluatetheactionanditsgaugeinvarian ceintheform(4.1.6), we'llneedtheexpressionsfor M M 2 ,and Q = M a p a + M m M givenbyusing

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158 8.BRSTQUANTUMMECHANICS i =( a; )in(7.1.14,15).Thus, M = i Z 0 ^ X b P = X n a y n ( a n ) ; M a = i Z 0 ^ X b P a = X n a y n [ a na ] ; M 2 = Z 0 ( b P i 2 2)=2 X na y n i a ni 1 =2( N 1) ; M m M = i 1 2 Z 0 ^ X b P j 2 = X a y n n y n a n ; n = i 1 p n 1 2 n 1 X m =1 q m ( n m ) a m i a n m;i 1 X m =1 q m ( n + m ) a y m i a n + m;i ; (8 : 2 : 1) whereagainthe i summationisoverboth a and ,representingmodescomingfrom boththephysical X a ( )(with x a identiedastheusualspacetimecoordinate)and theghostmodes X ( )(with x theghostcoordinatesofsect.3.4),asinthemode expansion(7.1.7). Tounderstandtherelationoftherst-quantizedBRSTquant ization[4.4,5]to thatderivedfromthelightcone(andfromtheOSp(1,1 j 2)),weshowtheSp(2)symmetryoftheghostcoordinates.Werstcombinealltheghost oscillatorsintoan \isospinor"[4.1]: C = 1 @=@ b C 0 ; b C : (8 : 2 : 2) Thisisospinordirectlycorresponds(exceptforlackofzer omodes)to ^ X ofthe OSp(1,1 j 2)formalismfromthelightcone:Weidentify ^ X =( x + p )+ C ; (8 : 2 : 3) andwecanthusdirectlyconstructobjectswhicharemanifes tlycovariantunderthe Sp(2)of M .Theperiodicinversederivativein(8.2.2)isdenedinter msofthe saw-toothfunction 1 @=@ f ( )= Z d 0 1 2 ( 0 ) 1 ( 0 ) f ( 0 ) ; (8 : 2 : 4) where ( )= 1for > 0.Theproductofthederivativewiththisinverse derivative,ineitherorder,issimplytheprojectionopera torwhichsubtractsout thezeromode.(Forexample, C + isjust b C minusitszeromode.)Alongwith b P a thiscompletestheidenticationofthenonzero-modesofth etwoformalisms.Wecan thenrewritetheotherrelevantoperators(8.1.10,12)inte rmsof C : p 2 + M 2 = Z d 2 ( b P a 2 + C 0 C 0 2) ;

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8.2.OSp(1,1 j 2) 159 Q = i 1 2 Z d 2 C ( b P a 2 + C 0 C 0 2) ; M = i 1 2 Z d 2 C ( C ) 0 : (8 : 2 : 5) Again,alldenitionsincludenormalordering.Thisrst-q uantizedIGL(1)canthen beseentoagreewiththatderivedfromOSp(1,1 j 2)in(8.2.1)byexpandinginzeromodes. Fortheclosedstring,theOSp(1,1 j 2)algebraisextendedtoanIOSp(1,1 j 2)algebra followingtheconstructionof(7.1.17):Asfortheopen-str ingcase,theD-dimensional indicesofthelight-coneformalismareextendedto(D+4)-d imensionalindices,but justthevalues A =( ; )arekeptfortheBRSTalgebra.Toobtaintheanalogof theIGL(1)formalism,weperformthetransformation(3.4.3 a)forbothleftandrighthandedmodes.TheextensionofIGL(1)toGL(1 j 1)analogoustothatofOSp(1,1 j 2) toIOSp(1,1 j 2)usesthesubalgebras( Q;J 3 ;p ;p ~ c = @=@c )oftheIOSp(1,1 j 2)'sofeach setofopen-stringoperators.Afterdroppingthetermscont aining @=@ ~ c x dropsout, andwecanset p + =1toobtain: Q ic 1 2 ( p a 2 + M 2 )+ M + i @ @c + Q + ; J 3 c @ @c + M 3 ; p 1 2 ( p a 2 + M 2 ) ; p ~ c @ @c : (8 : 2 : 6) ThesegeneratorshavethesamealgebraasN=2supersymmetry inonedimension, with Q and p ~ c correspondingtothetwosupersymmetrygenerators(actual lythe complexcombinationanditscomplexconjugate), J 3 totheO(2)generatorwhich scalesthem,and p the1Dmomentum.Theclosed-stringalgebraGL(1 j 1)isthen constructedinanalogytotheIOSp(1,1 j 2),takingsumsforthe J 'sanddierencesfor the p 's. Theapplicationofthisalgebratoobtainingthegauge-inva riantactionwillbe discussedinchapt.11.

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160 8.BRSTQUANTUMMECHANICS 8.3.Lorentzgauge WewillnextconsidertheOSp(1,1 j 2)algebrawhichcorrespondstorst-quantization intheLorentzgauge[3.7],asobtainedbythemethodsofsect .3.3whenappliedto theVirasoroalgebraofconstraints. From(3.3.2),fortheOSp(1,1 j 2)algebrawehave Q = i Z d 2 C ( 1 2 b P 2 1)+ 1 2 C ( C ) 0 i C Bi C + 1 2 ( C B 0 C 0 B ) i B + 1 4 C ( C 0 C ) 0 i B : (8 : 3 : 1) B isconjugatetothetime-componentsofthegaugeeld,which inthiscasemeans thecomponents g 00 and g 01 oftheunit-determinantpartoftheworld-sheetmetric (seechapt.6).Thisexpressioncanbesimpliedbytheunita rytransformation Q UQ U 1 with lnU = 1 2 Z ( 1 2 C C ) B 0 : (8 : 3 : 2) WethenhavetheOSp(1,1 j 2)(from(3.3.7)): J = Z C i 1 2 b P 2 + i + C 0 C + B 0 B B C ; J = Z C ( C ) ;J + =2 Z C B ; J + = Z 2 B B + C C : (8 : 3 : 3) Agauge-xedkineticoperatorforstringeldtheorywhichi sinvariantunderthe fullOSp(1,1 j 2)canbederived, K = 1 2 ( J ; J ; Z i B #) = Z 1 2 b P 2 1+ C 0 i C + B 0 i B = 1 2 ( p 2 + M 2 ) ; (8 : 3 : 4) asthezero-modeofthegenerators b G ( )from(3.3.10): b G ( )= 1 2 ( J ; J ; B #) = i ( 1 2 b P 2 1)+ C 0 C + C C 0 + B 0 B + B B 0 : (8 : 3 : 5) TheanalogintheusualOSp(1,1 j 2)formalismis 1 2 f J ; [ J ;p + 2 ] g = p A p A 2 p + p + p p = 2 M 2 : (8 : 3 : 6)

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8.3.Lorentzgauge 161 Thisdiersfromtheusuallight-conegauge-xedhamiltoni an p ,whichisnotOSp(1,1 j 2) invariant.UnliketheordinaryBRSTcase(butlikethelight -coneformalism),this operatorisinvertible,sinceit'softhestandardform K = 1 2 p 2 + .Thiswasmade possiblebytheappearanceofanevennumber(2)ofanticommu tingzero-modes.In ordinaryBRST(sect.4.4),thekineticoperatorisfermioni c: c ( 2 M 2 ) 2 @ @c M + is notinvertiblebecause M + isnotinvertible. Asusual,thepropagatorcanbeconvertedintoaformamenabl etorst-quantized path-integraltechniquesbyrstintroducingtheSchwinge rproper-timeparameter: 1 K = Z 1 0 de K ; (8 : 3 : 7) where isidentiedwiththe(Wick-rotated)world-sheettime.Att hefreelevel,the analysisofthispropagatorcorrespondstosolvingtheSchr odingerequationor,in theHeisenbergpicture(orclassicalmechanicsofthestrin g),tosolvingforthetime dependenceofthecoordinateswhichfollowfromtreating K asthehamiltonian: [ K;Z ]= iZ 0 ; Z [ K;Z ]=0 Z = Z ( + i )(8 : 3 : 8) for Z = P;C;B;=C;=B .Thusinthemodeexpansion Z = z 0 + P 11 ( z n y e in ( + i ) + z n e in ( + i ) )thepositive-energy z n y 'sarecreationoperatorswhilethenegative-energy z n 'sareannihilationoperators.(Rememberactivevs.passiv etransformations:Inthe Schrodingerpicturecoordinatesareconstantwhilestate shavetimedependence e tH ; intheHeisenbergpicturestatesareconstantwhilecoordin ateshavetimedependence e tH () e tH .) Whendoingstringeldtheory,inordertodenerealstring eldsweidentify complexconjugationoftheeldsasusualwithproper-timer eversalinthemechanics, which,inordertopreservehandednessintheworldsheet,me ansreversing aswell as .Asaresult,allreparametrization-covariantvariablesw ithanevennumberof 2Dvectorindicesareinterpretedasstring-eldcoordinat es,whilethosewithanodd numberaremomenta.(Seesect.8.1.)Thismeansthat X isacoordinate,while B and C aremomenta.Therefore,weshoulddenethestringeldas[ X ( ) ;G ( ) ;F ( )], where B = i=G and C = i=F .Thiseldisrealunderacombinedcomplex conjugationand\twist"( ),and Q isoddinthenumberoffunctionalplus derivatives.(Notethatthecorrespondingreplacementof B with G and C with F wouldnotberequiredifthe G i 'shadbeenassociatedwithaYang-Millssymmetry ratherthangeneralcoordinatetransformations,sinceint hatcase B and C carryno vectorindices.)

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162 8.BRSTQUANTUMMECHANICS ThisOSp(1,1 j 2)algebracanalsobederivedfromtheclassicalmechanicsa ction. The2Dgeneralcoordinatetransformations(6.1.2)generat edby = R d 2 m ( ) G m ( ) determinetheBRSTtransformationsby(3.3.2): Q X = C m @ m X; Q g mn = @ p ( C p g mn ) g p ( m @ p C n ) ; Q C m = 1 2 C n ( @ n C m ) C B m ; Q B m = 1 2 ( C n @ n B m B n @ n C m ) 1 8 h 2 C n ( @ n C p ) @ p C m + C n C p @ n @ p C m i : (8 : 3 : 9) Wethenredene e B m = B m 1 2 C n @ n C m Q C m = C n @ n C m C e B m ; Q e B m = C n @ n e B m : (8 : 3 : 10) TherestoftheOSp(1,1 j 2)followsfrom(3.3.7): J + ( X; g mn ;C m )=0 ;J + e B m =2 C m ; J ( X; g mn ; e B m )=0 ;J C m r = ( r C m ) ; J + ( X; g mn )=0 ;J + e B m =2 e B m ;J + C m = C m : (8 : 3 : 11) AnISp(2)-invariantgauge-xingtermis(droppingboundar yterms) L 1 = Q 2 1 2 pq g pq = pq h e B p @ m g qm + 1 2 g mn ( @ m C p )( @ n C q ) i ; (8 : 3 : 12) where istheratworld-sheetmetric.Thisexpressionistheanalog ofthegaugexingterm Q 2 1 2 A 2 forLorentzgaugesinYang-Mills[3.6,12].Variationof e B givesthe conditionforharmoniccoordinates.Theghostshavethesam eformoflagrangian as X ,butnotthesameboundaryconditions:Attheboundary,anyv ariablewith anevennumberof2Dvectorindiceswiththevalue1hasits -derivativevanish, whileanyvariablewithanoddnumbervanishesitself.These aretheonlyboundary conditionsconsistentwithPoincareandBRSTinvariance. Theyarepreservedbythe redenitionsbelow.

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8.3.Lorentzgauge 163 RatherthanthisLandau-typeharmonicgauge,wecanalsode nemoregeneral Lorentz-typegauges,suchasFermi-Feynman,byaddingto(8 .3.12)atermproportionalto Q 2 1 2 mn C m C n = 1 2 mn e B m e B n + .Wewillnotconsidersuchterms furtherhere. Althoughthehamiltonianquantummechanicalformof Q (8.3.3)alsofollows from(3.3.2)(withthefunctionalderivativesnowwithresp ecttofunctionsofjust insteadofboth and ),therelationtothelagrangianformfollowsonlyaftersom e redenitions,whichwenowderive.Thehamiltonianformtha tfollowsdirectlyfrom (8.3.9,10)canbeobtainedbyapplyingtheNoetherprocedur eto L = L 0 + L 1 :The BRSTcurrentis J m =( g np C m g m ( n C p ) ) 1 2 h ( @ n X ) ( @ p X )+( @ n C q )( @ p C q ) i e B n @ p ( g n [ m C p ] g mp C n ) ; (8 : 3 : 13) where(2D)vectorindiceshavebeenraisedandloweredwitht heratmetric.Canonicallyquantizing,with 1 0 P 0 = i X ; 1 0 e B m = i g 0m ; 1 0 m = i C m ; (8 : 3 : 14 a ) weapply X = 1 g 00 ( P 0 + g 01 X 0 ) ; C m = 1 g 00 ( m + g 01 C m 0 ) ; (8 : 3 : 14 b ) tothersttermin(8.3.13)and @ m g mn =0 ; g 0m @ m C n = n ; (8 : 3 : 14 c ) tothesecondtoobtain J 0 = 1 g 00 C 0 1 2 h ( P 0 2 + X 0 2 )+( m m + C m 0 C m 0 ) i + C 1 g 01 g 00 C 0 ( X 0 P 0 + C m 0 m ) e B m h m +( g m [ 0 C 1 ] ) 0 i ; (8 : 3 : 15) where Q R dJ 0 Bycomparisonwith(8.3.3),anobvioussimplicationistoa bsorbthe g factors intothe C 'sinthersttwoterms.Thisisequivalentto C m 1 m C 1 g 0m C 0 ; (8 : 3 : 16)

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164 8.BRSTQUANTUMMECHANICS andthecorrespondingredenitions(unitarytransformati on)ofand e B .Thisputs g -dependenceintothe e B terms, m e B m 0 1 g 00 e B 0 + 1 e B 1 g 01 g 00 e B 0 + ; (8 : 3 : 17) unlike(8.3.3),soweremoveitbythe g redenition g 01 = g 1 ; g 00 = q 1+2 g 0 +( g 1 ) 2 ; 1 0 B m = i g m : (8 : 3 : 18) Theseredenitionsgive J 0 = h C 0 1 2 ( P 0 2 + X 0 2 )+ C 1 X 0 P 0 i + n C 0 ( C 0 1 ) 0 + C 1 h C m 0 m +( C 0 0 ) 0 io n C 0 ( B 1 + g 1 B 0 ) 0 + C 1 h B 0 0 + g m B m 0 +( g 0 B 0 ) 0 io + C 0 h 1 2 C m 0 C m 0 +( g m C 0 ) 0 C m 0 i m B m : (8 : 3 : 19) Thequadraticterms CB 0 don'tappearin(8.3.3),andcanberemovedbytheunitary transformation Q 0 = UQ U 1 ;lnU = i 1 0 Z C 0 C 1 0 ; (8 : 3 : 20) giving J 0 = h C 0 1 2 ( P 0 2 + X 0 2 )+ C 1 X 0 P 0 i + n C 0 ( C 0 1 ) 0 + C 1 h C m 0 m +( C 0 0 ) 0 io n C 0 ( g 1 B 0 ) 0 + C 1 h g m B m 0 +( g 0 B 0 ) 0 io + C 0 ( g m C 0 ) 0 C m 0 m B m : (8 : 3 : 21) Finally,theremainingtermscanbexedbythetransformati on lnU = i 1 0 Z C 0 ( g 1 C 0 0 + g 0 C 1 0 )(8 : 3 : 22) toget(8.3.3),afterextending to[ ; ]bymakingthedenitions ^ P = 1 p 2 0 ( P 0 + X 0 ) ; C = 1 p 2 0 ( C 0 + C 1 ) ;i C = 1 p 2 0 ( 0 + 1 ) ;

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8.3.Lorentzgauge 165 G = 1 p 2 0 ( g 0 + g 1 ) ;i G = 1 p 2 0 ( B 0 + B 1 ) ; (8 : 3 : 23) wherethepreviouscoordinates,denedon[0 ; ],havebeenextendedto[ ; ]as Z ( )= Z ( ) ; (8 : 3 : 24) with\+"if Z hasanevennumberofvectorindiceswiththevalue1,and\ "foran oddnumber,inaccordancewiththeboundaryconditions,sot hatthenewcoordinates willbeperiodicin withperiod2 Todescribetheclosedstringwiththeworld-sheetmetric,w eagainuse2setsof open-stringoperators,asin(7.1.17),witheachsetofopen -stringoperatorsgivenas in(8.3.3,4),andthetranslationsarenow,intermsoftheze ro-modes b and c of B and C p + = s 2 i @ @b ; p = 1 p + i @ @c ; p = 1 p + K + p 2 : (8 : 3 : 25) TheOSp(1,1 j 2)subgroupoftheresultingIOSp(1,1 j 2),aftertheuseoftheconstraints p =0,reducestowhatwouldbeobtainedfromapplyingthegener alresult(3.3.2) toclosedstringmechanics.((3.2.6a),withoutthe B@=@ ~ C term,givestheusualBRST operatorwhenappliedtoclosed-stringmechanics,whichis thesameasthesumof twoopen-stringoneswiththetwosetsofphysicalzero-mode sidentied.) WenowputtheOSp(1,1 j 2)generatorsinaformanalogoustothosederived fromthelightcone[3.13].Let'srstconsidertheopenstri ng.Separatingoutthe dependenceonthezero-modes g and f intheOSp(1,1 j 2)operators, J = f ( @ @f ) + f M ; J + = 2 g @ @g f @ @f + f M + ; J + = 2 g @ @f + f M + ; J = f @ @g ig B + K @ @f + i C f + f M ; (8 : 3 : 26) wherethe f M 'sarethepartsofthe J 'snotcontainingthesezero-modes, K isgiven in(8.3.4),and B = Z BC 0 ;

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166 8.BRSTQUANTUMMECHANICS C = Z C C 0 = C : (8 : 3 : 27) From(3.3.8)and(8.3.26)wendthecommutatorsof f M AB B C ,and K ;the nonvanishingonesare: [ f M ; f M r ]= C ( r ( f M ) ) ; [ f M ; f M r ]= C r ( f M ) ; f f M ; f M + g = C f M + f M ; [ f M + ; f M ]= f M ; f f M ; f M g =2 iK C ; [ f M ; B r ]= C r ( B ) ; [ f M ; C r ]= C ( r ( C ) ) ; [ f M + ; B ]=3 B ; [ f M + ; C ]=2 C ; f f M + ; B g =2 C ; [ f M ; C r ]= C ( B r ) : (8 : 3 : 28) (Toshow f f M [ ; B ] g =0requiresexplicituseof(8.3.3),butitwon'tbeneeded below.) Wethenmaketheredenition(see(3.6.8)) g = 1 2 h 2 (8 : 3 : 29) asfortheparticle.Wenextredenethezero-modesasin(3.5 .2)byrstmakingthe unitarytransformation lnU =( lnh ) 1 2 @ @f ;f # f M + (8 : 3 : 30 a ) toredene h andthen lnU = f f M + (8 : 3 : 30 b ) toredene c .Thenetresultisthatthetransformedoperatorsare J + = @ @h h;J + = h @ @f ;J = f ( @ @f ) + f M ; J = f @ @h + 1 h @ @f ( K + f 2 ) f M f + b Q # ; (8 : 3 : 31 a ) where b Q = f M i 1 2 B + K f M + : (8 : 3 : 31 b )

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8.3.Lorentzgauge 167 Wealsohave p + = h;p = f ;p = 1 h ( K + f 2 ) : (8 : 3 : 32) Theseexpressionshavethecanonicalform(3.4.2a),withth eidentication f M $ M ; b Q $Q : (8 : 3 : 33) From(8.3.28)wethennd f b Q ; b Q g = 2 K f M ; (8 : 3 : 34) consistentwiththeidentication(8.3.33).ThustheIOSp( 1,1 j 2)algebra(8.3.31,32) takesthecanonicalformofchapt.3.Thisalsoallowstheclo sedstringformalismto beconstructed. Forexpandingtheelds,it'smoreconvenienttoexpandthec oordinatein real oscillators(preservingtherealityconditionofthestrin geld)as ^ P = p + 1 X 1 p n ( ia n y e in + ia n e in ) ; G = g + 1 X 1 p n ( g n y e in + g n e in ) ; B = b + 1 X 1 1 p n ( ib n y e in + ib n e in ) ; F = f + 1 X 1 p n ( f n y e in + f n e in ) ; C = c + 1 X 1 1 p n ( ic n y e in + ic n e in ) : (8 : 3 : 35) (Withourconventions,always z n y ( z n ) y ,andthus z n y = ( z n ) y .)Thecommutationrelationsarethen [ a m ;a n y ]=[ b m ;g n y ]=[ g m ;b n y ]= mn ; f c m ;f n y g = f f m ;c n y g = mn C ; (8 : 3 : 36) Wethenhave b Q = 1 X 1 h ( O n y c n c n y O n ) ( b n y f n f n y b n ) i ; O n = 1 p n e L n + 1 2 b n 2 g n K;

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168 8.BRSTQUANTUMMECHANICS K = e L 0 = 1 2 2 1+ 1 X 1 n ( a n y a n + b n y g n + g n y b n + c n y f n + f n y c n ) = 1 2 2 1+ N; e L n = p n@ a n + 1 X m =1 q m ( n + m )( a m y a n + m + b m y g n + m + c m y f n + m; + g m y b n + m + f m y c n + m; ) + n 1 X m =1 q m ( n m )( 1 2 a m a n m + b m g n m + c m f n m; ) : (8 : 3 : 37) TheLorentz-gaugeOSp(1,1 j 2)algebracanalsobederivedbythemethodofsect. 3.6.IntheGL(1)case,wegetthesamealgebraasabove,while intheU(1)caseweget adierentresultusingthesamecoordinates,suggestingas imilarrst-quantization withtheworld-sheetmetric.FortheGL(1)casewedenenewc oordinatesatthe GL(2 j 2)stageofreduction: X A =~ x A + e C A ;P A =~ p A + e C 0 A ; (8 : 3 : 38) where C isthegeneralizationof C of(8.2.3)toarbitraryindex.(Rememberthatin GL(2 j 2)alowerprimedindexcanbeconvertedintoanupperunprime dindex.) P A isthencanonicallyconjugateto X A .Wealsohaverelationsbetweenthecoordinates suchas e P A = P A ; e P A = p A + X 0 A ; f X A = p A + X A ; (8 : 3 : 39) where b P A =( e P A ; e P A 0 ),etc.However, f X A canbeexpressedintermsof P A only withaninverse -derivative.Asaresult,whenreexpressedintermsofthese new coordinates,ofalltheIOSp(2,2 j 4)generatorsonlytheIGL(2 j 2)oneshaveuseful expressions.Oftherest, ~ J AB isnonlocalin ,while ~ J AB and~ p A haveexplicitseparate termscontaining x A and p A .Explicitly,thelocalgeneratorsare p A p A ,and ~ J A B = i ( 1) AB x B p A i Z X A P B ; ~ J AB = ix [ A p B ) i Z X A X 0 B : (8 : 3 : 40) Intheseexpressionswealsousethelight-coneconstrainta ndgaugeconditionas translatedintothenewcoordinates: P = 1 2 p h b P a 2 2+2( p + + X 0 + ) P + +2( p + X 0 ) P i ;X =0 : (8 : 3 : 41)

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8.3.Lorentzgauge 169 Afterfollowingtheprocedureofsect.3.6(orsimplycompar ingexpressionstodeterminethe M ij ),thenal OSp (1 ; 1 j 2)generators(3.6.7c)are(droppingtheprimeson the J 's) J + =2 i Z X P + ;J + = i Z (2 X P + + X P ) ;J = i Z X ( P ) ; J = i Z [ X P + X ( 1 2 b P a 2 1+ X 0 P + + X 0 P )] : (8 : 3 : 42) Thisisjust(8.3.3),withtheidentication P + = G and X = C IntheU(1)casethereisnosuchredenitionpossible(which givesexpressions localin ).Thegeneratorsare J + = ix p + + i Z ^ X 0 b P + 0 ;J + = ix p + i Z ^ X 0 b P + 0 ; J = ix ( p ) i Z 0 ^ X b P i Z ^ X 0 b P 0 ; J = ix p + ix Z b P + i Z 0 ^ X b P i 1 p + p a Z 0 ^ X b P a p Z 0 ^ X b P i Z ^ X 0 b P 0 ; 2 p + b P = b P a 2 2+ b P b P + b P 0 b P 0 +2 b P + 0 b P 0 ; (8 : 3 : 43) where x A 0 = p A 0 =0.Afterperformingtheunitarytransformations(3.6.13) ,they become J + = ix p + ;J + = ix p + ;J = ix ( p ) i Z 0 ^ X b P i Z ^ X 0 b P 0 ; J = ix p + ix Z b P + i Z 0 ^ X b P i 1 p + p a Z 0 ^ X b P a p Z 0 ^ X b P + Z ^ X 0 b P 0 + Z ^ X 0 b P 0 + 1 2 Z ^ X + 0 b P 0 p a 2 2 p + Z 0 b P # : (8 : 3 : 44) Wecanstillinterpretthenewcoordinatesastheworld-shee tmetric,butdierent redenitionswouldbenecessarytoobtainthemfromthosein themechanicsaction, andthegaugechoicewillprobablydierwithrespecttothez ero-modes. Introducingextracoordinateshasalsobeenconsideredin[ 8.10],butinaway equivalenttousingbosonizedghosts,andrequiringimposi ng D =26byhandinstead

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170 8.BRSTQUANTUMMECHANICS ofderivingitfromunitarity.Adding4+4extradimensionst odescribebosonicstrings withenlargedBRSTalgebras OSp (1 ; 1 j 4)and OSp (2 ; 2 j 4)hasbeenconsideredby Aoyama[8.11]. Theuseofsuchextracoordinatesmayalsoproveusefulinthe studyofloopdiagrams:Inparticular,harmonic-typegaugescanbewell-de nedgloballyontheworld sheet(unlikeconformalgauges),andconsequentlycertain parameters(theworldsheetgeneralizationofproper-timeparametersforthepar ticle,(5.1.13))appearautomatically[8.12].Thissuggeststhatsuchcoordinatesma ybeusefulforclosedstring eldtheory(orsuperstringeldtheory,sect.7.3). Thegauge-invariantaction,itscomponentanalysis,andit scomparisonwiththat obtainedfromtheotherOSp(1,1 j 2)willbemadeinsect.11.2. Exercises (1)Provethattheoperators(8.1.9)satisfycommutationre lationslike(7.1.7a).Prove thattheyareconformallyinvariant. (2)Derive(8.1.20).Verifythattheusualfermionicantico mmutationrelationsfor (8.1.18)thenfollowfrom(8.1.19).Derive(8.1.22). (3)Derive(8.1.15).Derive(8.1.23bc).Prove Q 2 =0. (4)Derive(8.1.23ac)fromtheenergy-momentumtensorsof( 8.1.27,28ac). (5)Findthecommutationrelationsof 1 2 ^ Dd ^ D of(7.2.2),generalizing(8.1.2).Find theBRSToperator.Derivethegauge-xedVirasorooperator s,andshowthe conformalweightsof ^ is1/2,andofitsghostsare 1/2and+3/2.Use(8.1.23) toshowtheanomalycancelsfor D =10. (6)Findanalternaterst-orderformof(8.1.28b)byrewrit ingthelastterminterms of F and ,andshowhow(8.1.28ac)follow. (7)Showfromtheexplicitexpressionfor a that(8.1.27)canbemadevielbeinindependentbyeldredenitionintheconformalgauge. (8)ExplicitlyprovetheequivalenceoftheIGL(1)'sderive dinsect.8.1andfromthe lightcone,usingtheanalysisofsect.8.2. (9)Derive(8.3.1,3).Derive(8.3.9). (10)Derive(8.3.28).

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9.1.Externalelds 171 9.GRAPHS9.1.Externalelds OnewaytoderiveFeynmangraphsisbyconsideringapropagat orinanexternal eld: o o o o o o o o o o Forexample,forascalarparticleinanexternalscalareld L = 1 2 x 2 m 2 ( x ) (9 : 1 : 1 a ) [ p 2 + m 2 + ( x )] ( x )=0 (9 : 1 : 1 b ) propagator 1 p 2 + m 2 + ( x ) = 1 p 2 + m 2 1 p 2 + m 2 1 p 2 + m 2 + (9 : 1 : 1 c ) = + o o + o o o o + Forthe(open,bosonic)string,it'susefultousethe\one-h anded"versionof X ( ) (asin(7.1.7b))sothat ^ X and b P canbetreatedonan(almost)equalfooting,sowe willswitchtothatnotationhandandfoot.Thenthegenerali zationof(9.1.1b)(again jumpingdirectlytotherst-quantizedformforconvenienc e)is[7.5] D [ 1 2 b P 2 ( ) 1+ V ( )] E =0 : (9 : 1 : 2) (TheOSp(1,1 j 2)algebracanbesimilarlygeneralized.)However,forcons istencyof theseequationsofmotion,theymustsatisfythesamealgebr aas 1 2 b P 2 1((8.1.2)). (Ingeneral,thisisonlytrueincludingghostcontribution s,whichwewillignorefor

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172 9.GRAPHS theexamplesconsideredhere.)Expandingthisconditionor der-by-orderin V ,weget thenewrelations [ 1 2 b P 2 ( 1 ) 1 ; V ( 2 )]=2 i 0 ( 2 1 ) V ( 1 ) =2 i [ ( 2 1 ) V 0 ( 2 )+ 0 ( 2 1 ) V ( 2 )] ; (9 : 1 : 3 a ) [ V ( 1 ) ; V ( 2 )]=0 : (9 : 1 : 3 b ) Therstconditiongivestheconformaltransformationprop ertiesof V (ittransforms covariantlywithconformalweight1,like b P ),andthesecondconditionisoneof \locality".Asimpleexampleofsuchavertexisaphotoneld coupledtooneendof thestring: V ( )=2 ( ) A ( X (0)) b P; (9 : 1 : 4) where X (0)= ^ X (0).Graphsarenowgivenbyexpandingthepropagatorasthe inverseofthehamiltonian H = Z d 2 ( 1 2 b P 2 1+ V )= L 0 + Z d V L 0 + V: (9 : 1 : 5) Moregeneralverticescanbefoundwhennormalorderingisca refullytakeninto account[1.3,4],andonendsthat(9.1.3a)canbesatisedw hentheexternaleldis onshell.Forexample,considerthescalarvertex V ( )= 2 ( ) ( X (0)) : (9 : 1 : 6) Classically,scalareldshavethewrongconformalweight( zero): 1 2 h 1 2 b P 2 ( 1 ) 1 ; ^ X ( 2 ) i = i ( 2 1 ) @ @ 2 ^ X ( 2 ) ;(9 : 1 : 7 a ) butquantummechanicallytheyhaveweight\ 1 2 2 ": 1 2 h 1 2 b P 2 ( 1 ) 1 ; ^ X ( 2 ) i = i ( 2 1 ) @ @ 2 ^ X ( 2 ) + 1 2 @ 2 @ ^ X 2 ( 2 ) # i 0 ( 2 1 ) ^ X ( 2 ) : (9 : 1 : 7 b ) Therefore ^ X ( 2 ) ,andthus V of(9.1.6),satises(9.1.3a)if istheon-shellground state(tachyon): 1 2 2 =1 .(Similarremarksapplyquantummechanicallyfor themasslessnessofthephotoninthevertex(9.1.4).)

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9.1.Externalelds 173 AsanexampleofanS-matrixcalculation,considerastringi nanexternalplanewavetachyoneld,wheretheinitialandnalstatesofthest ringarealsotachyons: ( x )= e ik x : e ik X (0) :(9 : 1 : 8) Wethennd k 1 o o k 2 o o k N 1 k N = g N 2 h k N j V ( k N 1 ) ( p ) V ( k 3 )( p ) V ( k 2 ) j k 1 i = g N 2 D 0 e V ( k N 1 ) ( k 3 + k 4 + + k N ) e V ( k 2 ) 0 E ; V ( k )=: e ik X (0) := e V ( k ) e ik x ;X (0)= 1 X 1 1 p n ( a n y + a n ) ; ( p )= 1 1 2 p 2 +( N 1) ;N = 1 X 1 na n y a n ; (9 : 1 : 9) where g isthecouplingconstant,andwehavepulledthe x piecesoutofthe X 'sand pushedthemtotheright,causingshiftsintheargumentsoft he's(whichwere originally p ,themomentumoperatorconjugateto x ,nottobeconfusedwiththe constants k i ).WeuseSchwinger-likeparametrizations(5.1.13)forthe propagators: 1 1 2 p 2 +( N 1) = Z 1 0 dte t [ 1 2 p 2 +( N 1)] = Z 1 0 dx x x 1 2 p 2 +( N 1) ( x = e t ) ; (9 : 1 : 10) whereweuse t i for( k i + + k N ),asthedierenceinpropertimebetween e V ( k i 1 ) and e V ( k i ).Plugging(9.1.10)into(9.1.9),theamplitudeis g N 2 N 1 Y i =3 Z 1 0 dx i x i x i 1 2 ( k i + + k N ) 2 1 Y n 0 x 3 na y a e ik 2 1 p n a y 0 : (9 : 1 : 11) Toevaluatematrix-elementsofharmonicoscillatorsit'sg enerallyconvenienttouse coherentstates: j z i e za y j 0 i! a j z i = z j z i ;a y z E = @ @z z E ;e z 0 a y j z i = j z + z 0 i ;x a y a j z i = j xz i ; h z j z 0 i = e zz 0 ; 1= Z d 2 z e j z j 2 j z ih z j ;

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174 9.GRAPHS tr ( O )= Z d 2 z e j z j 2 h z jOj z i : (9 : 1 : 12) Using(9.1.12)andtheidentity Q 11 e cx n =n =(1 x ) c ,(9.1.11)becomes g N 2 Y Z dx i x i x i 1 2 ( k i + + k N ) 2 1 Y 2 i
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9.1.Externalelds 175 propertyiscalled\duality".(9.1.16c)showsthatthehigh -energybehaviorgoeslike s 1 2 t +1 insteadoftheusualxed-powerbehavior s j duetotheexchangeofaspin j particle,whichcanbeinterpretedastheexchangeofaparti clewitheectivespin j = 1 2 t +1.Thispropertyisknownas\Reggebehavior",and j ( t )= 1 2 t +1iscalled the\leadingReggetrajectory",whichnotonlydescribesth ehigh-energybehavior butalsothe(highest)spinatanygivenmasslevel(themassl evelsbeinggivenby integralvaluesof j ( t )). Insteadofusingoperatorstoevaluatepropagatorsinthepr esenceofexternal elds,wecanalsousetheotherapproachtoquantummechanic s,theFeynmanpath integralformalism.Inparticular,forthecalculationofp urelytachyonicamplitudes consideredabove,weevaluate(9.1.9)directlyintermsof V (ratherthan e V ),after using(9.1.10): g N 2 Z 1 0 dt 3 dt N 1 k N V ( k N 1 ) e t 3 ( 1 2 p 2 + N 1) V ( k 2 ) k 1 : (9 : 1 : 17) Using(9.1.14a),wecanrewritethisas g N 2 N 1 Y i =3 Z i +1 i 1 d i h k N j V ( k N 1 ; N 1 ) V ( k 3 ; 3 ) V ( k 2 ; 2 ) j k 1 i ; (9 : 1 : 18 a ) where V ( k; )=: e ik X (0 ; ) : ;X (0 ; )= e ( 1 2 p 2 + N 1) X (0) e ( 1 2 p 2 + N 1) ; (9 : 1 : 18 b ) isthevertexwhichhasbeen(proper-)time-translatedfrom 0to .(Rememberthatin theHeisenbergpictureoperatorshavetimedependence O ( t )= e tH O (0) e tH ,whereas intheSchrodingerpicturestateshavetimedependence j ( t ) i = e tH j (0) i ,sothat time-dependentmatrixelementsarethesameineitherpictu re.Thisisequivalentto therelationbetweenrst-andsecond-quantizedoperators .)Externalstatescanalso berepresentedintermsofvertices: j k i =lim !1 V ( k; ) e j 0 i ; h k j =lim !1 h 0 j e V ( k; ) : (9 : 1 : 19) Theamplitudecanthenberepresentedas,using(9.1.14c), g N 2 N 1 Y i =3 Z z i +1 z i 1 dz i lim z 1 !1 z N 0 ( z 1 ) 2 h 0 j V 0 ( k N ;z N ) V 0 ( k 1 ;z 1 ) j 0 i ; (9 : 1 : 20 a ) where V 0 ( k;z )= 1 z V ( k; ( z ))(9 : 1 : 20 b )

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176 9.GRAPHS accordingto(8.1.6),sinceverticeshaveweight w =1.Theamplitudewiththisformof theexternallinescanbeevaluatedbythesamemethodasthep reviouscalculation. (Infact,itdirectlycorrespondstothecalculationwith2e xtraexternallinesand vanishinginitialandnalmomenta.)However,beingavacuu mmatrixelement,it isofthesameformasthoseforwhichpathintegralsarecommo nlyusedineld theory.(Equivalently,itcanalsobeevaluatedbytheopera tormethodscommonly usedineldtheorybeforepathintegralmethodsbecamemore popularthere.)More detailswillbegiveninthefollowingsection,wheresuchme thodswillbegeneralized toarbitraryexternalstates. Couplingthesuperstringtoexternalsuper-Yang-Millsisa nalogoustothebosonic stringandsuperparticle[2.6]:Covariantize D D + ( ) P a P a + ( ) a n n + ( ) W .Assuming R d A askineticoperator(againignoringghosts), thevertexbecomes V = W D + a P a n (9 : 1 : 21) evaluatedat =0.Solvingtheconstraints(5.4.8)inaWess-Zuminogauge, wend W ; a A a +2 r a ; r a A a + 4 3 r a r ar r ; (9 : 1 : 22) evaluatedat =0,where\ "meansdroppingtermsinvolving x -derivativesofthe physicalelds A a and .Plugging(7.3.5)and(9.1.22)into(9.1.21)gives V A a b P a + r a b P a 1 6 ir a r ar r 0 : (9 : 1 : 23) Comparingwith(7.3.6),weseethatthevertices,inthisapp roximation,arethesame astheintegrandsofthesupersymmetrygenerators,evaluat edat =0.(Inthecase ofordinaryeldtheory,thevertices are justthesupersymmetrygenerators p a and q tothisorderin .)Exactexpressionscanbeobtainedbyexpansionofthesupe relds a ,and W in(9.1.21)toallordersin[7.6].Inpractice,supereldt echniques shouldbeusedevenintheexternaleldapproach,sosuchexp licitexpansion(oreven (9.1.22)and(9.1.23))isunnecessary.It'sinterestingto notethat,ifwegeneralize D P ,andntogauge-covariantderivatives r = D + r a = D a + a r =n + W with a ,and W nowfunctionsof ,describingthevectormultipletsofall masses,thenthefactthattheonlymodeofthe r 'smissingisthezero-modeofn ( R d n =0)directlycorrespondstothefactthattheonlygauge-inv ariantmode

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9.2.Trees 177 oftheconnectionsisthezero-modeof W (themasslessspinor,themassivespinors beingStueckelbergelds). Theexternaleldapproachhasalsobeenusedinthestringme chanicslagrangian methodtoderiveeldtheorylagrangians(ratherthanjustS -matrices)forthelower masslevels(tachyonsandmasslessparticles)[9.3,1.16]. Sincearbitraryexternalelds containarbitraryfunctionsofthecoordinates,thestring mechanicslagrangianisno longerfree,andloopcorrectionsgivetheeldtheorylagra ngianincludingeective termscorrespondingtoeliminatingthehigher-masseldsb ytheirclassicaleldequations.Thus,calculatingallmechanics-loopcorrectionsg ivesaneectiveeldtheory lagrangianwhoseS-matrixelementsarethetreegraphsofth estringeldtheorywith externallinescorrespondingtothelowermasslevels.Such eectivelagrangiansare usefulforstudyingtree-levelspontaneousbreakdownduet otheselower-masselds (vacuawheretheseeldsarenontrivial).Field-theory-lo opcorrectionscanbecalculatedbyconsideringmoregeneraltopologiesforthestri ng(mechanics-loopsare summedforonegiventopology).9.2.Trees Theexternaleldapproachislimitedbythefactthatittrea tsordinaryelds individuallyinsteadoftreatingthestringeldasawhole. Inordertotreatgeneral stringelds,astringgraphcanbetreatedasjustapropagat orwithfunnytopology: Forexample, @ @ @ @ @ @ & $ % @ @ @ @ @ @ canbeconsideredasapropagatorwheretheinitialandnal\ one-string"statesjust happentobedisconnected.Theholesintheworldsheetrepre sentloops.Whengroup

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178 9.GRAPHS theoryindicesareassociatedwiththeendsofthelines,the valuesoftheindicesare requiredtobethesamealongtheentireline,whichcorrespo ndstotracinginthematricesassociatedwiththestringstates.(Theendsofthest ringscanbeinterpretedas \quarks"whichcarrythe\ravor"quantumnumbers,boundbya stringof\gluons" whichcarryonly\color"canceledbythatofthequarks.)Suc hanapproachislimited toperturbationtheory,sincethestringisnecessarilygau ge-xed,andanyonegraph hasaxednumberofexternallinesandloops,i.e.,axedtop ology.Theadvantageofthisapproachtographsisthattheycanbeevaluatedb yrst-quantization, analogouslytothefreetheory.(Eventhesecond-quantized couplingconstantcanbe includedintherst-quantizedformalismbynotingthatthe powerofthecoupling constantwhichappearsinagraph,uptowavefunctionnormal izations,isjustthe Eulernumber,andthenaddingthecorrespondingcurvaturei ntegraltothemechanics action.) Werstconsiderthelight-coneformalism.WeWickrotateth epropertime i (seesects.2.5-6),sonowconformaltransformationsarear bitraryreparametrizationsof = + i (andthecomplexconjugatetransformationon )insteadof + (andof independently:see(6.2.7)),sincethemetricisnow d 2 = dd Therearethreepartstothegraphcalculation:(1)expressi ngtheS-matrixinterms oftheGreenfunctionforthe2DLaplaceequation,(2)nding anexplicitexpression fortheGreenfunctionforthe2Dsurfaceforthatparticular graph,byconformally transformingthe planetotheupper-halfcomplex( z )planewheretheGreenfunctiontakesasimpleform,and(3)ndingthemeasureforthein tegrationoverthe positionsoftheinteractionpoints. Therststepiseasy,andcanbedoneusingfunctionalintegr ation[9.4,1.4]or solvingfunctionaldierentialequations(thestringanal ogofFeynmanpathintegrals andtheSchrodingerequation,respectively).Sinceallbu tthezero-mode(theusual spacetimecoordinate)ofthefreestringisdescribedbyani nnitesetofharmonic oscillators,themostconvenientbasisisthe\number"basi s,wherethenonzero-modes arerepresentedintermsofcreationandannihilationopera tors.Thebasicideaisthen torepresentS-matrixelementsas A = h ext j V i = h ext j e j 0 i ; (9 : 2 : 1) where j V i representstheinteractionand h ext j representsallthestates(initialand nal)oftheexternalstrings,intheinteractionpicture.T hisissortofaspacetime symmetricversionoftheusualpicture,whereaninitialsta tepropagatesintoanal

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9.2.Trees 179 state:Instead,thevacuumpropagatesintoan\external"st ate.Theexponential e isthentheanalogoftheS-matrix exp ( H INT t ),whichpropagatesthevacuumat time0toexternalstatesattime t .Itthusconvertsannihilationoperatorsonitsleft (external,\out"states)intocreationoperators(forthe\ in"state,thevacuum,at \time" x + =0).itselfisthenthe\connected"S-matrix:Inthisrstquantizedpicture,whichlookslikeafree2Dtheoryinaspacewithfunnyge ometry,itcorresponds directlytothefreepropagatorinthisspace.Sinceweworki ntheinteractionpicture, wesubtractouttermscorrespondingtopropagationinan\or dinary"geometry. Intheformerapproach,theamplitudecanbeevaluatedasthe Feynmanpath integral A = Z 0@ N 1 Y i =3 d i 1A Z D X i ( ; ) Y r Z D P r ( )[ P r ] e i 1 0 R d 2 P r ( ) X r ( ; 1 ) # e P p r 1 r 1 0 R d 2 2 h 1 2 ( X 2 + X 0 2 )+ constant i ; (9.2.2) correspondingtothepicture(e.g.,for N =5) 1 1 2 3 4 wherethe 'shavebeenWick-rotated, 1 r istheendofthe r thstring(tobetaken to 1 ),the p 1 factoramputatesexternallines(convertsfromtheSchrod inger picturetotheinteractionpicture),thefactorinlargebra cketsistheexternal-linewave function[ X ](or [ X ]foroutgoingstates)writtenasaFouriertransform,andth e constantcorrespondstotheusualnormal-orderingconstan tinthefreehamiltonian.

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180 9.GRAPHS Forexplicitness,wehavewrittentheintegrationoverinte ractionpoints( Q d )for thesimplecaseofopen-stringtreegraphs.Planargraphsal waysappearasrectangles duetothestringlengthsbeing2 0 p + ,whichisconserved.Thefunctionalintegral (9.2.2)isgaussian,so,makingthedenition J ( ; )= i ( 1 ) 1 p 0 P ( ) ; (9 : 2 : 3) wend Z d 2 2 1 0 1 2 ( @X ) 2 + 1 p 0 JX # 1 2 Z d 2 2 d 2 0 2 J ( ) G ( ; 0 ) J ( 0 ) ; @ 2 G ( ; 0 )=2 2 ( 0 ) ; @ @n G ( ; 0 )= f ( ) Z d 2 J X p =0 ; G = X (2 m 0 )(2 n 0 ) G rs mn cos m r p + r cos n s 0 p + s e m r =p + r + n s 0 =p + s + G free +( zero mode ) 2 terms A = Z ( Y d i ) V ( ) D e 0 E ; = 1 4 X G rs mn r m s n ; (9 : 2 : 4) where G isthe2DGreenfunctionforthekineticoperator(laplacian ) @ 2 =@ 2 + @ 2 =@ 2 ofthatparticularsurface,and V ( )comesfrom det ( G ).WehaveusedNeumann boundaryconditions(correspondingto(6.1.5)),wherethe ambiguitycontainedin thearbitraryfunction f (necessaryingeneraltoallowasolution)isharmlessbecau se oftheconservationofthecurrent J (i.e.,themomentum p ).The G free termis droppedinconvertingtotheinteractionpicture:Infuncti onalnotation(see(9.2.2)), itproducestheground-statewavefunction.The(zero-mode ) 2 termsaredueto boundaryconditionsat 1 ,andappearwhenthemaptotheupper-halfplaneischosen sothattheendofonestringgoesto 1 ,givingadivergence.Theycorrespondtothe factor1 =z N inthesimilarmapusedfor(9.1.20a).Thefactors(2 m 0 ),whichdon't appearinthenaiveGreenfunction,aretocorrectforthefac tthatthegureabove isnotquitethecorrectone:Theboundariesoftheinitialan dnalstringsshould notgoto 1 beforethesourceterms(9.2.3)(i.e.,thewavefunctions)d o,becauseof theboundaryconditions.Thenetresultisthatnonzero-mod esappearwithanextra factorof2duetorerectionsfromtheboundary.However,the serelativefactorsof2 arecanceledinthe -integration,since R 0 d 2 cos 2 m = 1 4 (1+ m 0 ).Explicitly,after transformingthepartofthe planecorrespondingtothestringtothewholeofthe upper-half z plane,theGreenfunctionbecomes G ( z;z 0 )= ln j z z 0 j + ln j z z 0 j : (9 : 2 : 5)

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9.2.Trees 181 ThersttermistheusualGreenfunctionwithoutboundaries ,whoseuseinthe z plane(andnotjustthe plane)followsfromthefactthattheLaplaceequationis conformallyinvariant.Thesecondterm,whichsatisesthe homogeneousLaplace equationintheupper-halfplane,hasbeenaddedaccordingt othemethodofimages inordertosatisfytheboundaryconditionsattherealaxis, andgivesthererections whichcontributethefactorsof2. Inthelatter(Schrodingerequation/operator)approach, itallboilsdowntousing thegeneralexpression ( z )= I z dz 0 2 i 1 z 0 z ( z 0 )= X r I z r dz 0 2 i 1 z 0 z ( z 0 ) ; (9 : 2 : 6) where z r arethepointsinthe z -planerepresentingtheendsofthestrings(at = 1 ). ( z )isanarbitraryoperatorwhichhasbeenconformallytransf ormedtothe z plane: r ( z )= @ @z w ~ r ( ) ; ~ r ( )=( p + r ) w r ~ r p + r ; r ( )= 1 X 1 rn e n ; (9 : 2 : 7 a ) ~ r = i r 1 X s =1 p + s ; (9 : 2 : 7 b ) with( i )= ^ ( )intermsof b P ( )(sothe n 'sarethe n 'sof(7.1.7a)),and theconformaltransformations(9.2.7a)(cf.(8.1.6))ared eterminedbytheconformal weights w (=1for P ).The z mapisthemapfromtheaboveguretothe upper-halfplane.The mapisthemapfromthefree-string 2 [0 ; ]tothe r th interacting-string 2 [ P r 1 s =1 p + s ; P rs =1 p + s ].All integralsfrom to become contourintegralsinthe z plane.(Theupper-half z planecorrespondsto 2 [0 ; ], whilethelowerhalfis 2 [ ; 0].)Sincethestring(includingitsextensionto[ ; 0]) ismappedtotheentire z planewithoutboundaries,(9.2.6)getscontributionsfrom onlytheends,representedby z r .Weworkdirectlywith b P ( ),ratherthan X ( ), since b P dependsonlyon ,while X dependsonboth and .( ^ X hasacutinthe z plane,since p isn'tperiodicin .)Theopenstringresultscanalsobeapplied directlytotheclosedstring,whichhasseparateoperators whichdependon z instead of z (i.e.,+and modes,inthenotationofsect.6.2).Actually,thereisabit ofa cheat,since b P ( )doesn'tcontainthezero-mode x .However,thiszero-modeneeds specialcareinanymethod:Extrafactorsof2appearedinthe path-integralapproach becauseoftheboundaryconditionsalongthereal z -axisandatinnity.Infact,we'll seethatthelostzero-modetermscanbefoundfromthesameca lculationgenerally

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182 9.GRAPHS usedinbothoperatorandpathintegralapproachestondthe integrationmeasure ofstep3above,andthusrequiresnoextraeort. Wethereforelookdirectlyforapropagator e thatgives ( z )= e r ( z ) e = r ( z )+[ ; r ( z )]+ ; (9 : 2 : 8) where r correspondstoafreein-eldforthe r thstringintheinteractionpicture, and totheinteractingeld.Todothis,werstnda ~ forwhich [ ~ ; r ]= : (9 : 2 : 9) Wenextsubtractoutthefreepartof ~ (external-lineamputation): ~ =+ free ; [ free ; r ]= r : (9 : 2 : 10) Thisgivesawhichisquadraticinoperators,butcontainsn oannihilationoperators (whichareirrelevantanyway,sincethe j 0 i willkillthem).Asaresult,thereareno termswithmultiplecommutatorsintheexpansionoftheexpo nential.Wetherefore obtain(9.2.8).Whenwesubtractoutfreepartsbelow,wewil lincludethepartsof theexternal-lineamputationwhichcompensateforthefact that z and z r arenotat thesametime. Werstconsiderapplyingthismethodtooperatorsofarbitr aryconformalweight, asin(8.1.23).Thedesiredformofwhichgives(9.2.8)forb oth f and =f is 0 = X r;s I z r dz 2 i I z s dz 0 2 i 1 z z 0 f r ( z ) f s ( z 0 ) free stringterms; (9 : 2 : 11 a ) where" f r ( z 1 ) ; f s ( z 2 ) ) =2 i rs ( z 2 z 1 ) $ ^ f r ( 1 ) ; ^ f s ( 2 ) ) =2 rs ( 2 1 ) ; (9 : 2 : 11 b ) asfollowsfromthefactthattheconformaltransformations preservethecommutation relationsof f and =f .Theintegrationcontoursareorientedsothat I r d 2 ip + r = Z p + r p + r d 2 p + r =1 : (9 : 2 : 12) (9.2.11)caneasilybeshowntosatisfy(9.2.8).Thevalueof r ( !1 )forthe integrationcontourisxed,sothe in inthesecommutationrelationsisreallyjust a in ofthatcontour.Thefree-stringtermsaresubtractedasexp lainedabove.

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9.2.Trees 183 (Infact,theyarepoorlydened,sincetheintegrationcont oursfor r = s fallontop ofeachother.) Unfortunately,willprovediculttoevaluateinthisform .Wethereforerewrite itbyexpressingthe1 = ( z z 0 )asthederivativewithrespecttoeither z or z 0 ofa ln andperformanintegrationbyparts.Thenetresultcanbewri ttenas ( 1 ; 2 )= X r;s I z r dz 2 i I z s dz 0 2 i 0 ln ( z z 0 ) 1 r ( z ) 2 s ( z 0 ) free stringterms +( zero mode ) 2 terms; (9 : 2 : 13 a ) [ 2 r ( z 2 ) ; 1 s ( z 1 ) g = 2 i rs 0 ( z 2 z 1 ) ; (9 : 2 : 13 b ) The 0 onthecontourintegrationisbecausetheintegrationispoo rlydeneddue tothecutforthe ln :Wethereforedeneitbyintegrationbypartswithrespectt o either z or z 0 ,droppingsurfaceterms.Thisalsokillstheconstantparto fthe ln which contributesthe(zero-mode) 2 terms,whichwethereforeaddbackin.Actually,(9.2.11) hasno(zero-mode) 2 terms,butinthecase 1 = 2 = P ,thesetermsdetermine theevolutionofthezero-mode x ,whichdoesn'tappearin b P ,andthuscouldnotbe determinedby(9.2.8)anyway.( x doesappearin X and ^ X ,butthey'relessconvenient toworkwith,asexplainedabove.)These(quadratic-in-)ze ro-modecontributionsare mosteasilycalculatedseparatelybyconsideringthecasew henallexternalstatesare groundstatesofnonvanishingmomentum(seebelow).Inorde rforthecommutation relations(9.2.13b)tobepreservedbytheconformaltransf ormations,it'snecessary thattheconformalweights w 1 and w 2 of 1 and 2 bothbe1.Inthatcase, can bereplacedwith ~ in(9.2.13a)whilereplacing dz with d .However,oneimportant useofthisequationisfortheevaluationofvertices(S-mat riceswithnointernal propagators).Sincetheseverticesarejust functionalsinthestringeldcoordinates (seebelow),and functionalsareindependentofconformalweightexceptfor the transformation(sincethattransformationappearsexplic itlyintheargument ofthe functionals),wecanwritethisresult,forthecasesofS-ma triceswith w 1 = w 2 =1orvertices(with w 1 + w 2 =2)as ( 1 ; 2 )= X r;s I r d 2 i I s d 0 2 i 0 ln ( z z 0 ) 1 r ( ) 2 s ( 0 ) free stringterms +( zero mode ) 2 terms; (9 : 2 : 14 a ) [ 2 r ( 2 ) ; 1 s ( 1 ) g = 2 i rs 0 ( 2 1 ) ; (9 : 2 : 14 b ) wheretheappropriatefor X ( w 1 = w 2 =1)is = 1 2 ( P; P ) : (9 : 2 : 15)

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184 9.GRAPHS These 0 andfree-stringcorrectionsmayseemawkward,buttheywill automatically bexedbythesamemethodwhichgivesasimpleevaluationoft hecontourintegrals: i.e.,thetermswhicharediculttoevaluateareexactlytho sewhichwedon'twant. (Fornon-vertexS-matriceswith w 1 6 =1 6 = w 2 ,(9.2.11)or(9.2.13)canbeused,but theirevaluatedformsaremuchmorecomplicatedinthegener alcase.)Ingeneral (forcovariantquantization,supersymmetry,etc.)wealso needextrafactorswhich areevaluatedatinnitesimalseparationfromtheinteract ion(splitting)points,which followfromapplyingtheconformaltransformation(9.2.7) ,and(9.2.6)with z = z INT Onlycreationoperatorscontribute. Thecontour-integralform(9.2.14)canalsobederivedfrom thepath-integralform (9.2.4).Fortheopenstring[9.5],thesecontourintegrals canbeobtainedbyeither combiningintegralsoversemicirclesintheupper-half z plane( integralsfrom0to )withtheirrerections[9.6],ormoredirectlybyreformula tingtheopenstringas aclosedstringwithmodesofonehandednessonly,andwithin teractionsassociated withjustthepoints =0 ; ratherthanall Forthesecondstep, G rs mn unfortunatelyishardtocalculateingeneral.For open-stringtreegraphs,weperformthefollowingconforma lmappingtotheupperhalfcomplexplane[9.4],whereiseasytocalculate: = N X r =2 p + r ln ( z z r ) : (9 : 2 : 16) Theboundaryofthe(interacting)stringisnowthereal z axis,andtheinterioris theupperhalfofthecomplex z -plane.(Thebranchesinthe ln 'sin(9.2.16)arethus chosentorundownintothelower-halfplane.Whenweusethew holeplanefor contourintegralsbelow,we'llavoidintegralswithcontou rswithcutsinsidethem.) Asaresult,operatorssuchas b P ,whichwereperiodicin ,arenowmeromorphic at z r ,sothecontourintegralsareeasytoevaluate.Also,extend ing from[0 ; ] to[ ; ]extendstheupper-halfplanetothewholecomplexplane,so thereareno boundaryconditionstoworryabout.Toevaluate(9.2.14),w enotethat,sinceweare neglecting(zero-modes) 2 andfreestringterms( r = s m = n ),wecanreplace ln ( z z 0 ) e m=p + r n 0 =p + s 1 m p + r + n p + s @ @ + @ @ 0 ln ( z z 0 ) # e m=p + r n 0 =p + s (9 : 2 : 17) byintegrationbyparts.Wethenusetheidentity,forthecas eof(9.2.16), @ @ + @ @ 0 ln ( z z 0 )= N X r =2 p + r @ @ ln ( z z r ) #" @ @ 0 ln ( z 0 z r ) # : (9 : 2 : 18)

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9.2.Trees 185 Thenwecantriviallychangecompletelyto z coordinatesbyusing d@=@ = dz@=@z andconvertingthe exponentialsintoproductsofpowersof z monomials.Dierentiatingthe ln 'sgives(productsof)single-variablecontourintegralsw hichcaneasily beevaluatedasmultiplederivatives:= X rsmn 1 rm 2 sn ( p + r ) 1 w 1 ( p + s ) 1 w 2 1 np + r + mp + s N X t =2 p + t A rtm A stn +( zero mode ) 2 ; A rtm = I z r dz 2 i 1 z z t 24 ( z z r ) r 1 Y s =2 ( z s z ) p + s =p + r N Y s = r +1 ( z z s ) p + s =p + r 35 m : (9 : 2 : 19) Forthethirdstep,foropen-stringtrees,wealsoneedtheJa cobianfrom Q d i ( Q dz i ) V ( z ),whichfortreescaneasilybecalculatedbyconsideringth egraphwhere allexternalstatesaretachyonsandallbut2strings(onein comingandoneoutgoing) haveinnitesimallength.Wecanalsorestrictalltransver semomentatovanish,and determinedependenceonthemattheendofthecalculationby therequirementof Lorentzcovariance.(Alternatively,wecouldcomplicatet hecalculationbyincluding transversemomenta,andgetacalculationmoresimilartoth atof(9.1.9).)Wethen havetheamplitude(fromnonrelativistic-stylequantumme chanicalarguments,or specializing(9.2.2)) A = g N 2 f ( p + r ) Z N 1 Y i =3 d i e P N 1 r =2 p r r ; (9 : 2 : 20) where f isafunctiontobedeterminedbyLorentzcovariance,the 'saretheinteractionpoints,thestrings1and N arethosewhoselengthisnotinnitesimal,and wealsochoose z 1 = 1 z N =0inthetransformation(9.2.16).Wethensolvefor r (= Re ( r )),intermsof z r (inthisapproximationofinnitesimallengthsforallbut 2ofthestrings),asthenitevaluesof wheretheboundary\turnsaround": @ @z r =0 r = p + N lnz r + p + r ln p + r p + N z r 1 # + N 1 X s =2 ;s 6 = r p + s ln ( z r z s )+ O 24 p + r p + N 2 35 : (9 : 2 : 21) Wethennd,usingthemass-shellcondition p =1 =p + forthetachyon( p is H in nonrelativistic-stylecalculations) A = g N 2 fp + N 1 N 1 Y r =2 p + r e !# Z N 1 Y i =3 dz i z 2 N Y s>r =2 ( z r z s ) p + r =p + s + p + s =p + r

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186 9.GRAPHS e P p ir n rs ( p + ;z ) p is ; (9 : 2 : 22 a ) wherewehavenowincludedthetransverse-momentumfactor n rs ,whoseexponential formfollowsfrompreviousarguments.Itsexplicitvalue,a swellasthatof f ,can nowbedeterminedbythemanifestcovarianceofthetachyoni camplitude.However, (9.2.22a)isalsothecorrectmeasureforthe z -integrationtobeappliedto(9.2.1)(or (9.2.4)),usingfrom(9.2.19)(whichisexpressedinterms ofthesametransformation (9.2.16)).AtthispointwecanseethatLorentzcovarianced etermines f tobesuch thatthefactorinbracketsisaconstant.Wethennotethat,u sing p = H = ( 1 2 p i 2 + N 1) =p + ,wehave p r p s = p ir p is [( p + r =p + s )( 1 2 p is 2 + N 1)+ r $ s ]. Thisdeterminesthechoiceof n rs whichmakestheamplitudemanifestlycovariantfor tachyons: Z dzV ( z )= g N 2 Z N 1 Y i =3 dz i z 2 N Y s>r =2 ( z r z s ) p r p s ( p + r =p + s )( 1 2 p s 2 1) ( p + s =p + r )( 1 2 p r 2 1) : (9 : 2 : 22 b ) Notethat p dependencecancels,so p r p s and p r 2 canbechosentobethecovariant ones.Taking N =1tocomparewiththetachyonicparticletheory,weseethis agrees withtheresult(9.1.15)(afterchoosingalso z 2 =1).Italsogivesthe(zero-mode) 2 termswhichwereomittedinourevaluationof.(Thatis,weh avedeterminedboth ofthesefactorsbyconsideringthisspecialcase.)Forthec aseofthetachyon,we couldhaveobtainedthecovariantresult(9.2.22b)moredir ectlybyusingcovariant amputationfactors,i.e.,byusing p asanindependentmomentuminsteadofasthe hamiltonian(seesect.2.5).However,theresultlosesitsm anifestcovariance,even onshell,forexcitedstatesbecauseoftheusual1 =p + interactionswhichresultinthe light-coneformalismaftereliminatingauxiliaryelds. Asmentionedinsect.8.1,thereisanSp(2)invarianceoffre estringtheory.In termsofthetreegraphs,whichwerecalculatedbyperformin gaconformalmaptothe upper-halfcomplexplane,itcorrespondstothefactthatth isisthesubgroupofthe conformalgroupwhichtakestheupper-halfcomplexplaneto itself.Explicitly,the transformationis z ( m 11 z + m 12 ) = ( m 21 z + m 22 ),wherethematrix m ij isreal,and withoutlossofgeneralitycanbechosentohavedeterminant 1.Thistransformation alsotakesthereallinetoitself,andwhencombinedwith(9. 2.16)modiesitonly byaddingaconstantandchangingthevaluesofthe z r (butnottheirorder).In particular,the3arbitraryparametersallowarbitraryval ues(subjecttoordering)for z 1 z 2 ,and z N ,whichwere 1 ,1,and0.(Thisaddsatermfor z 1 to(9.2.16)whichwas previouslydroppedasaninniteconstant. !1 as z !1 in(9.2.16)corresponds

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9.2.Trees 187 totheendoftherststring.)BecauseoftheSp(2)invarianc e,(9.2.22)canbe rewritteninaformwithall z 'streatedsymmetrically:Thetachyonicamplitudeis then A = g N 2 Z Q Nr =1 dz r dz i dz j dz k ( z i z j )( z i z k )( z j z k ) Y 1 r
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188 9.GRAPHS For(9.2.19),weusetheintegral I 0 dz 2 i z n 1 ( z +1) u = 1 n d n dz n ( z +1) u j z =0 = u ( u 1) ( u n +1) n = u n ; u + n 1 n =( 1) n u n (9 : 2 : 25) toevaluate m> 0: A r 2 m = p +3 N rm ;A r 3 m = p +2 N rm ; N rm = 1 p + ;r +1 m p + ;r +1 p + r m ; m =0: A rt 0 = rt r 1 : (9 : 2 : 26) Theresultisthen(see,e.g.,[9.4]): ( 1 ; 2 )= 1 N 2 e 1 N 1 n 2 1 1 n N e 2 0 X p 2 + M 2 2 p + ; N rsmn = p +1 p +2 p +3 np + r + mp + s N rm N sn ; e = p +[ r ( 0 ) r +1] ; S = Z d 3 p + d 3 X p + X 0 D 1 2 3 e 0 E : (9 : 2 : 27) (Insomeplaceswehaveusedmatrixnotationwithindices r;s =1 ; 2 ; 3and m;n = 1 ; 2 ;:::; 1 implicit.)The 'sincludethe p 's.Forsimplicity,wehaveassumedthe 'shave w =1;otherwise,each shouldbereplacedwith p + 1 w .The 0 term comesfromshiftingthevalueof atwhichthevertexisevaluatedfrom =0tothe interactiontime = 0 (itgivesjustthepropagatorfactor e 0 P H r ,where H r isthe freehamiltonianoneachstring).Inmoregeneralcaseswe'l lalsoneedtoevaluate aregularized b atthesplittingpoint,whichisalsoexpressedintermsoft hemode expansionof ln ( z z r )(actuallyitsderivative1 = ( z z r ))whichwasusedin(9.2.14) toobtain(9.2.27): ( z 0 ) 1 p p +1 p +2 p +3 e + p p +1 p +2 p +3 p + w N : (9 : 2 : 28) (Again,matrixnotationisusedinthesecondterm.)Wehavea rbitrarilychosena convenientnormalizationfactorintheregularization.(A factorwhichdivergesas z z 0 mustbedividedoutanyway.)Thevertexiscyclicallysymmet ricinthe 3strings(eventhoughsomestringshave p + < 0).Besidestheconservationlaw P p =0,wealsohave P p + x =0,whichisactuallytheconservationlawforangular momentum J + i .Thesearespecialcasesofthe 0 conservationlawindicatedaboveby the function,afterincludingthe p + 1 w .(Rememberthatacoordinateofweight w is

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9.2.Trees 189 conjugatetoonewithweight1 w .)Thisconservationlawmakesthedenitionof ~ above r -independent.However,suchconservationlawsmaybeviola tedbyadditional vertexfactors(9.2.28). The3-stringvertexfortheclosedstringinoperatorformis essentiallyjustthe productofopen-stringverticesfortheclockwiseandcount erclockwisemodes,since the functionalscanbewrittenassuchaproduct,exceptforthez eromodes.However,whereasopenstringsmustjoinattheirends,closedst ringsmayjoinanywhere, andthe parametrizingthisjoiningisthenintegratedover.Equiva lently,thevertex mayincludeprojectionoperators N; 0 = R d 2 e i N whichperforma translation equivalenttotheintegration.(Theformerinterpretation ismoreconvenientfora rst-quantizedapproach,whereasthelatterismoreconven ientintheoperatorformalism.)Theseprojectionoperatorsareredundantina\Lan daugauge,"wherethe residual + constant gaugeinvarianceisxedbyintroducingsuchprojectors intothepropagator. Inthecovariantrst-quantizedformalismonecanconsider moregeneralgauges forthe reparametrizationinvarianceandlocalscaleinvariancet han g mn = mn Changingthegaugehastheeectof\stretching"thesurface in space.Sincethe 2Dmetriccanalwaysbechosentoberatinanysmallregionoft hesurface,it'sclear thattheonlyinvariantquantitiesareglobal.Thesearetop ologicalquantities(some integersdescribingthetypeofsurface)andcertainproper -lengthparameters(such astheproper-lengthofthepropagatorinthecaseofthepart icle,asin(5.1.13)).In particular,thisappliestothelight-coneformalism,whic hisjustacovariantgauge withstrongergaugeconditions(andsomevariablesremoved bytheirequationsof motion).Thus,theplanarlight-conetreegraphaboveisess entiallyaratdisc,and theproper-lengthparametersarethe i i =3 ;:::; N 1.However,therearemore generalcovariantgaugesevenforsuchsurfaceswithjustst raight-lineboundaries:For example,wecanidentify

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190 9.GRAPHS = 2 1 4 3 2 1 4 3 withtheproper-lengthparameterbeingtherelativepositi onofthe2splittingpoints (horizontalorverticaldisplacement,respectivelyforth e2graphs,withthevalueof theparameterbeingpositiveornegative).Moregenerally, theonlyinvariantsin thisgrapharetheproperlengthdistancesmeasuredalongth eboundarybetween the end points(thepointsassociatedwiththeexternalparticles) ,less3whichcanbe eliminatedbyremainingprojectiveinvariance(consider, e.g.,thesurfaceasadisc, withtheendpointsonthecircularboundary).Thus,wecanke epthesplittingpoints inthepositionsintheguresandvarytheproper-lengthpar ametersbymovingthe endsinstead.IfthisisinterpretedintermsofordinaryFey nmangraphs,therst graphseemstohaveintermediatestatesformedbythecollis ionofparticles1and2, whilethesecondoneisfrom1and3.Theidentityofthese2gra phsmeansthatthe sameresultcanbeobtainedbysummingoverintermediatesta tesinonly1ofthese 2channelsasintheother,aswesawforthecaseofexternalta chyonsin(9.1.16). Thus,dualityisjustamanifestationof reparametrizationinvarianceandlocal scaleinvariance.9.3.Loops Herewewillonlyoutlinetheprocedureandresultsofloopca lculations(fordetails see[0.1,1.3-5,9.7-10,5.4]andtheshelfofthisweek'spre printsinyourlibrary).Inthe rst-quantizedapproachtoloopstheonlyessentialdiere ncefromtreesisthatthe topologyisdierent.Thismeansthat:(1)It'snolongerpos sibletoconformallymap totheupper-halfplane,althoughonecanmaptotheupper-ha lfplanewithcertain linesidentied(e.g.,fortheplanar1-loopgraph,whichis topologicallyacylinder,we canchoosetheregionbetween2concentricsemicircles,wit hthesemicirclesidentied). (2)Theintegrationvariablesincludenotonlythe 'softheinteractionpointswhich

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9.3.Loops 191 denethepositionoftheloopinthestring,butalsothe 's,whicharejustthe p + 's oftheloop.Incovariantgaugesit'salsonecessarytotaket heghostcoordinates intoaccount.Inthesecond-quantizedapproachtheloopgra phsfollowdirectlyfrom theeldtheoryaction,asinordinaryeldtheory.However, forexplicitcalculation, thesecond-quantizedexpressionsneedtobetranslatedint orst-quantizedform,as forthetrees.1-loopgraphscanalsobecalculatedintheext ernaleldapproachby \sewing"togetherthe2endsofthestringpropagator,conve rtingthematrixelement in(9.1.9)intoatrace,usingthetraceoperatorin(9.1.12) Aninterestingfeatureofopenstringtheoriesisthatclose dstringsaregenerated asboundstates.Thiscomesfromstretchingtheone-loopgra phwithintermediate statesof2180 -twistedopenstrings: & $ % = $ & % $ & % = Thus,aclosedstringisaboundstateof2openstrings.Thecl osed-stringcoupling canthenberelatedtotheopen-stringcoupling,eitherbyex aminingmoregeneral

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192 9.GRAPHS graphs,orbynoticingthattheGauss-Bonnettheoremsaysth attwicethenumberof \handles"(closed-stringloops)plusthenumberof\window s"(openstringloops)is atopologicalinvariant(theEulernumber,uptoaconstant) ,andthus1closed-string loopcanbeconvertedinto2open-stringloops.Specically hg closed =( hg open ) 2 g closed = hg open 2 : (9 : 3 : 1) Thus,forconsistent h countingtheopenstringsmustbethoughtofasfundamentali s suchatheory(whichsofarmeansjusttheSO(32)superstring ),andtheclosedstrings asboundstates.Since(known)closedstringsalwayscontai ngravitons,thismakesthe SO(32)superstringtheonlyknownexampleofatheorywheret hegravitonappears asaboundstate.Thegravitonpropagatoristheresultofult ravioletdivergencesdue toparticlesofarbitrarilyhighspinwhichsumtodivergeon lyatthepole: Z 1 0 dk 2 (1 0 2 p 2 k 2 + 1 2 0 4 p 4 k 4 )= Z dk 2 e 0 2 p 2 k 2 = 1 0 2 p 2 : (9 : 3 : 2) (Ingeneral, p 2 + M 2 appearsinsteadof p 2 ,sotheentireclosed-stringspectrumis generated.) Asmentionedintheintroduction,thetopologyofa2Dsurfac eisdenedbya fewintegers,correspondingto,e.g.,thenumberofholes.B ychoosingthecoordinates ofthesurfaceappropriately(\stretching"itinvariouswa ys),thesurfacetakesthe formofastringtreegraphwithone-loopinsertions,eachon eloopinsertionhaving thevalue1ofoneofthetopologicalinvariants(e.g.,1hole ).(Actually,someofthese insertionsare1-loopclosed-stringinsertions,andthere forearecountedas2-loopin anopen-stringtheorydueto(9.3.1).)Forexample,aholein anopen-stringsheet maybepushedaroundsothatitrepresentsaloopasinaboxgra ph,apropagator correction,atadpole,oranexternallinecorrection.Such dualitytransformations canalsoberepresentedinFeynmangraphnotationasaconseq uenceoftheduality propertiesofsimplergraphssuchasthe4-pointtreegraph( 9.1.16): @ @ @ @ @ @ @ @ @ @ @ @ =

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9.3.Loops 193 @ @ @ @ @ @ = @ @ @ & $ % @ @ @ = @ @ @ @ @ @ = @ @ @ @ @ @ & $ % Bydoingstretchingofsuchplanargraphsoutoftheplane,th eseloopscanevenbe turnedintoclosed-stringtadpoles: = n Stretchingrepresentscontinuousworld-sheetcoordinate transformations.However,therearesomecoordinatetransformationswhichcan' tbeobtainedbycombining innitesimaltransformations,andthusmustbeconsidered separatelyinanalyzing gaugexingandanomalies[9.11].Thesimplestexampleisfo raclosed-stringloop (vacuumbubble),whichisatorustopologically.Thegroupo fgeneralcoordinate transformationshasasasubgroupconformaltransformatio ns(whichcanbeobtained asaresidualgaugeinvarianceuponcovariantgaugexing,s ect.6.2).Conformal transformations,inturn,haveasasubgroupthe(complex)p rojectivegroupSp(2,C): Thedeningrepresentationofthisgroupisgivenby2 2complexmatriceswithdeterminant1,sothecorrespondingrepresentationspacecon sistsofpairsofcomplex

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194 9.GRAPHS numbers.Ifweconsiderthetransformationpropertyofacom plexvariablewhichis theratioofthe2numbersofthepair,wethennd: z 1 z 2 0 = ab cd z 1 z 2 ;z 0 = z 1 z 2 (9 : 3 : 3 a ) z 0 0 = az 0 + b cz 0 + d ; (9 : 3 : 3 b ) where ad bc =1.Finally,theprojectivegrouphasasadiscretesubgroup the \modular"groupSp(2,Z),where a;b;c;d are(real)integers(stillsatisfying ad bc = 1).Toseehowthisrelatestothetorus,denethetorusasthe complexplanewith theidenticationofpoints z z + n 1 z 1 + n 2 z 2 (9 : 3 : 4) foranyintegers n 1 ;n 2 ,for2particularcomplexnumbers z 1 ;z 2 whichpointindierent directionsinthecomplexplane.Wecanthenthinkofthetoru sastheparallelogram withcorners0 ;z 1 ;z 2 ;z 1 + z 2 ,withoppositesidesidentied,andthecomplexplanecan bedividedupintoaninnitenumberofequivalentcopies,as impliedby(9.3.4).The conformalstructureofthetoruscanbecompletelydescribe dbyspecifyingthevalue of z 0 = z 1 =z 2 .(E.g., z 1 and z 2 bothchangeunderacomplexscaletransformation, butnottheirratio.Withoutlossofgenerality,wecanchoos etheimaginarypartof z 0 tobepositivebyordering z 1 and z 2 appropriately.)However,ifwetransform( z 1 ;z 2 ) underthemodulargroupasin(9.3.3a),then(9.3.4)becomes z z + n i z 0 i ;z 0 i = g i j z j ; (9 : 3 : 5 a ) where g i j istheSp(2,Z)matrix,orequivalently z z + n 0 i z i ;n 0 i = n j g j i : (9 : 3 : 5 b ) Inotherwords,anSp(2,Z)transformationgivesbackthesam etorus,sincetheidenticationofpointsinthecomplexplane(9.3.4)and(9.3.5b)i sthesame(sinceitholds areallpairsofintegers n i ).Wethereforedenethetorusbythecomplexparameter z 0 ,moduloequivalenceundertheSp(2,Z)transformation(9.3 .3b).Itturnsoutthat themodulargroupcanbegeneratedbyjustthe2transformati ons z 0 1 z 0 andz 0 z 0 +1 : (9 : 3 : 6) Therelevanceofthemodulargrouptothe1-loopclosed-stri ngdiagramisthatthe functionalintegraloverallsurfacesreduces(fortori)to anintegralover z 0 .Gauge

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9.3.Loops 195 xingforSp(2,Z)thenmeanspickingjustoneoftheinniten umberofequivalent regionsofthecomplexplane(under(9.3.3b)).However,for aclosedstringinlessthan itscriticaldimension,thereisananomalyinthemodularin variance,andthetheory isinconsistent.Modularinvariancealsorestrictswhatty pesofcompacticationare allowed. Ifthe2Dgeneralcoordinateinvarianceisnotviolatedbyan omalies,it'sthen sucienttoconsiderthese1-loopobjectstounderstandthe divergencestructureof thequantumstringtheory.However,whilethestringcanbes tretchedtoseparateany two1-loopdivergences,weknowfromeldtheorythatoverla ppingdivergencescan't befactoredinto1-loopdivergences.Thissuggeststhatany 1-loopdivergences,since theywouldleadtooverlappingdivergences,wouldviolatet he2Dreparametrization invariancewhichwouldallowthe1-loopdivergencestobedi sentangled.Hence,it seemsthatastringtheorymustbeniteinordertoavoidsuch anomalies.Conversely, weexpectthatnitenessat1loopimpliesnitenessatalllo ops.Somedirectevidence ofthisisgivenbythefactthatallknownstringtheorieswit hfermionshave1-loop anomaliesintheusualgaugeinvariancesofthemasslesspar ticlesifandonlyifthey alsohave1-loopdivergences.Aftertherestrictionsplace dbytree-levelduality(which determinestheground-statemassandrestrictstheopen-st ringgaugegroupstoU(N), USp(N),andSO(N)),supersymmetryinthepresenceofmassle ssspin-3/2particles, and1-loopmodularinvariance,thislastanomalyrestricti onallowsonlySO(32)as anopen-stringgaugegroup(althoughitdoesn'trestrictth eclosed-stringtheories) [1.11]. Ofthenitetheories,theclosed-stringtheoriesarenite graph-by-graph,whereas theopen-stringtheoryrequirescancellationbetweenpair sof1-loopgraphs,withthe exceptionofthenonplanarloopdiscussedabove.The1-loop closed-stringgraphs (correspondingto2-loopgraphsintheopen-stringtheory) are(1)thetorus(\handle") and(2)theKleinbottle,withexternallinesattached.Thep airsof1-loopopen-string graphsare(1)theannulus(planarloop,or\window")+Mobi usstrip(nonorientable loop)withexternalopenand/orclosedstrings,and(2)thed isk+agraphwiththe topologyofRP 2 (adiskwithoppositepointsidentied)withexternalclose dstrings. TheKleinbottleisallowedonlyfornonorientedclosedstri ngs,andtheMobiusstrip andRP 2 areallowedonlyfornonorientedopenstrings. Itshouldbepossibletosimplifycalculationsandgivesimp leproofsofniteness bytheuseofbackgroundeldmethodssimilartothosewhichi ngravityandsupersymmetrymadehigher-loopcalculationstractableandallo wedsimplederivationsof

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196 9.GRAPHS no-renormalizationtheorems[1.2].However,theuseofarb itrarybackgroundstring eldswillrequirethedevelopmentofgauge-covariantstri ngeldtheory,thepresent statusofwhichisdiscussedinthefollowingchapters.Exercises (1)Generalize(9.1.3)tothespinningstring,using(7.2.2 )insteadof 1 2 b P 2 .Writing 1 2 ^ Dd ^ D ( ) 1 2 ^ Dd ^ D ( )+ ( ) V V = W + V ,showthat V isdeterminedexplicitly by W (2)Derive(9.1.7b).(3)Fillinallthestepsneededtoobtain(9.1.15)from(9.1. 6,8).Deriveallpartsof (9.1.16).Derive(9.1.19). (4)Evaluate(9.1.20)byusingtheresult(9.1.15)with N N +2(butdropping2 d integrations)andletting k 0 = k N +1 =0. (5)Derive(9.2.18).(6)Generalize(9.2.16)toarbitrary z 1 z 2 z N andderive(9.2.22c)bythemethod of(9.2.20,21).TaketheinnitesimalformoftheSp(2)tran sformationandshow thatit'sgeneratedby L 0 L 1 withthecorrespondence(8.1.3),where z = e i (7)Derive(9.2.27).Evaluate G rsmn of(9.2.4)using(9.2.19,26,27).

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10.LIGHT-CONEFIELDTHEORY 197 10.LIGHT-CONEFIELDTHEORY Inthischapterweextendthediscussionofsect.2.1tothest ringandconsider interactingcontributionstothePoincarealgebraofsect .7.1alongthelinesofthe Yang-Millscasetreatedinsect.2.3. Forthestring[10.1,9.5,10.2],it'sconvenienttouseael d[ X i ( ) ;p + ; ],since p + isthelengthofthestring.This X for 2 [0 ;p + ]isrelatedtothatin(7.1.7)for p + =1by X ( ;p + )= X ( =p + ; 1).The hermiticity conditiononthe(open-string) eldis [ X i ( ) ;p + ; ]= y [ X i ( p + ) ; p + ; ] : (10 : 1 a ) Thesamerelationholdsfortheclosedstring,butwemayrepl ace p + withjust ,sincetheclosedstringhastheresidualgaugeinvariance + constant whichisxedbytheconstraint(orgaugechoice) N =0.(See(7.1.12).In loops,thisgaugechoicecanbeimplementedeitherbyprojec tionoperatorsorby Faddeev-Popovghosts.)Asdescribedinsect.5.1,thischar ge-conjugationcondition correspondstoacombinationofordinarycomplexconjugati on( reversal)witha twist(matrixtranspositioncombinedwith reversal).Thetwisteectivelyactsasa charge-conjugationmatrixin space,inthesensethatexpressionsinvolving tr y acquiresuchafactorifreexpressedintermsofjustandnot y (and(10.1a)looks likearealityconditionforagroupforwhichthetwististhe groupmetric).Here isanN Nmatrix,andtheoddmasslevelsofthestring(includingthe massless Yang-Millssector)areintheadjointrepresentationofU(N ),SO(N),orUSp(N)(for evenN)[10.3],whereinthelatter2casestheeldalsosatis esthe reality condition =( ) ; (10 : 1 b ) where isthegroupmetric(symmetricforSO(N),antisymmetricfor USp(N)).The factthatthelattercasesusetheoperationof reversalseparately,or,bycombining with(10.1a),thetwistseparately,meansthattheydescrib enonorientedstrings:The stringeldisconstrainedtobeinvariantunderatwist.The sameistrueforclosed strings(althoughclosedstringshavenogrouptheory,soth echoiceoforientedvs.

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198 10.LIGHT-CONEFIELDTHEORY nonorientedisarbitrary,and =1in(10.1b)).Thetwistoperatorcanbedened similarlyforsuperstrings,includingheteroticstrings. Forgeneralopenstringsthe twistismostsimplywrittenintermsofthehattedoperators ,onwhichitactsas ^ O ( ) ^ O ( )(i.e.,as e iN ).Forgeneralclosedstrings,ittakes ^ O ( ) ( ) ^ O ( ) ( ).(Forclosedstrings, isirrelevant,since N =0.) Light-conesuperstringelds[10.2]alsosatisfythereali tycondition(inplaceof (10.1b),generalizing(5.4.32))thattheFouriertransfor mwithrespectto a isequal tothecomplexconjugate(whichistheanalogofacertaincon ditiononcovariant superelds) Z D e R a a p + = 2 [ a ]= [ a ] : (10 : 1 c ) a hasamodeexpansionlikethatoftheghost b C orthespinningstring's ^ (of thefermionicsector).Theground-stateoftheopensuperst ringisdescribedbythe light-conesupereldof(5.4.35),whichisafunctionofthe zero-modesofallthe abovecoordinates.Thus,thelowest-mass(massless)secto roftheopensuperstring issupersymmetricYang-Mills. Thefreeactionofthebosonicopenstringis[10.1,9.5] S 0 = Z D X i Z 1 1 dp + Z 1 1 dtr y p + i @ @ + H ; H = Z 2 0 p + 0 d 2 1 2 0 2 X i 2 + 1 0 X i 0 2 1 # = Z p + p + d 2 ( 1 2 b P i 2 1)= p i 2 + M 2 2 p + : (10 : 2) Thefreeeldequationisthereforejustthequantummechani calSchrodingerequation. (The p + integralcanalsobewrittenas2 R 1 0 ,duetothehermiticitycondition.This formalsoholdsforclosedstrings,with H thesumof2open-stringones,asdescribed insec.7.1.)Similarremarksapplytosuperstrings(using( 7.3.13)). Asintherst-quantizedapproach,interactionsaredescri bedbysplittingand joiningofstrings,butnowthegraphgetschoppedupintopro pagatorsandvertices: 12 3 4 5 67

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10.LIGHT-CONEFIELDTHEORY 199 The3-open-stringvertexisthenjusta functionalsetting1stringequalto2others, representedbyaninnitesimalstripintheworldsheet.The interactiontermin theactionisgivenby(9.2.23)for Z = X .The3-closed-stringvertexisasimilar functionalfor3closedstrings,whichcanberepresentedas theproductof2openstring functionals,sincetheclosed-stringcoordinatescanbere presentedasthe sumof2open-stringcoordinates(oneclockwiseandonecoun terclockwise,butwith thesamezero-modes).Thisvertexgenerallyrequiresanint egrationoverthe of theintegrationpoint(sinceclosedstringscanjoinanywhe re,nothavinganyends, correspondingtothegaugeinvariance + constant ),buttheequivalentoperation ofprojectiononto N =0canbeabsorbedintothepropagators. Generalverticescanbeobtainedbyconsideringsimilarsli cingsofsurfaceswith generalglobaltopologies[10.2,9.7].Thereare2oforder g ,corresponding locally toa splittingorjoining: $ & % $ & % $ $ Theformeristhe3-open-stringvertex.Theexistenceofthe latterisimpliedby theformerviathenonplanarloopgraph(seesect.9.3).Ther estareorder g 2 ,and correspondlocallyto2stringstouchingtheirmiddlesands witchinghalves: @ @ @ @ @ @ $ @ @ @ @ @ @ & % $ & $ % $

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200 10.LIGHT-CONEFIELDTHEORY & % $ & % $ $ & & $ % $ % & $ % & $ % $ & & $ % $ % & $ % & $ % $ Thetype-I(SO(32)open-closed)theoryhasallthesevertic es,butthetype-IIABand heterotictheorieshaveonlythelastone,sincetheyhaveon lyclosedstrings,and theyareoriented(clockwisemodesaredistinguishablefro mcounterclockwise).Ifthe type-Itheoryistreatedasatheoryoffundamentalopenstri ngs(withclosedstrings asboundstates,so h canbedened),thenwehaveonlytherstoftheorderg verticesandtherstoftheorderg 2 Thelight-conequantizationofthespinningstringfollows directlyfromthecorrespondingbosonicformalismbythe1Dsupersymmetrizatio ndescribedinsect.7.2. Inparticular,insuchaformalismtheverticesrequirenofa ctorsbesidesthe functionals[10.4].(Inconvertingtoanon-supereldformalis m,integrationofthevertex over producesavertexfactor.)However,theprojection(7.2.5) mustbeputinby hand.Also,thefactthatboundaryconditionscanbeeithero f2types(forbosons vs.fermions)mustbekeptinmind. Theinteractionsofthelight-conesuperstring[10.2]ared oneasforthelight-cone bosonicstring,butthereareextrafactors.Forexample,fo rthe3-open-stringvertex

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10.LIGHT-CONEFIELDTHEORY 201 (theinteractingcontributionto p ),wehavetheusual -functionalstimes V ( p )= P L + 1 2 P ab a b + 1 24 P L C abcd a b c d ; (10 : 3 a ) evaluatedatthesplittingpoint.Theinteractingcontribu tionsto q aregivenbythe sameoverlap -functionalstimesthevertexfactors V ( q a )= a ;V ( q a )= 1 6 C abcd b c d : (10 : 3 b ) (Theeuphoricnotationfor q isasinsect.5.4for d .)Theseareevaluatedasin(9.2.28), where P and = haveweight w =1.Theirformisdeterminedbyrequiringthatthe supersymmetryalgebrabemaintained.The functionalpartisgivenasin(9.2.27), butnowtheof(9.2.14)insteadofjust(9.2.15)is = 1 2 ( P; P ) 0 ; : (10 : 4) Asforthebosonicstring,theclosed-superstringvertexis theproductof2openstringones(including2factorsoftheform(10.3a)fortype IorIIbutjust1for theheterotic,andintegratedover ).Forthegeneralinteractionsabove,allorderg interactionshaveasingleopen-stringvertexfactor,whil eallorderg 2 have2,since theinteractionsofeachorderarelocallyallthesame.Thev ertexfactoriseither1 or(10.3a),dependingonwhetherthecorrespondingsetofmo desisbosonicorsupersymmetric.Whenthesesuperstringtheoriesaretruncatedt otheirgroundstates,the factor(10.3a)keepsonlythezero-modecontributions,whi chistheusuallight-cone, 3-pointvertexforsupersymmetricYang-Mills,andtheprod uctof2suchfactors(for closedstrings)istheusualvertexforsupergravity. Thesecond-quantizedinteractingPoincarealgebraforth elight-conestringcan beobtainedperturbatively.Forexample, [ p ;J i ]=0 [ p (2) ;J (2) i ]=0 ; (10 : 5 a ) [ p (3) ;J (2) i ]+[ p (2) ;J (3) i ]=0 ; (10 : 5 b ) etc.,where( n )indicatestheorderinelds(seesect.2.4).Thesolutiont o(10.5a) isknownfromthefreetheory.Thesolutionto(10.5b)canbeo btainedfromknown resultsfortherst-quantizedtheory[1.4,10.5]:Therst termrepresentstheinvarianceofthe3-pointinteractionofthehamiltonianunderfre eLorentztransformations.

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202 10.LIGHT-CONEFIELDTHEORY Thefactthatthisinvarianceholdsonlyon-shellisanexpec tedconsequenceofthe factthatthesecondtermin(10.5b)issimplythecommutator ofthefreehamiltonian withtheinteractioncorrectiontotheLorentzgenerator.T hus,thealgebraofthe completeinteractinggeneratorsclosesoshellaswellaso n,andtheexplicitformof J (3) i followsfromtheexpressionfornonclosuregivenin[1.4,10 .5]: J (3) i (1 ; 2 ; 3)= 2 igX r i ( int ) X p + [ X i ] ; (10 : 6) where\ r "denotesanyofthethreestringsandrepresentstheusualo verlap-integral -functionalswithsplittingpoint INT .Thisistheanalogofthegeneralizationof (2.3.5)totheinteractingscalarparticle,where p (1 = 2 p + )( p i 2 + 2 ).Since p alsocontainsa4-pointinteraction,thereisasimilarcont ributionto J (2) i (i.e., (10.6)withreplacedbythecorresponding3-stringproduc tderivedfromthe4stringlight-conevertexin p ,and g replacedwith g 2 ,butotherwisethesamenormalization).Explicitsecond-quantizedoperatorcalcula tionsshowthatthisclosesthe algebra[10.6].Similarconstructionsapplytosuperstrin gs[10.7]. Covariantstringrulescanbeobtainedfromthelight-conef ormalisminthesame wayasinsect.2.6,and p + nowalsorepresentsthestringlength[2.7].Thus,from (10.2)wegetthefreeaction,intermsofaeld[ X a ( ) ;X ( ) ;p + ; ], S 0 = Z D X a D X Z 1 1 dp + Z 1 1 dtr y p + i @ @ + H ; H = Z p + 0 d 2 ( 1 2 0 2 X a 2 + X X + 1 0 X a 0 2 + X 0 X 0 # 1 ) : (10 : 7) Thevertexisagaina functional,inallvariables. Thelight-coneformalismforheteroticstringeldtheoryh asalsobeendeveloped, andcanbeextendedtofurthertypesofcompactications[10 .8]. Unfortunately,theinteractinglight-coneformalismisno tcompletelyunderstood, evenforthebosonicstring.Therearecertainkindsofconta cttermswhichmust beaddedtothesuperstringactionandsupersymmetrygenera torstoinsurelowerboundednessoftheenergy(supersymmetryimpliespositivi tyoftheenergy)and canceldivergencesinscatteringamplitudesduetocoincid enceofvertexoperator factors[10.9],andsomeofthesetermshavebeenfound.(Sim ilarproblemshave appearedinthecovariantspinningstringformulationofth esuperstring:seesect. 12.2.)Thisproblemisparticularlyevidentforclosedstri ngs,whichwerethoughtto haveonlycubicinteractionterms,whichareinsucienttob oundthepotentialin

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Exercises 203 aformalismwithonlyphysicalpolarizations.Furthermore ,theclosed-stringbound stateswhichhavebeenfoundtofollowatoneloopfromopen-s tringtheoriesby explicitlyapplyingunitaritytotreegraphsdonotseemtof ollowfromthelightconeeldtheoryrules[10.1].Sinceunitarityrequirestha t1-loopcorrectionsare uniquelydeterminedbytreegraphs,theimplicationisthat thepresentlight-cone eldtheoryactionisincomplete,orthattherulesfollowin gfromithavenotbeen correctlyapplied.Itisinterestingtonotethatthetypeof graphneededtogive thecorrectclosed-stringpolesresemblestheso-calledZgraphofordinarylight-cone eldtheory[10.10],whichcontainsalinebackward-moving in x + whenthelight-cone formalismisobtainedasanultrarelativisticlimit,becom inganinstantaneousline whenthelimitisreached.Exercises Ican'tthinkofany.

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204 11.BRSTFIELDTHEORY 11.BRSTFIELDTHEORY11.1.Closedstrings Sincethegauge-invariantactionsforfreeopenstringsfol lowdirectlyfromthe methodsofsects.3.4-5usingthealgebrasofchapt.8(forth ebosonicandfermionic cases,usingeitherOSp(1,1 j 2)orIGL(1)algebras),wewillconsiderherejustclosed strings,afterafewgeneralremarks. Otherstringactionshavebeenproposedwhichlackthecompl etesetofStueckelbergelds[11.1],andasaresulttheyareexpectedtosue rfromproblemssimilar tothoseofcovariant\unitary"gaugesinspontaneouslybro kengaugetheories:no simpleKlein-Gordon-typepropagator,nonmanifestrenorm alizability,andsingularityofsemiclassicalsolutions,includingthoserepresent ingspontaneousbreakdown. Furtherattemptswithnonlocal,higher-derivative,orinc ompleteactionsappeared in[11.2].Equivalentgauge-invariantactionsforthefree Ramondstringhavebeen obtainedbyseveralgroups[4.13-15,11.3].Theactionof[1 1.3]isrelatedtotherest byaunitarytransformation:Ithasfactorsinvolvingcoord inateswhichareevaluated atthemidpointofthestring,whereastheothersinvolvecor respondingzero-modes. Foropenorclosedstrings,thehermiticitycondition(10.1 a)nowrequiresthat theghostcoordinatesalsobetwisted:IntheIGL(1)formali sm(wheretheghost coordinatesaremomenta) [ X ( ) ;C ( ) ; e C ( )]= y [ X ( ) ; C ( ) ; e C ( )](11 : 1 : 1 a ) ( e C getsanextra\ "becausethetwistis reversal,and e C carriesa indexinthe mechanicsaction),andintheOSp(1,1 j 2)formalismwejustextend(10.1a): [ X a ( ) ;X ( ) ;p + ]= y [ X a ( p + ) ;X ( p + ) ; p + ] : (11 : 1 : 1 b ) Asinthelight-coneformalism(seesect.10.1),fordiscuss ingfreetheories,wescale by p + in(11.1.1b),sothetwistthentakes .(Fortheclosedstringthe

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11.1.Closedstrings 205 twistis ,soinboth(11.1.1a)and(11.1.1b)theargumentsofthecoor dinates arejust ontheright-handside.) Intherestofthissectionwewillconsiderclosedstringson ly.Firstweshowhow toextendtheOSp(1,1 j 2)formalismtoclosedstrings[4.10].Byanalogyto(4.1.1) thekineticoperatorfortheclosedstringisa functioninIOSp(1,1 j 2): S = Z d D xd 2 x dx d 2 p d p + y p + 2 ( J AB ) ( p A ) ; = 1 2 J AB BA + p A A ; (11 : 1 : 2) where,asin(7.1.17),thePoincaregenerators J AB and p A aregivenassums,and p A asdierences,oftheleft-handedandright-handedversion softheopen-string generatorsof(7.1.14).TheHilbert-spacemetricnecessar yforhermiticityisnow p + 2 (orequivalently p + (+) p + ( ) ),sinceafactorof p + isneededfortheopen-string modesofeachhandedness.Forsimplicity,wedonottakethep hysicalmomenta p a tobedoubledhere,sincetheIOSp(1,1 j 2)algebraclosesregardless,buttheycanbe doubledifthecorresponding functionsandintegrationsareincludedin(11.1.2). Moreexplicitly,the functioninthePoincaregroupisgivenby ( J AB ) ( p A )= ( J 2 ) i ( J + ) 2 ( J + ) 2 ( J ) ( p ) 2 ( p ) ( p + ) : (11 : 1 : 3) Toestablishtheinvarianceof(11.1.2),thefactthat(4.1. 1)isinvariantindicatesthat it'ssucienttoshowthat ( J AB )commuteswith ( p A ).Thisfollowsfromthefact thateachofthe J AB 'scommuteswith ( p A ).Weinterpret ( p )= p + ( N ) inthepresenceoftheother ( p )'s,where ( N )isaKronecker ,andtheother ( p )'sareDirac 's. Allthenontrivialtermsarecontainedinthe 2 ( J ).Asintheopen-stringcase, dependenceonthegaugecoordinates x and x iseliminatedbythe 2 ( J + )and ( J + )ontheleft,andfurthertermsarekilledby ( J 2 ).Similarly,dependence on p and p + iseliminatedbythecorresponding functionsontheright,and furthertermsarekilledby ( p ).Forconvenience,thelattereliminationshouldbe donebeforetheformer.Aftermakingtheredenition 1 p + andintegratingout theunphysicalzero-modes,theactionissimilartotheOSp( 1,1 j 2)case: S = Z d D x 1 2 y ( N ) ( M 2 ) h 2 M 2 +( M a p a + M m M ) 2 i ; =( M a p a + M m M ) + 1 2 M + N : (11 : 1 : 4) Thisistheminimalformoftheclosed-stringaction.

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206 11.BRSTFIELDTHEORY ThenonminimalformisobtainedbyanalogytotheIGL(1)form alism,inthe samewayOSp(1,1 j 2)wasextendedtoIOSp(1,1 j 2):Usinga functionintheclosedstringgroupGL(1 j 1)ofsect.8.2(foundfromsumsanddierencesoftheexpress ions in(8.2.6),asin(7.1.17)),weobtaintheaction(with p N ) S = Z d D xdcd p ~ c y iQ ( J 3 ) ( p ~ c ) ( N ) ; = Q + J 3 + N ~ + p ~ c ; (11 : 1 : 5) or,afterintegratingout p ~ c ,with= + p ~ c S = Z d D xdc y i b Q ( b J 3 ) ( N ) ; = b Q + b J 3 + N ~ ; (11 : 1 : 6) wherethe c 'sindicatethatalltermsinvolving p ~ c anditscanonicalconjugatehave beendropped.Theeld = + c iscommuting. Forthegauge-xingintheGL(1 j 1)formalismabove(ortheequivalentonefrom rst-quantization),wenowchoose[4.5] O = 2 p ~ c c; @ @c # K = p ~ c c ( 2 M 2 ) 4 M + @ @c # 2 N c; @ @c # ; (11 : 1 : 7) wherewehaveused Q = i 1 4 c ( p 2 + M 2 )+ i 1 2 M + @ @c i N @ @ p ~ c + i 1 2 M + p ~ c + Q + : (11 : 1 : 8) Expandingthestringeldovertheghostzero-modes, =( + ic )+ i p ~ c ( ^ + c ^ ) ; (11 : 1 : 9) wesubstituteintothelagrangian L = 1 2 y K andintegrateovertheghostzero modes: @ @c @ @ p ~ c L = 1 2 y ( 2 M 2 ) +2 y M + +4 i ( ^ y N + i ^ y N ) : (11 : 1 : 10) containspropagatingelds, containsBRSTauxiliaryelds,and ^ and ^ contain lagrangemultiplierswhichconstrain N =0fortheotherelds.Althoughthe propagatingeldsarecompletelygauge-xed,theBRSTauxi liaryeldsagainhave thegaugetransformations = ;M + =0 ; (11 : 1 : 11 a ) andthelagrangemultipliershavethegaugetransformation s ^ = ^ ; ^ = ^ ; N ^ = N ^ =0 : (11 : 1 : 11 b )

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11.2.Components 207 11.2.Components Togetabetterunderstandingofthegauge-invariantstring actionintermsofmore familiarparticleactions,wenowexpandthestringactiono versomeofthelower-mass componentelds,usingthealgebrasofchapt.8intheformal ismofsect.4.Allof theseresultscanalsobederivedbysimplyidentifyingther educiblerepresentations whichappearinthelight-cone,andthenusingthecomponent methodsofsect.4.1. However,herewe'llworkdirectlywithstringoscillators, andnotdecomposethe reduciblerepresentations,forpurposesofcomparison. AsanexampleofhowcomponentsappearintheIGL(1)quantiza tion,themasslessleveloftheopenstringisgivenby(cf.(4.4.6)) = h ( A a a a y + C a y )+ icBa ~ c y i j 0 i ; (11 : 2 : 1) where a =( a c ;a ~ c ),alloscillatorsarefortherstmode,andwehaveused(4.4 .5). ThelagrangianandBRSTtransformationsthenagreewith(3. 2.8,11)for =1.In ordertoobtainparticleactionsdirectlywithouthavingto eliminateBRSTauxiliary elds,fromnowonweworkwithonlytheOSp(1,1 j 2)formalism.(Bythearguments ofsect.4.2,theIGL(1)formalismgivesthesameactionsaft ereliminationofBRST auxiliaryelds.) Asdescribedinsect.4.1,auxiliaryeldswhichcomefromth eghostsectorare crucialforwritinglocalgauge-invariantactions.(These auxiliaryeldshavethesame dimensionasthephysicalelds,unliketheBRSTauxiliary elds,whichare1unit higherindimensionandhavealgebraiceldequations.)Let 'sconsiderthecounting oftheseauxiliaryelds.Thisrequiresndingthenumberof Sp(2)singletsthatcan beconstructedoutoftheghostoscillatorsateachmassleve l.TheSp(2)singlet constructedfromtwoisospinorcreationoperatorsis a m y a n y ,whichwedenoteas ( mn ).Ageneralauxiliaryeldisobtainedbyapplyingtothevac uumsomenonzero numberofthesepairstogetherwithanarbitrarynumberofbo soniccreationoperators. Therstfewindependentproductsofpairsoffermionicoper ators,listedbyeigenvalue ofthenumberoperator N ,are: 0: I 1: 2:(11)3:(12)

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208 11.BRSTFIELDTHEORY 4:(13) ; (22) 5:(14) ; (23) 6:(15) ; (24) ; (33) ; (11)(22)(11 : 2 : 2) where I istheidentityandnooperatorexistsatlevel1.Thenumbero findependent productsofsingletsateachlevelisgivenbythepartitionf unction 1 Q 1n =2 (1 x n ) =1+ x 2 + x 3 +2 x 4 +2 x 5 +4 x 6 +4 x 7 +7 x 8 +8 x 9 +12 x 10 +14 x 11 +21 x 12 + ; (11 : 2 : 3) correspondingtothestatesgeneratedbyasinglebosonicco ordinatemissingitszeroth andrstmodes,asdescribedinsect.8.1. Wenowexpandtheopenstringuptothethirdmasslevel(conta iningamassive, symmetric,rank-2tensor)andtheclosedstringuptothesec ondmasslevel(containingthegraviton)[4.1].Themodeexpansionsoftherelevant operatorsaregivenby (8.2.1).Sincethe ( M 2 )projectorkeepsonlytheSp(2)-singletterms,wendthat uptothethirdmassleveltheexpansionof is =[ 0 + A a a y 1 a + 1 2 h ab a y 1 a a y 1 b + B a a y 2 a + ( a 1 y ) 2 ] j 0 i : (11 : 2 : 4) Here 0 isthetachyon, A a isthemasslessvector,and( h ab ;B a ; )describethemassive, symmetric,rank-twotensor.It'snowstraightforwardtous e(8.2.1)toevaluatethe action(4.1.6): L = L 1 + L 0 + L 1 ;(11 : 2 : 5) L 1 = 1 2 0 ( 2 +2) 0 ; (11 : 2 : 6 a ) L 0 = 1 2 A 2 A + 1 2 ( @ A ) 2 = 1 4 F 2 ; (11 : 2 : 6 b ) L 1 = 1 4 h ab ( 2 2) h ab + 1 2 B ( 2 2) B 1 2 ( 2 2) + 1 2 ( @ b h ab + @ a B a ) 2 + 1 2 ( 1 4 h a a + 3 2 + @ B ) 2 : (11 : 2 : 6 c ) Thegaugetransformationsareobtainedbyexpanding(4.1.6 ).Thepiecesinvolving aretrivialinthecomponentviewpoint,sincetheyaretheon esthatreduce thecomponentsof totheSp(2)singletsgivenin(11.2.4).SincethenonlytheS p(2)

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11.2.Components 209 singletpartof Q cancontribute,onlythe( M )isospinorsectorof isrelevant. Wethereforetake =( a 1 y + a a 1 a y a 1 y + a 2 y ) j 0 i : (11 : 2 : 7) Thentheinvariancesarefoundtobe A a = @ a ;(11 : 2 : 8 a ) h ab = @ ( a b ) 1 p 2 ab ; B a = @ a + p 2 a ; = @ + 3 p 2 : (11 : 2 : 8 b ) (11.2.6b)and(11.2.8a)aretheusualactionandgaugeinvar ianceforafreephoton; however,(11.2.6c)and(11.2.8b)arenotinthestandardfor mformassive,symmetric ranktwo.Letting h ab = b h ab + 1 10 ab b ; = 1 2 b h a a 3 10 b ;(11 : 2 : 9) onends b h ab = @ ( a b ) ;B a = @ a + p 2 a ; b = 5 p 2 : (11 : 2 : 10) Inthisformit'sclearthat b and B a areStueckelbergeldsthatcanbegaugedaway by and a .(Thiswasnotpossiblefor ,sincethepresenceofthe @ termin itstransformationlaw(11.2.8b)wouldrequirepropagatin gFaddeev-Popovghosts.) Inthisgauge L 1 reducestotheFierz-Paulilagrangianforamassive,symmet ric, rank-twotensor: L = 1 4 b h ab 2 b h ab + 1 2 ( @ b b h ab ) 2 1 2 ( @ b b h ab ) @ a b h c c 1 4 b h a a 2 b h b b 1 4 ( b h ab b h ab b h a a 2 ) : (11 : 2 : 11) Theclosedstringistreatedsimilarly,sowe'llconsideron lythetachyonand masslesslevels.Theexpansionofthephysicalclosed-stri ngeldtothesecondmass levelis =( 0 + h ab a + 1 a y a 1 b y + A ab a + 1 a y a 1 b y + a + 1 y a 1 y ) j 0 i ; (11 : 2 : 12) where 0 isthetachyonand h ab A ab ,and describethemasslesssector,consisting ofthegraviton,anantisymmetrictensor,andthedilaton.W ehavealsodropped

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210 11.BRSTFIELDTHEORY eldswhicharekilledbytheprojectionoperatorfor N .Wethenndfortheaction (11.1.4): L = L 2 + L 0 ; L 2 = 1 2 0 ( 2 +4) 0 ; L 0 = 1 4 h ab 2 h ab + 1 4 A ab 2 A ab 1 2 2 + 1 2 ( @ b h ab + @ a ) 2 + 1 2 ( @ b A ab ) 2 : (11 : 2 : 13) Thenontrivialgaugetransformationsarefoundfrom(11.1. 4): h ab = @ ( a b ) ;A ab = @ [ a b ] ; = @ : (11 : 2 : 14) Theseleadtotheeldredenitions h ab = b h ab + ab b ; = b + 1 2 b h a a ;(11 : 2 : 15) whichresultintheimprovedgaugetransformations b h ab = @ ( a b ) ;A ab = @ [ a b ] ; b =0 : (11 : 2 : 16) Substitutingbackinto(11.2.13),wendthecovariantacti onforatachyon,linearized Einsteingravity,anantisymmetrictensor,andadilaton[4 .10]. Theformulationwiththeworld-sheetmetric(sect.8.3)use smoregaugeand auxiliarydegreesoffreedomthaneventheIGL(1)formulati on.Webeginwiththe openstring[3.13].Ifweevaluatethekineticoperatorfort hegauge-invariantaction (4.1.6)betweenstateswithoutfermionicoscillators,we nd b Q 2 P 11 ( b n y O n + O n y b n ),sothekineticoperatorreducesto ( M 2 )( 2 K + b Q 2 ) 2 K (1 N GB )+ 1 X 1 b n y b n + 1 p n ( b n y e L n + e L n y b n ) # ; N GB = 1 X 1 ( b n y g n + g n y b n ) ; (11 : 2 : 17) droppingthe f and c termsin K and e L n .Theoperator N GB countsthenumberof g y 's plus b y 'sinastate(withoutfactorsof n ).Wenowevaluatetherstfewcomponent levels.Theevaluationofthetachyonactionistrivial: = ( x ) j 0 i!L = 1 2 ( 2 +2) '; (11 : 2 : 18)

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11.2.Components 211 where S = R d 26 x L .Forthephoton,expandinginonlySp(2)singlets, =( A a 1 y + B g 1 y + G b 1 y ) j 0 i! L = 1 2 A 2 A 1 2 B 2 B @ A = 1 4 F ab 2 1 2 ( B + @ A ) 2 : (11 : 2 : 19) Thedisappearanceof G followsfromthegaugetransformations =( f 1 y c + c 1 y f ) j 0 i! ( A; B ; G )=( @ c ; 2 c ; f 1 2 c ) : (11 : 2 : 20) Since G istheonlyeldgaugedby f ,nogauge-invariantcanbeconstructedfrom it,soitmustdropoutoftheaction. Forthenextlevel,weconsiderthegaugetransformationsr stinordertodeterminewhicheldswilldropoutoftheactionsothatitscalcul ationwillbesimplied. TheSp(2)singletpartoftheeldis =( 1 2 h ab a 1 a y a 1 b y + h a a 2 a y + B a g 1 y a 1 a y + G a b 1 y a 1 a y + hg 1 y b 1 y + b B g 1 y 2 + G b 1 y 2 + B g 2 y + b G b 2 y + + f 1 y 2 + c 1 y 2 + 0 f 1 y c 1 y ) j 0 i : (11 : 2 : 21) Theonlytermsin whichcontributetothetransformationoftheSp(2)singlet sare = ( cp a y 1 + cb g y 1 + cg b y 1 ) f y 1 +( fp a y 1 + fb g y 1 + fg b y 1 ) c y 1 + c f y 2 + f c y 2 0 E : (11 : 2 : 22) Thegaugetransformationsofthecomponentsarethen h ab = @ ( a cp b ) ab 1 p 2 c ; h a = p 2 cp a + @ a c ; B a = @ a cb +2 K cp a ; G a = 1 2 cp a + fp a + @ a cg ; h = 1 p 2 c 1 2 cb + fb +2 K cg ; b B =2 K cb ; G = 1 2 cg + fg ;

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212 11.BRSTFIELDTHEORY B = p 2 cb +2 K c ; b G = 1 2 c + f + p 2 cg ; + = cb ; = p 2 f 1 2 fb @ fp +2 K fg ; 0 = p 2 c 1 4 cb + 1 2 fb 1 2 @ cp + K cg : (11 : 2 : 23) Wethengaugeaway G a =0 fp a = 1 2 cp a @ a cg ; G =0 fg = 1 2 cg ; + =0 cb =0 ; =0 f = 1 2 p 2 fb + 1 2 p 2 @ cp 1 p 2 ( 2 + K ) cg ; 0 =0 fb = 2 p 2 c + @ cp 2 K cg : (11 : 2 : 24) Thetransformationlawsoftheremainingeldsare h ab = @ ( a b ) ab ; h a = p 2( a + @ a ) ; h = 3 + @ ; B a = ( 2 2) a ; B = p 2( 2 2) ; b B =0 ; b G = 1 p 2 ( 3 + @ ) ; (11 : 2 : 25) where a = cp a and = c = p 2. cg dropsoutofthetransformationlawasaresult ofthegaugeinvarianceforgaugeinvariance = b Q = b Q ( ) : (11 : 2 : 26) Thelagrangiancanthenbecomputedfrom(4.1.6)and(11.2.1 7)tobe L = 1 4 h ab ( 2 2) h ab + 1 2 h a ( 2 2) h a 1 2 h ( 2 2) h 1 2 B a 2 1 2 B 2 +2 b B ( p 2 b G h ) + B a ( @ b h ab + @ a h + p 2 h a )+ B ( @ h + 3 p 2 h 1 2 p 2 h a a ) : (11 : 2 : 27)

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11.2.Components 213 Thelagrangianof(11.2.19)isequivalenttothatobtainedf romtheIGL(1)action, whereasthelagrangianof(11.2.27)isliketheIGL(1)onebu tcontainsinaddition the2gauge-invariantauxiliaryelds b B and p 2 b G h .BothreducetotheOSp(1,1 j 2) lagrangiansaftereliminationofauxiliaryelds. Theresultsfortheclosedstringaresimilar.Hereweconsid erjustthemassless levelofthenonorientedclosedstring(whichissymmetricu nderinterchangeof+and modes).TheSp(2)and N invariantcomponentsare =( h ab a y + a a y a + B a g y (+ a y ) a + G a b y (+ a y ) a + hg y (+ b y ) + Bg y + g y + Gb y + b y + + f y + f y + c y + c y + 0 f y (+ c y ) ) j 0 i ; (11 : 2 : 28) wherealloscillatorsarefromtherstmode( n =1),and h ab issymmetric.Thegauge parametersare = h ( cp a y + cb g y + cg b y ) (+ f y ) +( fp a y + fb g y + fg b y ) (+ c y ) i j 0 i : (11 : 2 : 29) Thecomponenttransformationsarethen h ab = @ ( a cp b ) ; B a = @ a cb 1 2 2 cp a ; G a =2 fp a cp a + @ a cg ; h =2 fb cb 1 2 2 cg ; B = 2 cb ; G =4 fg 2 cg ; + =2 cb ; = fb @ fp 1 2 2 fg ; 0 = fb 1 2 cb 1 2 @ cp 1 4 2 cg : (11 : 2 : 30) Wechoosethegauges G a =0 fp a = 1 2 cp a 1 2 @ a cg ; G =0 fg = 1 2 cg ; + =0 cb =0 ; 0 =0 fb = 1 2 @ cp + 1 4 2 cg : (11 : 2 : 31)

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214 11.BRSTFIELDTHEORY Theremainingeldstransformas h ab = @ ( a b ) ; B a = 1 2 2 a ; h = @ ; B =0 ; = @ ; (11 : 2 : 32) where a = cp a ,and cg dropsout.Finally,thelagrangianis L = 1 2 h ab 2 h ab h 2 h 4 B a 2 8 B ( + h ) 4 B a ( @ b h ab @ a h ) : (11 : 2 : 33) Againwendtheauxiliaryelds B and + h inadditiontotheusualnonminimal ones.Aftereliminationofauxiliaryelds,thislagrangia nreducestothatof(11.2.13) (forthenonorientedsector,uptonormalizationoftheeld s). Exercises (1)Derivethegauge-invariantactions(IGL(1)andOSp(1,1 j 2))forfreeopenstrings. DothesamefortheNeveu-Schwarzstring.DerivetheOSp(1,1 j 2)actionforthe Ramondstring. (2)FindalltheSp(2)-singleteldsatthelevelsindicated in(11.2.2).Separatethem intosetscorrespondingtoirreduciblerepresentationsof thePoincaregroup(includingtheirStueckelbergandauxiliaryelds). (3)Derivetheactionforthemasslessleveloftheopenstrin gusingthebosonized ghostsofsect.8.1. (4)Derivetheactionforthenextmassleveloftheclosedstr ingafterthosein (11.2.13).Dothesamefor(11.2.33). (5)Rederivetheactionsofsect.11.2forlevelswhichinclu despin2byrstdecomposingthecorrespondinglight-conerepresentationsintoirr educiblerepresentations, andthenusingtheHilbert-spaceconstructionsofsect.4.1 foreachirreducible representation.

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12.1.Introduction 215 12.GAUGE-INVARIANT INTERACTIONS 12.1.Introduction Thegauge-invariantformsoftheinteractingactionsforst ringeldtheoriesarefar fromunderstood.Interactingclosedstringeldtheorydoe snotyetexist.(Although aproposalhasbeenmade[12.1],itisnotevenatthepointwhe recomponentactions canbeexamined.)Anopen-stringformulationexists[4.8]( seethefollowingsection), butitdoesnotseemtorelatetothelight-coneformulation( chapt.10),andhasmore complicatedvertices.Furthermore,alltheseformulation sareintheIGL(1)formalism, sorelationtoparticlesislessdirectbecauseoftheneedto eliminateBRSTauxiliary elds.Mostimportantly,theconceptofconformalinvarian ceisnotclearinthese formulations.Ifaformulationcouldbefoundwhichincorpo ratedtheworld-sheet metricascoordinates,asthefreetheoryofsect.8.3,itmig htbepossibletorestore conformaltransformationsasalargergaugeinvariancewhi challowedthederivation oftheotherformulationsas(partial)gaugechoices. Inthissectionwewillmostlydiscussthestatusofthederiv ationofaninteracting gauge-covariantstringtheoryfromthelightcone,withint eractionssimilartothoseof thelight-conestringeldtheory.Thederivationfollowst hecorrespondingderivation forYang-Millsdescribedinsects.3.4and4.2[3.14],butth eimportantstep(3.4.17) ofeliminating p + dependencehasnotyetbeenperformed. AsperformedforYang-Millsinsect.3.4,thetransformatio n(3.4.3a)with U 1 istherststepinderivinganIGL(1)formalismfortheinte ractingstringfrom thelightcone.Sincethetransformationisnonunitary,the factorof p + in(2.4.9)is canceled.In(2.4.7),usingintegrationbyparts,(3.4.3a) inducesthetransformation ofthevertexfunction V ( n ) 1 p +1 p + n U (1) U ( n ) V ( n ) ; (12 : 1 : 1)

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216 12.GAUGE-INVARIANTINTERACTIONS whereweworkinmomentumspacewithrespectto p + .Thelowest-orderinteracting contributionto Q thenfollowsfromapplyingthistransformationtotheOSp-e xtended formofthelight-conevertex(10.6): J (3) c = 2 ig p + r p +1 p +2 p +3 X r c ( int ) X p + [ X a ][ p + X c ][ p + 1 X ~ c ] ; (12 : 1 : 2) eectivelygivingconformalweight-1to X c andconformalweight1to X ~ c .A functionalthatmatchesacoordinatemustalsomatchthe -derivativeofthecoordinate,andwithnozero-modesonemusthave p + 1 X ~ c p + ~ c =0 # = @ p + 1 X ~ c p + !# = p + 2 X ~ c 0 p + !# : (12 : 1 : 3) (Eventhenormalizationisunambiguous,sincewithoutzero -modescanbenormalizedto1betweengroundstates.)Wenowrecognize X c and X ~ c 0 tobejusttheusual Faddeev-Popovghost C ( )of -reparametrizationsandFaddeev-Popov anti ghost = e C ( )of -reparametrizations(asin(8.1.13)),ofconformalweight s-1and2,respectively,whichisequivalenttotherelation(8.2.2,3)( asseenbyusing(7.1.7b)). Finally,wecan(functionally)Fouriertransformtheantig hostsothatitisreplaced withthecanonicallyconjugateghost.Ournalvertexfunct ionistherefore J (3) c = 2 ig p + r p +1 p +2 p +3 C r ( int ) X p + [ X a ][ p + C ][ p + e C ] ; (12 : 1 : 4 a ) orintermsofmomenta f J (3) c = 2 igp + r C r ( int ) X p + [ p + 1 P a ] p + 2 C # p + 2 e C # : (12 : 1 : 4 b ) Theextra p + 'sdisappearduetoFouriertransformationofthezero-mode s c : 1 p + Z dce cp ~ c f ( p + c )= Z dce cp ~ c =p + f ( c )= ~ f p ~ c p + : (12 : 1 : 5) Equivalently,theexponentof U by(3.4.3a)is c@=@c + M 3 ,sothezero-modepart justscales c ,but c@=@c =1 ( @=@c ) c ,sobesidesscaling @=@c thereisanextra factorof p + foreachzero-mode,cancelingthosein(12.1.1,2).Thereis noeecton thenormalizationwithrespecttononzero-modesbecauseof theabove-mentioned normalizationinthedenitionofwithrespecttothecreat ionandannihilation operators.Asimilaranalysisappliestoclosedstrings[12 .2]. Hata,Itoh,Kugo,Kunitomo,andOgawa[12.3]proposedanint eractingBRST operatorequivalenttothisone,andcorrespondinggaugexedandgauge-invariant

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12.2.Midpointinteraction 217 actions,butwith p + treatedasanextracoordinateasin[2.7].(Asimilarearlie r attemptappearedin[4.9,7.5],with p + xed,asaconsequenceofwhichtheBRST algebradidn'tclosetoallorders.Similarattemptsappear edin[12.4].)However, asexplainedin[2.7],suchaformalismrequiresalsoanaddi tionalanticommuting coordinateinorderfortheloopstowork(aseasilycheckedf ortheplanar1-loop graphwithexternaltachyons[2.7]),andcanleadtoproblem swithinfraredbehavior [4.4]. Theusualfour-pointvertexofYang-Mills(andeven-higher -pointverticesofgravityfortheclosedstring)willbeobtainedonlyaftereldre denitionsofthemassive elds.Thiscorrespondstothefactthatitshowsupinthezer o-slopelimitofthe S-matrixonlyaftermassivepropagatorshavebeenincluded andreducedtopoints. IntermsoftheLagrangian,forarbitrarymassiveelds andmasslesselds ,the terms,forexample, L = 1 2 [ 2 + M 2 + M 2 U ( )] + M 2 V ( )(12 : 1 : 6 a ) (where U ( )and V ( )representsomeinteractionterms)become,inthelimit M 2 1 L = M 2 [ 1 2 (1+ U ) 2 + V ] : (12 : 1 : 6 b ) Thecorrespondingeldredenitionis =~ V 1+ U ; (12 : 1 : 6 c ) whichmodiestheLagrangianto L = 1 2 ~ ( 2 + M 2 + M 2 U )~ + 1 2 M 2 V 2 1+ U + O ( M 0 ) : (12 : 1 : 6 d ) Theredenedmassiveelds~ nolongercontributeinthezero-slopelimit,andcan bedroppedfromtheLagrangianbeforetakingthelimit.Howe ver,theredenition hasintroducedthenewinteractionterm 1 2 M 2 V 2 1+ U intothe -partoftheLagrangian. 12.2.Midpointinteraction WittenhasproposedanextensionoftheIGL(1)gauge-invari antopenbosonic stringactiontotheinteractingcase[4.8].Althoughthere maybecertainlimitations withhisconstruction,itsharescertaingeneralpropertie swiththelight-cone(and covariantizedlight-cone)formalism,andthusweexpectth esepropertieswillbecommontoanyfutureapproaches.Theconstructionisbasedonth euseofavertexwhich

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218 12.GAUGE-INVARIANTINTERACTIONS consistsmainlyof -functionals,asinthelight-coneformalism.Althoughthe geometryoftheinnitesimalsurfacecorrespondingtothese -functionalsdiersfromthat ofthelight-conecase,theyhavecertainalgebraicfeature sincommon.Inparticular, byconsideringastructureforwhichthe -functional(timescertainvertexfactors) isidentiedwiththeproductoperationofacertainalgebra ,theassociativityofthe productfollowsfromtheusualpropertiesof -functionals.Thisissucienttodene aninteracting,nilpotentBRSToperator(orLorentzgenera torswith[ J i ;J j ]=0), whichinturngivesaninteractinggauge-invariant(orLore ntz-invariant)action. Thestringeldsareelementsofanalgebra:avectorspacewi thanouterproduct Wewriteanexplicitvectorindexonthestringeld i ,where i = Z ( )isthecoordinates( X;C; e C forthecovariantformalismand X T ;x forthelightcone),and excludesgroup-theoryindices.Thentheproductcanbewrit tenintermsofarank-3 matrix ( ) i = f i jk k j : (12 : 2 : 1) Inordertoconstructactions,andbecauseoftherelationof aeldtoarst-quantized wavefunction,werequire,inadditiontotheoperationsnec essarytodeneanalgebra, aHilbert-spaceinnerproduct h j i = Z D Ztr y = tr i y i ; (12 : 2 : 2) where tr isthegroup-theorytrace.Furthermore,inordertogivethe hermiticity conditionontheeldwerequireanindenite,symmetriccha rge-conjugationmatrix nonthisspace: i =n ij j y ; (n)[ X ( ) ;C ( ) ; e C ( )] [ X ( ) ;C ( ) ; e C ( )] : (12 : 2 : 3 a ) nisthe\twist"of(11.1.1).Toallowarealityconditionor, combiningwith(12.2.3a), asymmetryconditionforrealgrouprepresentations(forSO (N)orUSp(N)),wealso requirethatindicescanbefreelyraisedandlowered: ( ) i =n ij jk ( ) t k ; (12 : 2 : 3 b ) where isthegroupmetricand t isthegroup-indextranspose.Inordertoperform theusualgraphicalmanipulationsimpliedbyduality,thet wistmusthavetheusual eectonvertices,andthusontheinnerproduct: n( )=(n) (n) : (12 : 2 : 4)

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12.2.Midpointinteraction 219 Furtherpropertiessatisedbytheproductfollowfromthen ilpotenceoftheBRST operatorandintegrabilityoftheeldequations Q =0:Dening Q = Q 0 + ( Q ) i = Q 0 i j j + f i jk k j ; (12 : 2 : 5) S = Z D Z tr y ( 1 2 Q 0 + 1 3 )= tr h 1 2 (n Q 0 ) ij j i + 1 3 (n f ) ijk k j i i ; (12 : 2 : 6) wendthathermiticityrequires (n Q 0 ) ij =( Q 0 n) ji ; (n f ) ijk =( f nn) kji ; (12 : 2 : 7) integrabilityrequiresantisymmetryof Q 0 andcyclicityof : (n Q 0 ) ij = (n Q 0 ) ji ; (n f ) ijk =(n f ) jki (12 : 2 : 8) (wherepermutationofindicesisinthe\graded"sense,butw ehaveomittedsome signs:e.g.,(n Q 0 ) ij j i =+(n Q 0 ) ij j i when[ Z ]and[ Z ]are anticommuting but i and i includecomponentsof either statistics),andnilpotencerequires,besides Q 0 2 =0,that isBRSTinvariant(i.e., Q 0 isdistributiveover )andassociative: Q 0 ( )=( Q 0 ) + ( Q 0 ) $ Q 0 i l ( f nn) ljk + Q 0 j l ( f nn) ilk + Q 0 k l ( f nn) ijl =0 ; (12 : 2 : 9) ( )=( ) $ (n f ) ijm f m kl =(n f ) jkm f m li (12 : 2 : 10) (wherewehaveagainignoredsomesignsduetograding). shouldalsobeinvariant underalltransformationsunderconservedquantities,and thustheoperators @=@ z R d@=@ Z mustalsobedistributiveover Atthispointwehavemuchmorestructurethaninanordinarya lgebra,and onlyonemorethingneedstobeintroducedinordertoobtaina matrixalgebra:an identityelementfortheouterproduct I = I = $ f i jk I k = i j : (12 : 2 : 11) It'snotclearwhystringeldtheorymusthavesuchanobject ,butboththelight-cone approachandWitten'sapproachhaveone.Inthelight-conea pproachtheidentity elementisthegroundstate(tachyon)atvanishingmomentum (includingthestring length,2 0 p + ),whichisrelatedtothefactthatthevertexforanexternal tachyon takesthesimpleform: e ik X (0) :.Wenowconsidertheeldsasbeingmatricesin Z ( )-spaceaswellasingroupspace,althoughthevectorspaceo nwhichsuchmatrices

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220 12.GAUGE-INVARIANTINTERACTIONS actmightnotbe(explicitly)dened.(Suchaformalismmigh tbeaconsequenceof thesamedualitypropertiesthatrequiregeneralmatrixstr ucture,asopposedtojust adjointrepresentation,inthegroupspace.) isnowthematrixproduct.(12.2.7,10) thenexpressjusttheusualhermiticityandassociativityp ropertiesofthematrix product.Thetraceoperation Tr ofthesematricesisimpliedbytheHilbert-space innerproduct(12.2.2): h j i = Tr y $ Tr = h I j i : (12 : 2 : 12) (12.2.8)statestheusualcyclicityofthetrace.Finally,t hetwistmetric(12.2.3)is identiedwiththematrixtranspose,inadditiontotranspo sitioninthegroupspace, asimpliedby(12.2.4).Usingthistranspositionincombina tionwiththeusualhermitianconjugationtodenethematrixcomplexconjugate,t hehermiticitycondition (12.2.3a)becomesjusthermiticityinthegroupspace:Deno tingthegroup-space matrixindicesas a b a b = b a : (12 : 2 : 13) Asaresultof I t = I $ n ij I j = I i (12 : 2 : 14) andthefactthat Q 0 and @=@ z aredistributiveaswellasbeing\antisymmetric"(odd undersimultaneoustwistingandintegrationbyparts),we nd Q 0 I = @ @ z I =0 : (12 : 2 : 15) Givenone product,it'spossibletodeneotherassociativeproducts bycombiningitwithsomeoperators d whicharedistributiveoverit.Thus,theconditionof associativityof ? and implies A?B = A dB d 2 =0 A?B =( dA ) ( dB ) d 2 = dord 2 =0 : (12 : 2 : 16) Theformerallowstheintroductionofconservedanticommut ingfactors(asforthe BRSTopen-stringvertex),whilethelatterallowstheintro ductionofprojectionoperators(asexpectedforclosedstringswithrespectto N ). Thegaugetransformationsandactioncomedirectlyfromthe interactingBRST operator:Usingtheanalysisof(4.2.17-21), = Z ;Q c ; c = Q 0 +[ ; ] ; (12 : 2 : 17 a )

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12.2.Midpointinteraction 221 wherethelastbracketisthecommutatorwithrespecttothe product,and S = iQ = Z 1 2 y Q 0 + 1 3 y ( ) ; (12 : 2 : 17 b ) whereforphysicaleldswerestrictto J 3 =0 ;J 3 = : (12 : 2 : 17 c ) Gaugeinvariancefollowsfrom Q 2 = 1 2 [ Q;Q ] c =0.Actually,theprojectiononto J 3 =0issomewhatredundant,sincetheothereldscanberemove dbygauge transformationsornondynamicaleldequations,atleasta ttheclassicallevel.(See (3.4.18)forYang-Mills.) Apossiblecandidateforagauge-xedactioncanbewritteni ntermsof Q as S = Q; 1 2 Z y O c + 1 6 Z y ( Q INT )(12 : 2 : 18 a ) = 1 2 Z ( Q 0 ) y c; @ @c # 1 2 Z ( Q INT ) y c; @ @c # 1 3 ; (12 : 2 : 18 b ) where Q 0 and Q INT arethefreeandinteractiontermsof Q ,and O = 1 2 [ c;@=@c ]. Eachtermin(12.2.18a)isseparatelyBRSTinvariant.TheBR STinvarianceofthe secondtermfollowsfromtheassociativitypropertyofthe product.Duetothe 1 = 3in(12.2.18b)onecaneasilyshowthatall 2 termsdropout,duejusttothe c dependenceofandthecyclicityof Q INT .Suchtermswouldcontainauxiliaryelds whichdropoutofthefreeaction.Wewouldliketheseeldsto occuronlyinaway whichcouldbeeliminatedbyeldredenition,correspondi ngtomaintainingagauge invarianceofthefreeactionattheinteractinglevel,sowe couldchoosethegauge wheretheseeldsvanish.Unfortunately,thisisnotthecas ein(12.2.18),soallowing thisabeliangaugeinvariancewouldrequireaddingsomeadd itionalcubic-interaction gauge-xingterm,eachtermofwhichcontainsauxiliary-e ldfactors,suchthatthe undesiredauxiliaryeldsareeliminatedfromtheaction. Whereasthe -functionalsusedinthelight-coneformalismcorrespondt oa\rat" geometry(seechapt.10),withallcurvatureintheboundary (specically,thesplittingpoint)ratherthanthesurfaceitself,thoseusedinWit ten'scovariantformalism correspondtothegeometry(withthe3externallegsamputat ed)

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222 12.GAUGE-INVARIANTINTERACTIONS C C C C C C C C C C C C C C C X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X X (Theboundariesofthestringsareontop;thefoldsalongthe bottomdon'taect theintrinsicgeometry.)Allthecurvatureisconcentrated inthecuspatthebottom (asmallcirclearounditsubtendsanangleof3 ),withnointrinsiccurvatureinthe boundary(allpartsoftheboundaryformanglesof withrespecttothesurface). Eachstringis\folded"inthemiddle( = 2),andthusthevertex -functionalsequate thecoordinates Z ( )ofonestringwith Z ( )forthenextstringfor 2 [0 ;= 2] (andthereforewith Z ( )forthepreviousstringfor 2 [ = 2 ; ]).These functionsareeasilyseentodeneanassociativeproduct:T wosuccessive products produceacongurationliketheoneabove,butwith4strings ,andassociativityis justthecyclicityofthis4-stringobject(see(12.2.10)). (However,vertexfactorscan ruinthisassociativitybecauseofdivergencesofthecoinc idenceoftwosuchfactors intheproductoftwovertices,asinthesuperstring:seebel ow.) Similarremarksapplytothecorrespondingproductimplied bythe functionals of(9.2.23)ofthelight-coneformalism,butthereassociat ivityisviolatedbyanamount whichisxedbythelight-cone4-pointinteractionvertex. Thisisduetothefactthat inthelight-coneformalismthereisa4-pointgraphwhereas tringhasastringsplit ofromoneside,followedbyanincomingstringjoiningonto thesameside.Ifno conformaltransformationismade,thisgraphisnonplanar, unlikethegraphwhere thissplittingandjoiningoccuronoppositesides.Thereis asimilargraphwhere thesplittingandjoiningoccurontheoppositesidefromthe rstgraph,andthese 2graphsarecontinuouslyrelatedbyagraphwitha4-pointve rtexasdescribedin chapt.10.The -positionoftheinteractionpointinthesurfaceofthestri ngvaries fromoneendofthestringtotheother,withthevertexhaving thispointatanend beingthesameasthelimitofthe1oftheother2graphswheret hepropagatorhas vanishinglength.Ontheotherhand,inthecovariantformal ismthelimitsofthe

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12.2.Midpointinteraction 223 2correspondinggraphsarethesame,sono4-pointvertexisn eededtointerpolate betweenthem. Theappropriatevertexfactorforthemidpointinteraction followsdirectlyfrom thequantummechanicswithbosonizedghosts.From(8.1.28a ),since w = 1for b C ,thereisan R d 2 e 1 3 2 R termwhichcontributesonlywherethereiscurvature. Thus,thetotalcontributionofthecurvatureatthecusptot hepathintegralisan extrafactorofanexponentialwhoseexponentis3/2times evaluatedatthecusp. TheBRSTinvarianceoftherst-quantizedactionthenguara nteesthatthevertex conserves Q 0 ,andthecoecientofthecurvaturetermcompensatesforthe anomaly inghost-numberconservationatthecusp.(Similarremarks applyforfermionized ghostsusingtheLorentzconnectionterm.)Alternatively, thecoecientfollowsfrom consideringtheghostnumberoftheeldsandwhatghostnumb erisrequiredforthe vertextogivethesamematrixelementsforphysicalpolariz ationsasinthelightconeformalism:Intermsofbosonized-ghostcoordinates,a nyphysicalstatemustbe e ^ q= 2 by(8.1.19,21a).Sincethe functionalandfunctionalintegrationin have nosuchfactors,andthevertexfactorcanbeonlyatthecusp( otherwiseitdestroys theabovepropertiesofthe functionals),itmustbe e 3 = 2 evaluatedatthecuspto cancelthe^ q -dependenceofthe3elds.Intermsoftheoriginalfermioni cghosts, weusethelatterargument,sincetheanomalouscurvaturete rmdoesn'tshowup intheclassicalmechanicslagrangian(althoughasimilara rgumentcouldbemade byconsideringquantummechanicalcorrections).Thenthep hysicalstateshaveno dependenceon c ,whilethevertexhasa dc integrationforeachofthe3coordinates, andasingle functionforoverallconservationofthe\momentum" c .(Asimilar argumentfollowsfromworkingintermsofFouriertransform edeldswhichdepend insteadonthe\coordinate" @=@c .)Theappropriatevertexfactoristhus C ( 2 ) ~ C ( 2 ) ^ C ( 2 ) ^ C ( 2 ) ; (12 : 2 : 19 a ) or,intermsofbosonizedghosts(butstillforeldswithfer mionicghostcoordinates) e 2 ( = 2) e ^ ( = 2) e ^ ( = 2) ; (12 : 2 : 19 b ) where = 2,themidpointofeachstring,isthepositionofthecusp.(T hedierencein thevertexfactorfordierentcoordinatesisanalogoustot hefactthatthe\scalar" ( g ) 1 = 2 D ( x x 0 )ingeneralrelativityhasdierentexpressionsfor g indierent coordinatesystems.)Thevertex,includingthefactor(12. 2.19),canbeconsidered aHeisenberg-picturevacuuminthesamewayasinthelight-c oneformalism,where

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224 12.GAUGE-INVARIANTINTERACTIONS thevertex j V i = e j 0 i in(9.2.1)wastheeectofactingontheinteraction-pictur e vacuumwiththeS-matrix(oftherst-quantizedtheory).Ho wever,inthiscasethe vacuumincludesvertexfactorsbecausetheappropriatevac uumisnotthetachyon onebutrathertheoneleftinvariantbytheSp(2)subalgebra oftheVirasoroalgebra [12.5](seesect.8.1).ThisisaconsequenceoftheSp(2)sym metryofthetreegraphs. Becauseofthemidpointformoftheinteractionthereisaglo balsymmetrycorrespondingtoconformaltransformationswhichleavethemi dpointxed.Insecondquantizednotation,anyoperatorwhichisthebracketof(th einteracting) Q with somethingitselfhasavanishingbracketwith Q ,andisthereforesimultaneously BRSTinvariantandgeneratesaglobalsymmetryoftheaction (because Q isthe action).Inparticular,wecanconsider Q; Z y b C ( ) b C ( ) # c Z y b G ( ) b G ( ) ; (12 : 2 : 20) wherethe2 = b C termscancelintheinteractiontermbecauseoftheformofth e overlapintegralforthevertex(andthelocationofthevert exfactor(12.2.19)atthe midpoint),andthesurvivingfreetermcomesfromtherst-q uantizedexpressionfor b G = f Q 0 ;= b C g .Thus,thissubalgebraoftheVirasoroalgebraremainsaglo bal invarianceattheinteractinglevel(withoutbecominginho mogeneousintheelds). ThemodeexpansionofWitten'svertexcanbeevaluated[12.6 ,7]asinthelightconecase(sect.9.2).(PartialevaluationsweregivenandB RSTinvariancewasalso studiedin[12.8].)Now = 1 2 ( P; P ) C 0 ; C : (12 : 2 : 21) ( b C hasweight w =2.)Themapfromthe planetothe z planecanbefoundfrom thefollowingsequenceofconformaltransformations: x i= 2 21,3 = ln x i 31 2 = i 1 1+ = e

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12.2.Midpointinteraction 225 & $ % x 0 2 1,3 = 3 = 2 = i i + & $ % x 0 2 13 = i z i + z = 2 = 3 x i 2 3 1 z = i 1 1+ (Thebold-facenumberslabeltheendsofthestrings.)Thism apsthestringfroman inniterectangle( )totheupper-halfplane( )totheinterioroftheunitcircle( )to adierentcirclewithallthreestringsappearingonthesam esheet( )totheupperhalfplanewithallstringsononesheet( z ).Ifthecutfor ( )ischosenappropriately (thepositiveimaginaryaxisofthe plane),thecutunderwhichthethirdstringis hiddenisalongthepartoftheimaginary axisbelow i= 2.(Moreconveniently,if thecutistakeninthenegativerealdirectioninthe plane,thenit'sinthepositive realdirectioninthe plane,withhalvesof2stringshiddenunderthecut.)Sincet he lasttransformationisprojective,wecandropit.(Project ivetransformationsdon't aectequationslike(9.2.14).) Unfortunately,althoughthecalculationcanstillbeperfo rmed[12.7],thereisnow nosimpleanalogto(9.2.18).It'seasiertouseinsteadamap similarto(9.2.16)by consideringa6-string functionalwithpairsofstringsidentied[12.6]:Specic ally, wereplacethelast2mapsabovewith & $ % x 0 6 4 2 1 3 5 ^ = i ^ z i +^ z = 1 = 3 x i 2 4 6 3 1 5 ^ z = i 1 ^ 1+ ^

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226 12.GAUGE-INVARIANTINTERACTIONS andthus,relabeling r r +1(orperforminganequivalentrotationofthe ^ circle) = 5 X r =1 p + r ln (^ z z r ) ln 3 ; p + r =( 1) r +1 ;z r =( p 3 ; 1 p 3 ; 0 ; 1 p 3 ; p 3 ; 1 ) : (12 : 2 : 22) Wecanthenusethesameprocedureasthelightcone.However, itturnsouttobe moreconvenienttoevaluatethecontourintegralsintermso f ratherthan^ z .Also, insteadofapplying(9.2.18)to(12.2.22),weapplyittothe correspondingexpression for ^ : = 5 X r =1 r ln ( ^ r ) 1 4 i; p + r =( 1) r ; r = e i ( r 2) = 3 : (12 : 2 : 23) Reexpressing(9.2.18)intermsof ^ ,wend @ @ + @ @ 0 ln ( ^ ^ 0 )= 1 6 ( ^ 3 + ^ 0 3 )+( ^ ^ 0 2 + ^ 2 ^ 0 )+ 1 ^ ^ 0 2 + 1 ^ 2 ^ 0 : (12 : 2 : 24) Usingtheconservationlaws,therstsetoftermscanbedrop ped.Sinceit'sactually = ^ 2 (or z ),andnot ^ ,forwhichthestringismappedtothecomplexplane ( ^ describesa6-stringvertex,andthusdoublecounts),the ln weactuallywantto evaluateis ln ( 0 )= ln ( ^ ^ 0 )+ ln ( ^ + ^ 0 ) : (12 : 2 : 25) Thisjustsaysthatthegeneralcoecients N rs inmultiplyingoscillatorsfrom string r timesthosefromstring s isrelatedtothecorrespondingctitious6-string coecients f N rs by N rs = f N rs + f N r;s +3 : (12 : 2 : 26) Thecontourintegralscannowbeevaluatedover intermsof 1+ x 1 x 1 = 3 = 1 X 0 a n x n ; 1+ x 1 x 2 = 3 = 1 X 0 b n x n ; (12 : 2 : 27) Thesecoecientssatisfytherecursionrelations ( n +1) a n +1 = 2 3 a n +( n 1) a n 1 ; ( n +1) b n +1 = 4 3 b n +( n 1) b n 1 ; (12 : 2 : 28) whichcanbederivedbyappropriatemanipulationsofthecor respondingcontour integrals:e.g., a n = I 0 dx 2 ix 1 x n 1+ x 1 x 1 = 3 = I 0 dx 2 ix 1 x n 3 2 (1 x 2 ) 1+ x 1 x 1 = 3 # 0 : (12 : 2 : 29)

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12.2.Midpointinteraction 227 Becauseof i 'srelativeto(12.2.27)appearingintheactualcontourint egrals,weuse insteadthecoecients A n = a n ( ( 1) n= 2 ( neven ) ( 1) ( n 1) = 2 ( nodd ) ; (12 : 2 : 30) andsimilarlyfor B n .Wenallyobtainanexpressionsimilarto(9.2.27),except that wemustuse(12.2.26),and f N rsmn = 1 m p + r + n p + s M rsmn ; M r;r + t;mn = 1 3 c mnt h A m B n +( 1) m + n + t B m A n i ; c mnt = ( ( 1) m Re ( e it 2 = 3 )( m + neven ) Im ( e it 2 = 3 )( m + nodd ) : (12 : 2 : 31) Thetermsfor n = m 6 =0or n =0 6 = m canbeevaluatedbytakingtheappropriate limit( n m or n 0). m = n =0canthenbeevaluatedseparately,using (9.2.22b),(12.2.22),and(12.2.26).Thenalresultis ( 1 ; 2 )= X 0 1 N 2 1 4 ln 3 3 2 4 X p 2 ; (12 : 2 : 32) where P 0 isover r;s =1 ; 2 ; 3and m;n =0 ; 1 ;:::; 1 exceptfortheterm m = n =0. Asfor(9.2.27), referstoallsetsofoscillators,with replacedwith p + 1 w for oscillatorsofweight w .Inthiscaseweuse(12.2.21),andthe p + 'sareall 1,sofor theghoststhereisanextrasignfactor p + r p + s for f N rsmn Thereareanumberofproblemstoresolveforthisformalism: (1)Incalculating S-matrixelements,the4-pointfunctionisconsiderablymo rediculttocalculatethan inthelight-coneformalism[12.9],andtheconformalmapsa resocomplicatedthat it'snotyetknownhowtoderiveeventhe5-pointfunctionfor tachyons,althoughargumentshavebeengivenforequivalencetothelight-cone/e xternal-eldresult[12.10]. (2)Itdoesn'tseempossibletoderiveanexternal-eldappr oachtointeractions,since thestringlengthsareallxedtobe .Inthelight-coneformalismtheexternal-eld approachfollowsfromchoosingtheLorentzframewhereallb ut2ofthestringlengths (i.e., p + 's)vanish.(Thus,e.g.,inthe3-stringvertex1stringredu cestoapointon theboundary,reducingtoavertexasinsect.9.1.)Thisisre latedtothefactthat I of(12.2.11)isjusttheharmonicoscillatorgroundstateat vanishingmomentum(and length)forthelight-coneformalism,butforthisformalis mit's [ X ( ) X ( )]. (3)Thefactthatthegauge-invariantvertexissodierentf romthelight-conevertex indicatesthatgauge-xingtothelight-conegaugeshouldb edicult.Furthermore,

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228 12.GAUGE-INVARIANTINTERACTIONS thelight-coneformalismrequiresa4-pointinteractionin theaction,whereasthiscovariantformalismdoesn't.Perhapsaformalismwithalarge rgaugeinvarianceexists suchthatthese2formalismsarefoundby2dierenttypesofg augechoices.(4) Thereissomedicultyinextendingthediscussionofsect.1 1.1fortheclosedstring totheinteractingcase,sincetheusualformofthephysical -statevertexrequiresthat thevertexberelatedtotheproductofopen-stringvertices fortheclockwiseand counterclockwisestates,multipliedbycertainvertexfac torswhichdon'texistinthis formalism(althoughtheywouldinaformalismmoresimilart othelight-coneone, sincethelight-coneformalismhasmorezero-modeconserva tionlaws).Thisisparticularlyconfusingsinceopenstringsgenerateclosedonesat the1-looplevel.However, someprogressinunderstandingtheseclosedstringshasbee nmade[12.11].Also,a generalanalysishasbeenmadeofsomepropertiesofthe3-po intclosed-stringvertex requiredbyconsistencyofthe1-looptadpoleand4-stringt reegraphs[12.12],using techniqueswhichareapplicabletoverticesmoregeneralth an -functionals[12.13]. Thegauge-xingofthisformalismwithaBRSTalgebrathatcl osesonshellhas beenstudied[12.14].Ithasbeenshownbothintheformalism oflight-cone-likeclosed stringtheory[12.15]andforthemidpoint-interactionope nstringtheory[12.16]that thekinetictermcanbeobtainedfromanactionwithjustthec ubictermbygiving anappropriatevacuumvaluetothestringeld.However,whe reasintheformercase (barringdicultiesinloopsmentionedabove)thisvacuumv alueisnaturalbecause ofthevacuumvalueofthecovariantmetriceldforthegravi ton,inthelattercase thereisnoclassicalgravitonintheopenstringtheory,sot heexistence(orusefulness) ofsuchamechanismissomewhatconfusing. Themidpoint-interactionformulationoftheopensuperstr ing(asatruncated spinningstring)hasalsobeendeveloped[11.3,12.17].The supersymmetryalgebra closesonlyonshell,andtheactionapparentlyalsoneeds(a tleast)4-pointinteractions tocanceldivergencesin4-pointamplitudesduetocoincide nceofvertexoperator factors(bothofwhichoccuratthemidpoint)[12.18].Suchi nteractionsmightbeof thesametypeneededinthelight-coneformulation(chapt.1 0). Exercises (1)ChecktheBRSTinvarianceof(12.2.18).(2)Findthetransformationof ln ( z z 0 )undertheprojectivetransformation z az + b cz + d (andsimilarlyfor z 0 ).Usetheconservationlaw P r 0 r =0toshowthat(9.2.14) isunaected.

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Exercises 229 (3)Derivethelasttermof(12.2.32).

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240 REFERENCES S.Samuel,Nucl.Phys.B296(1988)187. [12.18]M.B.GreenandN.Seiberg,Nucl.Phys.B299(1988)55 9; C.Wendt,Scatteringamplitudesandcontactinteractionsi nWitten'ssuperstringeld theory,SLACpreprintSLAC-PUB-4442(Nov.1987).

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INDEX 241 INDEX Action see BRST;Lightcone;Yang-Mills Anomalies see Graphs,anomalies Auxiliaryelds2.1,4.1Bosonicstrings6.1,7.1,8.1Bosonization8.1Boundstates9.3Boundaryconditions6.1,6.3Bracket,covariant3.4,4.2BRST3.1,4.1,11.1,12.1 action,gauge-invariant4.1,11.1,12.2BRST13.2,3.4,4.1BRST23.2,3.3closedstrings11.1extramodes3.2,4.3gaugexing3.2,4.4,5.5,12.2GL(1 j 1)4.2,8.2 GL(2 j 2)3.6,4.1,8.3 IGL(1)3.2,4.2,8.1,12.1,12.2innitetowerofghosts5.5interacting2.4,3.2,3.3,3.4,4.2, 12.1,12.2 IOSp(1,1 j 2)3.3,11.1 IOSp(2,2 j 4)3.6,4.1 IOSp(D,2 j 2)2.6,3.4 IOSp(D+1,3 j 4)3.6,4.1,5.5,8.3 Lorentzgauge3.2Lorentzgauge,string8.3openstrings8.1,8.3,12.1,12.2OSp(1,1 j 2)3.3,3.4,3.5,3.6,4.1,4.5, 8.2,8.3 OSp(D,2 j 2)2.6,3.4 particles5.2,5.5supersymmetry5.5,7.3temporalgauge3.1,3.2U(1,1 j 1,1)3.6,4.1,4.5,5.5 Canonicalquantization covariant3.4,4.2lightcone2.4 Chan-Patonfactors see Grouptheoryindices Chiralscalars3.1,6.2Classicalmechanics conformalgauge6.2gaugecovariant6.1lightcone6.3particle5.1 Closedstrings1.1,1.2,6.1,6.3,7.1,8.2, 10,11.1,11.2,12.2 Coherentstates5.5,9.1Compactication1.2,1.3Componentexpansions11.2Conformaltransformations D=26.2,8.1,9.1,9.2D > 22.2,2.3 Sp(2)8.1,9.2,9.3,12.2superconformal5.4 Conjugation,charge,complex,orhermitian see Propertimereversal Constraints see BRST Cosetspacemethods2.2,5.4Covariantizedlightcone see Lightcone,covariantized Criticaldimension1.2,7.1,8.1DDFoperators8.1Dimension,spacetime D =11.1,2.6,5.1 D =21.1,1.3,3.1,6.1 D =35.4,7.3 D =41.1,1.3,5.4,7.3 D =61.3,5.4,7.3 D =101.2,1.3,5.4,7.3 D =261.2,7.1,8.1 Dimensionalreduction2.3,5.4Divergences see Graphs,loops Divisionalgebras5.4Duality9.1Externalelds see Graphs,externalelds Fermionization8.1Fermions3.5,4.5,5.3Field,string10.1,11.1,12.1Fieldequations eldstrengths2.2,5.4,7.3gaugeeldssee BRST;Yang-Mills Finiteness see Loops 4+4-extension see BRST:GL(2 j 2),IOSp(2,2 j 4),U(1,1 j 1,1) Functionalintegrals9.2r matrices2.1,3.5,5.3,5.4,7.2,7.3 Gauge

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242 INDEX see BRST;Lightcone Gaugexing see BRST;Lightcone Ghosts see BRST Graphs anomalies1.2,8.1,9.3covariantizedlightcone2.6externalelds9.1lightcone2.5loops1.1,1.2,9.3trees9.2 Gravity4.1,11.2 super1.1 Grouptheoryindices7.1,9.2,10Gupta-Bleuler3.1,3.2,5.4,7.3,8.1Hadrons1.1Hamiltonianquantization3.1Heteroticstrings see Superstrings IGL(1) see BRST Interactingstringpicture see Graphs Interactions see BRST;Lightcone IOSp see BRST;Lightcone,covariantized Kleintransformation7.2Koba-Nielsenamplitude9.1Lagrangemultipliers3.1Length,string6.3Lightcone2.1,7.1,10 actions2.1,5.5,10bosonicparticles5.1bosonicstrings6.3,7.1,10covariantized2.6,10graphs2.5,9.2interactions2.1,2.4,9.2,10Poincarealgebra2.3,7.1,7.2,10spinningstrings7.2spinors5.3superparticles5.4,5.5superstrings7.3Yang-Mills2.1,2.4 Loops see Graphs,loops Metric D =15.1 D =26.2,8.3 D> 2 see Gravity rat-space2.1,6.2 Neveu-Schwarz-Ramondmodel see Spinningstrings No-ghosttheorem4.4,4.5Octonions5.4Orientation see Twists OSp see BRST;Lightcone,covariantized Particle1.1,5.1 bosonic5.1,5.2isospinning5.3spinning5.3supersymmetric5.4 Pathintegrals9.2Perturbationtheory see Graphs Poincarealgebra see Lightcone,Poincarealgebra Polyakovapproach see Graphs Projectivegroup8.1,9.3Propertime see Dimension,spacetime, D =1 Propertimereversal5.1,6.3,8.1,8.3,10QCD 1.1 Quantummechanics see BRST;Canonicalquantization; Lightcone;Pathintegrals Quaternions5.4Ramond-Neveu-Schwarzmodel see Spinningstrings Reggetheory1.1,9.1Reparametrizationinvariance see Metric, D =1and D =2 Scaleinvariance,2Dlocal(Weyl) see Metric, D =2 reversal see Twists S-matrix see Graphs Spectrum,mass7.1Spinningstrings7.2Spinors see Fermions; r matrices;Supersymmetry

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INDEX 243 String seethewholebook Stueckelbergelds4.1Superconformaltransformations5.4,7.2Supergravity see Gravity,super Superspin5.4,5.5Superstrings(I,IIAB,heterotic)1.2,7.3,9.3, 10 Supersymmetry D =15.3 D =27.2 D> 21.2,1.3,5.4,5.5,7.3 Tension,string( 0 )6.1 Trees see Graphs,trees Twistors5.4Twists7.1,10Venezianomodel see Bosonicstring Vertices see BRST;Lightcone Virasorooperators see Conformaltransformations,2D Worldsheet see Dimension,spacetime,D=2 Yang-Mills2.1,2.4,3.1,3.2,3.3,3.4,4.1, 4.2,4.4,4.6,11.2 super5.4,9.1