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Noncommutative field theories, solitons and superalgebra

University of Florida Institutional Repository
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NONCOMMUTATIVESCALARFIELDTHEORIES, SOLITONSANDSUPERALGEBRA By XIAOZHENXIONG ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2002

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Copyright2002 by XiaozhenXiong

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Idedicatethisworktomyfamily.

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ACKNOWLEDGMENTS Whenonethinksabouthisorhersuccess,onewould“ndthatanindividuals successalwaysreliesonthekindnessandhelpfrommanyotherpeople.Itisno exceptionforthePhDdegreethatIwillbereceiving.Listedinthefollowingare thepersonstowardswhomIfeeldeepestappreciation.Certainlytheyarenotthe onlyonesthatdeservethecreditformysuccess.WiththisopportunityIwould alsoliketoacknowledgemythankstoallotherpeoplewhohavehelpedme. Iwouldliketobeginbythankingmyparents,YateXiongandQifangZhou. WhenIwaslittleandnaughty,theyalwaysledmethroughhardshiptohappiness. Whenolderandambitious,theywerealwaysverysupportiveofme.Theyalways leavemethebest,butneveraskmetodoanythinginreturn.EverytimeI succeed,itisalwaysbeenagreatjoytothemaswell.ThistimecertainlyIwould liketosharethewonderfulmomentwiththemagain. AspecialthankyoualsogoestomysisterXiaolinXiong.Asaneldersister, sheisalwaysreadytolookafterme.Shealwayswalksonestepaheadofme,and shareswithmehersuccessandfailure.Ifeelveryluckytohavesuchanicesister. Ifitwasnotbecauseofher,IwouldhaveexperiencedmuchmorefailurewhileI wasgrowingup.IamsureshewillfeelveryproudtoseeherbrothertogetaPhD oneday. OfcourseIoweagreatdebtofgratitudetomyadvisor,Prof.PierreRamond. Ithasbeenagreathonorandpleasuretohavehimasmysupervisor,andwork withhimthroughthePhD.Henotonlyledmethroughtheacademicresearch work,butalsofromtimetotimegavemesuggestionsonmentalstrengthfor beingadedicatedscientist.Someofhiscriticismswereevenblunt,butIamsure iv

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Iwillrememberallofthembecausetheyhavecontributedalottomysuccessin obtainingmyPhDandwillhaveatremendousamountofin”uenceinmyfuture workandlifeaswell. AlotofthanksgotoProf.RichardWoodard.Heisthevery“rstonewho introducedmetothe“eldtheory,andwasalwayswillingtoanswermyquestions indetail.TheknowledgeIhavelearnedinthe“eldtheoryclasshasalwaysbeena greatsupporttomyPhDresearch.Hisprinciplesforphysicistandfornerdhave alwaysbeenanencouragementformetomoveforwards.Iadmirehiscourageand determinationinholdinghisprinciples,becauseoftenIfoundtheyarefarmore easytounderstandthantostickto. VeryspecialthanksgotomyverygoodfriendVivianGuo.Shehasencouragedmealotduringthehardesttimei nmyPhDwork,whenIfeltexhausted physicallyandmentally.Thetimewespenttogetherisalwaysapreciousand beautifulmemorytome.AtthismomentofmysuccessIwouldliketosaythanks toherandwishherthebest. Manythanksgotomygirlfriend,QiangMei.ShegavemealotofencouragementswhenImetdicultiesinthere search,andfeltthatIwasnotableto “nishtheresearchwork.Itisagreatpleasuretotalktohereveryday.Iamlooking forwardtocontinuingtodevelopourrelationshipafterI“nishthePhD. IwouldalsoliketothankSusanRizzoandDarleneLatimer.Theirworkhas alwaysbeen,andwillalwaysbeveryhelpfultograduatestudentsandfacultyin thedepartment.CertainlyIappreciatetheconveniencetheybringtomeduring thelongtimeIhavespenthere. Finallyalotofthankswouldhavetogotothefaculty,postdoctoralsand fellowstudentsinIFT.Theyarealwaysveryfriendlyandreadytohelpanddiscuss thequestions.Theresearchenvironmen thereiswonderful.Itwouldbeagreat memoryofminespendingthebesttimeinmylifewiththepeoplehere.Particular v

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thanksgotomyotherPhDcommitteemembers,Dr.PierreSikivie,Dr.Charles Thorn,Dr.ZonganQiuandDr.PaulEhrlich.Iappreciatetheirtimeinguiding myprocesstowardsthePhD. vi

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TABLEOFCONTENTS page ACKNOWLEDGMENTS.............................iv ABSTRACT....................................ix CHAPTER 1INTRODUCTION..............................1 2NONCOMMUTATIVEPERTURBATIVEDYNAMICS... .......8 2.1NoncommutativePerturbationTheory...............8 2.2Noncommutative 4Theory.....................10 2.3RenormalizationinWess-ZuminoModel..............14 3DEFORMEDSUPERPOINCAR EALGEBRA...............19 3.1UnitaryRepresentationsofSuperPoincar eAlgebra.........19 3.2DeformedSuperPoincar eAlgebra..................25 3.2.1NotationsandIdentities....................25 3.2.24Theory...........................26 3.2.3Wess-ZuminoModel......................29 3.3Discussions..............................33 4QUANTIZATIONOFNONCOMMUTATIVESOLITONS. .......34 4.1Introduction..............................34 4.2NoncommutativeSolitonsandD-branes...............35 4.2.1NoncommutativeSolitonsinScalarFieldTheory......35 4.2.2NoncommutativeSolitonsinGaugeTheory.........37 4.3ClassicalNoncommutative Q -ballSolution.............40 4.3.1HamiltonianandEquationofMotion............41 4.3.2 Q -ballSolutions........................43 4.3.3VirialRelation.........................46 4.4QuantizationofNoncommutative Q -ball..............47 4.4.1CanonicalQuantization....................47 4.4.2EnergyCorrectionsatVerySmall .............51 4.5Finite andNoncommutativeGMSSolitons............58 4.6ConclusionandDiscussion......................60 vii

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5SOLUTIONOFKOSTANTEQUATION..................62 5.1EulerTripletfor SU (3) /SU (2) U (1)...............63 5.1.1The N =2Hypermultipletin4Dimensions.........63 5.1.2CosetConstruction......................65 5.1.3GrassmannNumbersandDiracMatrices..........66 5.1.4SolutionsofKostantsEquation...............68 5.2SupergravityinElevenDimensions.................70 5.2.1Superalgebra..........................70 5.2.2RepresentationsofGrassmanVariables...........73 5.2.3 F 4 /SO (9)OscillatorandDierentialformRepresentations78 5.2.4SolutionofKostantEquationin F 4 /SO (9).........81 6SUMMARY..................................89 APPENDIXCOMPUTERCODE .......................91 REFERENCES...................................107 BIOGRAPHICALSKETCH............................112 viii

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFul“llmentofthe RequirementsfortheDegreeofDoctorofPhilosophy NONCOMMUTATIVESCALARFIELDTHEORIES, SOLITONSANDSUPERALGEBRA By XiaozhenXiong December2002 Chair:PierreRamond MajorDepartment:Physics Thisdissertationpresentsperturbativeandnonperturbativeaspectsof noncommutative(NC)“eldtheories,aswellassuperalgebrasinNC“eldtheory andhigherdimensionaltheories.Inpa rticular,theperturbativestructuresof theNCWess-Zuminomodelareinvestigatedindetail,aswellasthedeformed superalgebrarelationsofthemodel.NCsolitonsinscalar“eldtheoryarequantized andquantumcorrectionstotheenergya recalculated,whereUV-IRdivergences arefoundsimilartothoseintheperturbativetheory.Kostantequationsinhigher dimensionsareconstructedwithdierent ialformrepresentations,inwhichthe solutionsarealsoexpressed. ix

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CHAPTER1 INTRODUCTION Thisdissertationisbasedonthefollowingpapersonnoncommutative(NC) “eldtheoryandsuperalgebra[1…4]. ThedissertationresearchfocusesonthequantumbehaviorofNC“eldtheory, includingrenormalizationoftheperturbat iveandnonperturbativestructuresinthe theory.Thechaptersareorganizedasfollows. FollowinganintroductiontoNCgeome tryascertainlimitofstringtheoryin thebeginningchapter,chapter2and3covertheworkonNCperturbative“eld theory[1].Inchapter2,wediscussthep erturbativedynamicsofNC“eldtheory. RenormalizationoftheWess-Zuminomodelisstudiedindetail.Inchapter3,a slightlydigressedtopic,thedeformedsuperPoincar ealgebrainNC“eldtheory, isdiscussed.Therepresentationoftheconservedgeneratorsisalsousefulforthe solitontheorylater. Therearecomprehensivediscussionsin literatureaboutNCsolitonsandtheir interpretationsasD-branesinNCscala rorgauge“eldtheories[5,6].However, quantizationofNCsolitonshasonlybeendi scussedinlimitedplaces[7,8],where onlylargeNClimitor1 / correctionsareconsidered.Animportantfeatureof NC“eldtheories,UV/IRmixing,isomi ttedinthelargeNClimit.Chapter4is dedicatedtoquantizationofN Csolitonsinthesmallenough limit.Inparticular quantizationofNC Q -ballsolitonsisdiscussedindetail.Quantumcorrections tothesolitonenergyarethencalculatedandUV/IRmixingstructuressimilarto thoseinperturbationtheoryarerecovered.EnergyofNCGMSsolitonsisfoundto beUVrenormalizableatone-loopwithUV/ IRtermsincludedinthecorrections. 1

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2 ThestructureissuggestedastheconsequenceofinteractionsbetweenD-branesand strings,whichcouldbeafuturedirectiontopursue. Chapter5summarizesthework[4]on11dimensionalsuperPoincar ealgebra, whereanalternativerepresentationofthesolutionsofKostantequationsin cosetspace F 4 /SO (9)isgiven.The“nalchaptersummarizestheresultsinthis dissertationanddiscussesfutureresearchdirections. Recentlytherehasbeenarevivalofinterestinthestudyofnoncommutative (NC)geometry,duetothediscoveryth atNC“eldtheoryappearstobecertain limitoftheeectiveactionoftheopenstringmodeslivingonbranes[9,10].NC geometryhasbeenformulatedinstrictmathematicalfashion[11].Theideacan becapturedasfollows:Oncommutativemanifold M thereexistsanalgebra A = C( M )ofcommutativesmoothfunctions,withtheproductbeingfunction multiplication.NCalgebraisadeformationofcommutativealgebrawithdeformed product,ormorespeci“cally,starproductfortheconcernsofthisdissertation, fg ( x ) exp( i 2 ij ix jy ) f ( x ) g ( y ) x = y, (1.1) where ij= Š jiisanon-degenerateconstantantisymmetricmatrix.NCgeometry isthende“nedintermsofNCgeneralizatio nsofthealgebraicconstructsde“nedin theordinarycommutativegeometry. ThemotivationforstudyingNCgeome tryhasbeenmanyfold.Theideacan comefromthefundamentalprincipleo fquantummechanics,wherethephase variables,positionandmomentumdonotcommute.Onecanjustconjecturethat thepositionsthemselvesmightnotcommute,leadingto [ xi,xj]= iij, (1.2) where ijisanantisymmetricparametermeasuringnoncommutativity.

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3 Quantum“eldtheorycanbewellformulatedwithNCgeometryconcept, calledNC“eldtheories.OnesimplereasontoinvestigateNC“eldtheoryisthat noncommutativitywouldintroducephase factorsthatcouldbetterregularize ultraviolet(UV)divergencepresentinordinary“eldtheories.However,aswewill see,oftenpartofUVdivergenceassociatedwithplanardiagramsarestillpresent andotherUVdivergencesassociatedwithnonplanardiagramsbecomeUV/IR divergence,whichstillneedsfurtherinterpretation. Anothermotivationisfromtheuncertaintyprincipleinquantumgravity, wherepositionisnotexpectedtobemeasuredaccuratelyatthePlanckscale. Peoplealsobelievequantumgravityshouldbenonlocalingeneral.Oneexpects byinvestigatingNC“eldtheoryasnonlocaltheorythatabetterunderstandingof nonlocalitycanbeachievedconceptuallyandpractically. StringtheoryactuallyprovidesstrongermotivationforstudyingNC“eld theory.Yang-Millstheoryisprovedtoar iseinanaturallimitinthecontextof thematrixmodelof M -theory[9,12,13],withthenoncommutativityarisingfrom theexpectationvalueofabackground “eld.NCgeometryhasalsobeenusedas aframeworkforopenstring“eldtheory[14].LaterSeibergandWittenstudied openstringsinthepresenceofaconstantNeveu-Schwarz B “eldnonzeroonthe Dp -brane[10].Inthezeroslopelimit( 0),NCgeometryarisesasalimitof stringtheory.Theeectiveactionoftheopenstringmodesonthebranebecomes NC“eldtheoryduetothepresenceof B “eld.Thesamepaperactuallyshowsthe equivalencebetweenNCYang-MillstheoryandordinaryYang-Millstheory. NC“eldtheoryhasalsoappearednat urallyincondensedmattertheory.A simpleexamplehasbeenshown[15]inwhichnoncommutativityariseswhenthe theoryofelectronsmovinginamagnetic“eldiskeptinthelowestLandaulevel incertainlimit.TheideaisgeneralizedinthetheoryofthequantumHalleect [16].Basicallyanobservablealgebra,whichiswellde“nedinperiodiccase,canbe

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4 generalizedtonon-periodicalbackground (presenceofmagnetic“eld)andactually becomesaNCmanifoldcalledNCBrillouinzone. Despitethereasonsabove,experimentalsupportforNC“eldtheoryas realisticlowenergytheoryislimited,due tononlocalityandviolationofLorentz symmetryintroducedbyuncertaintyrelationsbetweenthecoordinates.Anupper boundofNCparameter,(10 Tev )Š 2,isobtainedintheLorentzviolatingextension ofstandardmodel[17]. InthisdissertationwemainlyconsiderNC“eldtheoryfromstringtheory perspective.Inthefollowingwebrie”yillustratetheidea[10,15]thatNCYangMillstheoryarisesfromcertain limitofopenstringtheory. Theworldsheetactionforanopenstringwithnonzero B “eldon Dp -brane boundaryintheEuclideansignatureis S = 1 4 gijaxiaxjŠ i 2 Bijxitxj, (1.3) where tisatangentialderivativealongtheworldsheetboundary .For xialong thebrane,wehavetheequationofmotionattheboundary gijnxj+2 iBijtxj| =0 (1.4) TheaboveboundaryconditionsactuallyinterpolatefromNeumannboundary conditions( B =0)toDirichletboundaryconditions( B or gij 0).Witha specialboundaryconditionwhentheworldsheetisadiscconformallymappedto upperhalfplane,thepropagatorbecomes[18…20] xi( z ) xj( z ) = Š [ gijln | z Š z|Š gijln | z Š z| + Gijln | z Š z|2+ 1 2 ijln z Š z z Š z+ Dij. (1.5)

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5 where Gij= 1 g +2 B ij S= 1 g +2 B g 1 g Š 2 B ij, (1.6) Gij= gijŠ (2 )2( BgŠ 1B )ij, (1.7) ij=2 1 g +2 B ij A= Š (2 )2 1 g +2 B B 1 g Š 2 B (1.8) Thepropagatorofopenstringvertexope ratorsinsertedontheboundaryofis xi( ) xj( ) = Š Gijln( Š )2+ i 2 ij ( Š ) (1.9) where Gij,thecoecientthatdeterminestheanomalousdimensionsofopenstring vertexoperators,isreferredtoasopenstringmetric. Tofocusonthelowenergybehaviorwhiledecouplingthestringbehavior,take thezeroslopelimit( 0)oftheopenstringsystem, 1 2 0 ,gij 0 (1.10) where i,j refertothedirectionsalongthebrane.Then G and become Gij= Š1 (2 )2(1 Bg1 B)ijfor i,j alongthebrane gijotherwise (1.11) Gij= Š (2 )2( BgŠ 1B )ijfor i,j alongthebrane gijotherwise (1.12) ij= 1 Bijfor i,j alongthebrane 0otherwise (1.13) Inthislimit,theactionhasonlythetopologicalterm, Š i 2 Bijxitxj, (1.14)

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6 andthepropagator(1.9)becomes xi( ) xj( ) = i 2 ij ( Š ) (1.15) Interpreting astime,NCgeometryarisesbyevaluatingthecommutator, [ xi( ) ,xj( )]= T ( xi( ) xj( Š) Š xi( ) xj( +))= iij. (1.16) Withtheaboveequation,onecanthenarguethatforgeneraloperatorproducts O ( ) O( ),theleadingtermswouldbeindependentof Š for ,and wouldhavetogivestarproducts(1.1)ofoperators,becauseoftheassociativityand translationinvariance.Explicitly,normalorderedoperatorssatisfy : eipixi( ):: eiqixi(0):= eŠi 2ijpiqj ( ): eipx ( )+ iqx (0): (1.17) ormoregenerally, : f ( x ( )):: g ( x (0)):=: ei 2 ( ) ij x i ( ) x j (0)f ( x ( )) g ( x (0)): (1.18) wheretherighthandsideisexactlythestarproduct(1.1)ofthefunctionsonthe NCspace. Throughthegeneralprocedureforreduc tionofopenstring“eldtheorywith nonzero B “eldalongthebrane[10],taking 0andkeeping G andtheeective openstringcoupling Gs“xed,itissuggestedthattheeectiveactionisgauge invariantNCYang-Millstheorywith“eld F L ( F )= c Gs det( G +2 ( F +)) (1.19) where c = Tp/gsisindependentof gsand TpistheD p -branetensionfor B =0.In thisform appearsonlyinthestarproductaectedonlyby B ,andcertaindegrees

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7 offreedomexistinthechoiceoftheparameter,whichisgivenby 1 G +2 = Š 2 + 1 g +2 B (1.20) Todeterminetheeectiveopenstringcoupling Gs,take F =0to“ndtheconstant term, L ( F =0)= c Gs det( G +2 ) (1.21) AlsofortheDirac-Born-InfeldLagrangian(see[21]forareview)forslowingvarying “elds, LDBI= c gs det( g +2 ( F + B )) (1.22) take F =0, L ( F =0)= c gs det( g +2 B ) (1.23) Theequivalenceoftheabovetwotermsgives Gs= gs det( G +2 ) det( g +2 B ) 1 2. (1.24)

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CHAPTER2 NONCOMMUTATIVEPERTURBATIVEDYNAMICS NC“eldtheory,afterquantization,showsdierentultravioletstructures fromtheordinary“eldtheory[22].Basicallynoncommutativityintroducesphase factorsinthevertices,whichintheloopintegrationbecomeconvergentfactors thatregularizetheUVdivergence.However,someUVdivergencesarestillleft, andadditionalUV/IRdivergencesareintroduced.TheintriguingUV/IRmixing termscanbereproducedbyintegratingoutsomenewlightdegreesoffreedomwith specialpropagatorsandinteractions.Thesenewlightdegreesoffreedomcanbe interpretedasclosedstringmodeswithchannelduality[23].Evenfor“eldtheory concern,renormalizationofNC“eldtheoryneedstobereexaminedbecauseofthe changeinthedivergencestructure. InthischapterfollowingabasicintroductiontotheNC“eldtheory,perturbationdynamicsinNC 4theory[22],aswellastheimplicationsfromstringtheory, arereviewed.Thenwediscussrenormaliz ationofsupersymmetricNCWess-Zumino model. ThecommutativeWess-Zuminomodelisthesimplestsupersymmetrical“eld theorymodelin(3+1)dimension.Itin cludesascalarandafermion“eldwith supersymmetrybetweenthem.Becauseofthesupersymmetry,cancelationofthe divergenceoccursgenerally.Theonlymass renormalizationisduetowavefunction renormalization,andthevertexand masscorrectionsareabsent[24]. 2.1 Noncommutative Perturbation Theory Asexplainedintheintroductionchapte r,aspeci“cNCalgebracanbede“ned bythestarproductofthecommutativefunctions,Eqn.(1.1).Theunderlining 8

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9 noncommutativespacehasphase spacequantizationstructure, [ xi, xj]= iij, (2.1) where i,j isassumedtolabelthespacedimensionsonlyinthisdissertation. Noncommutativityassociatedwithtimecoordinatebringstheproblemsofcausality andunitaritywhenthetheoryiscanonicallyquantized[25].WeylproposedEqn. (2.1)asLiealgebraofagroupwithgroupelements, U ( p )=exp( ip x ) (2.2) InthefunctionrepresentationoftheNCalgebrawithWeylordering, f ( x ) dnp (2 )n dnxf ( x )eŠ ipxU ( p )= dnp (2 )n f ( p ) U ( p ) (2.3) where n isthenumberofthespacedimensions,theoperatorproducts f ( x ) g ( x )= dnp (2 )n dnq (2 )n f ( p ) g ( q ) U ( p ) U ( q ) = dnp (2 )n dnq (2 )n f ( p Š q ) g ( q )eŠi 2piqjijU ( p )(2.4) NCstarproduct(1.1)isjustthefunctionrepresentationoftheaboveoperator products,for f ( x ) g ( x )= dnp (2 )ndnq (2 )neŠ i ( p + q ) x f ( p ) g ( q ) eŠi 2piqjij. (2.5) TheWeylrepresentationisinitiallyus edinphasespacequantization[26].NC “eldtheoryis“rstproposedbyFilk[27],replacingtheordinaryproductsofthe “eldsbyNCstarproducts.Thepropagatorremainsthesamesince f ( x ) g ( x )=0 (2.6) Theinteractionverticesnowdependontheexternalmomentumthroughphasefactors,whichisinducedbythestarproduc ts.Thephasefactors,whileindependent

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10 oftheoverallpermutationofthemoment um,distinguishtheFeynmandiagramsto beplanarandnonplanarones.Thephasefactorsforplanardiagramdependonly onexternalmomenta,anddonotaecttheUVdivergenceintheloopintegration. Butfornonplanardiagram,thedependenceoftheinternalmomentabythephase factorintroducesregularizationinthemomentumintegral,andUVdivergenceis generallyconvertedintoUV/IRdivergen ce.Renormalizationofthetheoryneeds tobereexaminedcasebycase.ItisinterestingthatrenormalizationoftheNC “eldtheoriesdierssigni“cantlyfromtheircommutativeanalog.Forexample, NCQED,asasimpleextensionofNC U (1)YMtheory,isrenormalizableatone loopduetoSlavnov-Tayloridentityfor SU (2)likesymmetry,butthe functions includecontributionsfromelectronsin U (1)facet[28].Generalconsiderationof convergencetheoremandrenormalizationinNC“eldtheoryhasbeendiscussed [29,30].TherestofthechapterdiscussesrenormalizationofNC4theoryand Wess-Zuminomodelindetail. 2.2 Noncommutative 4 Theory TheNC4theoryinthefour-dimensionalspace-time,isdescribedby L = 1 2 Š 1 2 m2 Š 4! (2.7) Itiswellknown[10,22,27]thatundertheintegrationthestarproductofthe “eldsdoesnotaectthequadraticpartsoftheLagrangians,whereasitmakesthe interactionLagrangianbecomenonlocal.Hence,theFeynmanrulesinmomentum spaceofNC“eldtheoryaresimilartotho seofcommutativetheoryexceptthatthe verticesoftheNCtheoryaremodi“edbyaphasefactor.FortheLagrangian(2.7),

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11 theFeynmanruleforthedeformedvertexis Š i 3 (cos 1 2 ( p1 p2+ p1 p3+ p2 p3) +cos 1 2 ( p1 p2+ p1 p3Š p2 p3) +cos 1 2 ( p1 p2Š p1 p3Š p2 p3)) (2.8) where pis, i =1 ... 4,aremomentacomingoutofthevertexand pi pj= pipj.When 0,thedeformedvertexbecomesthenon-deformedone. Byusingtheabovevertex,oneyieldsawavefunctionrenormalizationofthescalar “eld atone-looporderthathasonlyonediagramasfollows: ()( p2) = Š 6 d4k i (2 )4(2+cos( p k )) ( k2+ m2) = Š 48 2 0d 2eŠ im2 1+ 1 2 eŠ i p 2 4 ei 2 = Š 48 2 2Š m2ln ( 2 m2) Š 96 2 2 effŠ m2ln ( 2 eff m2) + (2.9) TheSchwingerparametrizationtechniquetodealwiththeaboveintegrationscan befoundinItzyksonandZuber[31]andHayakawa[28].Inthesecondline,the termisproportionaltoexp( Š i p2/ 4 ),where p = p,isduetothenonplanar contributionandtheexp( i/ 2)factorisintroducedtoregulatethesmall divergenceintheplanarcontribution.Note thatthenonplanarcontributionisonehalfoftheplanarone.Inthethirdline,wekeeponlythedivergenttermsandthe eectivecuto,2 eff=1 / (1 / 2+( p2) / 4),showsthemixingofUVdivergenceand IRsingularity[22].Theaboveintegrationcanalsobedonebyusingdimensional regularizationmethod[32].Inthecasethat isacomplexscalar“eld,thereare twowaysoforderingthe“elds and inthequarticinteraction( )2.So,the

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12 mostgeneralpotentialoftheNCcomplexscalar“eldactionis A + B. Thepotentialisinvariantunderglobaltransformationsincethestarproduct hasnothingtodowiththeconstantphasetransformation.Itwasshownby Arefeva,BelovandKoshelev[33]thatthetheoryisnotgenerallyrenormalizable forarbitraryvaluesof A and B andisrenormalizableatone-looplevelonlywhen B =0or A = B Theoneloop1PIquadraticeectiveactionis S(2) 1 PI= d4p 1 2 p2+ M2+ 96 2(1 4 p2+1 2) Š M2 96 2ln( 1 M2(1 4 p2+1 2) )+ ( p ) ( Š p ) (2.10) where M2= m2+ 2 48 2Š m2 48 2ln( 2 m2)+ (2.11) istherenormalizedmass. TheappearanceofUV/IRmixingtermss uggeststhepresenceofnewdegrees ofthefreedom.IndeedthecorrectIRsingularityintheeectiveactioncanbe systematicallyreproducedbytheintroductionofnewlightdegreesoffreedom [22,23].Abriefreviewoftheideaisasfollows. Considerthemodi“edeectiveaction S eff()= Seff+ d4p 1 2 4 p2Š 1 1 4 p2+1 2Š 11( p ) 1( Š p ) + 1 2 ln( 1 4 p2+ 1 2) Š ln( 1 4 p2) Š 12( p ) 2( Š p ) + 1 96 2 ( i1+ M22) (2.12)

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13 Uponintegratingout intheaboveaction,thecorrectquadraticandlogarithmic divergencescanbereproduced,andtheUV/IRmixingtermsarecancelledbythe exchangediagrams. shavespecialpropagators, 1( p ) 1( Š p ) 4 p2Š 1 1 4 p2+1 2(2.13) 2( p ) 2( Š p ) ln( 1 4 p2+ 1 2) Š ln( 1 4 p2)(2.14) Apossibleinterpretationtothepresenceofthosenewdegreesoffreedom isthat theyareactuallytransversemodesofparticles swhichpropagatefreelyinmore dimensions.Forexample,aparticle propagatesintwoextradimensionsand coupleslinearlyto onthebranewillproducethelogarithmicpropagator.De“ne ( x )= ( x,x=0)andwritetheactionof withaLagrangemultiplier ( x ), exp Š d4x ( x ) ( x,x=0) Š d4xd2x1 2 ( )2 = [ d ][ d ]exp Š d4x { ( x ) ( x )+ i ( x )[ ( x ) Š ( x,x=0)] } Š d4xd2x1 2 ( )2 = [ d ][ d ]exp Š d4p [ ( p ) ( Š p )+ i ( Š p ) ( p )] Š d4pd2q [ 1 2 ( Š p, Š q )( 1 4 p2+ q2) ( p,q ) Š i ( Š p ) ( p,q )] (2.15) Integrateout “rst,leaving exp Š d4p ( p ) ( Š p )+ i ( p ) ( Š p )+ 1 2 ( p ) ( Š p ) dq 1 / 4 p2+ q2 (2.16) ThenintegrateouttheLagrangemultiplier ,givingthedesiredaction, exp Š d4p ( p ) ( Š p )+ 1 2 ( Š p ) ( p )lnŠ 1 1 / 4 p2+1 2 1 / 4 p2 (2.17)

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14 Thedualitybetweenthehighmomentumdegreesoffreedomin andpropagationof intheextradimensionssuggeststhat sareassociatedwithopenstrings modesand sor sareassociatedwithclosedstringmodes,sinceinthestring theorythelowenergyclosedstringmodesarerelatedtothehighenergymodes oftheopenstringsbychannelduality.Anonplanarloopdiagramistopologically equivalenttoastringdiagraminwhichanumberofopenstringsbecomesaclosed stringthatfreelypropagatesinthebulkandturnsbackintoopenstrings.ConnectionsbetweenNC“eldtheoriesandstringtheoriessuggestedbythoseanalogies havenotbeenclearlyunderstoodyet. 2.3 Renormalization in Wess-Zumino Model Inthissection,weinvestigaterenormalizationatoneloopintheNCWZ theory.NCWZLagrangianisgivenbyintro ducingstarproductsintheinteraction termsandpermutatingthosetermstopre servesupersymmetrytransformations. HerewefollowtheconventionsbySohnius[34].TheNCWZmodelisdescribedby thesumofthefreeo-shellLagrangianandofthetwoinvariants, Ltot= L0+ Lm+ Lg, (2.18) where L0= 1 2 AA + BB + i + F2+ G2 (2.19) Lm= Š m ( FA + GB + 1 2 ) (2.20) Lg= Š g 3 AAF Š BBF + ABG + ( A Š 5B ) +permutationterms] (2.21)

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15 Theo-shellLagrangians L0, Lmand Lgareseparatelyinvariantunderthe supersymmetrytransformations: A = ,B = 5 ,F = i ,G = i 5 = Š ( F + 5G ) Š i ( A + 5B ) (2.22) where and aretheglobalin“nitesimalMajoranaspinorparameters. TheFeynmanrulesinmomentumspacecanbeextractedoutdirectlyfromthe Lagrangians(2.18).Onegetsasfollows: 1.Propagators Thepropagatorsofthe“eldsandthemixed“eldsontheNCspacearethe sameasthoseonthecommutativeone. 2.Deformedvertices €Šg 3( AAF +permutationterms) Š 2 ig cos( 1 2 pAi pAf) €g 3( BBF +permutationterms) ig cos( 1 2 pBi pBf) €Šg 3( ABG +permutationterms) Š 2 ig cos( 1 2 pA pB) €Šg 3( A +permutationterms) Š igI cos( 1 2 pi pf) €g 3( 5B +permutationterms) 2 ig5cos( 1 2 pi pf) .

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16 Thedeformedverticesweobtaindierf romthenon-deformedonesbyafactor cos(1 2pi pf).ByusingtheaboveFeynmanrules,onecanstudytherenormalization oftheNCWess-Zuminomodel.Theresultsaresummarizedasfollows: 1.Wavefunctionrenormalization € Majorana“eld ForMajorana“eld,atonelooptherearetwodiagrams.Thesumof themgivesacontribution ( )( p )=8 g2 d4k i (2 )4cos2( 1 2 p k ) k ( k2Š m2)(( k + p )2Š m2) = Š p g2 4 21 0d (1 Š ) 0d eŠ i(m2Š (1 Š ) p2) 1+ eŠ1 i p 2 4 e1 i 2= Š p g2 8 2 ln ( 2 m2)+ ln ( 2 eff m2) + .... (2.23) € Scalar“elds A,B Foreachscalar“eld,atoneloopthereare“vediagrams.Thesumof themgivesacontribution ( AA )( p2)=( BB )( p2) =8 g2 d4k i (2 )4cos2( 1 2 p k ) k p ( k2Š m2)(( k + p )2Š m2) = Š p2g2 8 2 ln ( 2 m2)+ ln ( 2 eff m2) + .... (2.24) € Auxiliary“elds F,G For F “eld,atonelooptherearetwodiagramsWhile,for G “eld, atoneloopthereisonlyonediagram.However,theyhavethesame

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17 contribution ( FF )( p2)=( GG )( p2) = Š 4 g2 d4k i (2 )4cos2( 1 2 p k ) 1 ( k2Š m2)(( k + p )2Š m2) = Š g2 8 2 ln ( 2 m2)+ ln ( 2 eff m2) + .... (2.25) € Mixed“elds ( FA )( p2)=( GB )( p2)=0 (2.26) Again,alltheintegrationscanbedonedirectlybyusingtheSchwinger parametrizationtechnique[28,31].Thedivergenttermsofthewavefunction renormalizationsofall“eldsaret hesame,whereasthe“nitetermsof( FF )and( GG )aredierentfromthoseoftheothers.NotethatintheNC Wess-Zuminomodeltheplanarandnonplanarcontributionshavethesame multiplicativefactor.Renormalizatio niscutbyhalfcomparedtothatofthe ordinarycase. 2.Massrenormalizations € Since,atone-loop( )( p )isproportionaltoonly p andbothFAand GBarezero,thereisnomassrenormalization. 3.Vertexcorrections € FA2, FB2, ABG Foreachvertex,atonelooptherearetwodiagramsandtheyaddupto zero.So,thereisnocorrectionforeachvertex. € A 5B Similarly,thereisnocorrectionforeachofthesetwovertices.Since,at onelooptherearetwodiagramsandtheyaddupto“nitevalues. Justasinthe4theory,theUV/IRmixingalsoappearsintheNCWZ theory,whichisthegeneralconseque nceoftheuncertaintyrelationsamong

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18 NCcoordinates[10].RenormalizationintheNCWZtheoryisverysimilarto thecommutativeone.ComparedwiththeordinaryWess-Zuminotheory,the countertermforthewavefunctionrenorm alizationreducesbyone-half,butthe cancelations,inparticulartheabsenceofmassandvertexcorrections,persist duetosupergaugeinvariance.Therenormalizationofthewavefunctionofthe commutativetheorycanberecoveredbysettingequaltozero.Supergauge invariancesustainsgenerallyinNC“eldtheories.Inthenextchapterwewill studysuperPoincar ealgebrainNC“eldtheoriesandagainverifythesupergauge invariancefromanalgebraicpointofview.

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CHAPTER3 DEFORMEDSUPERPOINCAR EALGEBRA Inthischapterfollowinganintroductiononclassi“cationofrepresentations ofPoincar eandsuperPoincar ealgebra,algebraofNCFTsarestudied.Conserved currentsarederivedbyNoethersprocedure,thenarepresentationofthegeneratorsofdeformedPoincar eorsuperPoincar ealgebraissuggested,andcommutation relationsarecalculatedexplicitly.NC 4andNCWZtheoryarestudiedasthe examples. 3.1 Unitary Representations of SuperPoincar e Algebra Poincar einvarianceisconsideredasafundamentalpropertyofmoderntheory sincethediscoveryofspecialrelativity.InrecentdecadessuperPoincar einvariance, asanenlargedinvariance,isalsoconsideredasapropertyoffundamentaltheory fortheoreticalconcerns(see[35]forareview),althoughthereisnoexperimental evidenceclearlysupportingtheconject ureyet.TheimportanceoftheunitaryrepresentationsofthePoincar ealgebraandtheirclassi“cationisoriginallyrecognized byWigner[36].ThefollowingisareviewaboutthetheoryofunitaryPoincar eand superPoincar erepresentationin3+1dimensionalspace. Poincar ealgebraincludesLorentzgenerators Mandtranslationgenerators P,satisfyingcommutationrelations, [ P,P]=0 [ M,P]= i ( PŠ P) [ M,M]= i ( M+ M+ M+ M) 19

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20 where =( Š 1 1 1 1)and =0 1 2 3.Therepresentationsarecharacterizedby thevaluesoftheCasimiroperators, PP,andthesquaresofthePauli-Lubanski formsbuiltoutoftheLevi-Civit asymbols.In d =4space-timedimensions,the Pauli-Lubanskivectoris W= 1 2 MP. (3.1) Modernphysicstheory,builtintheframeworkofquantummechanics,assumes existenceofaHilbertspaceinwhichphysicalparticlesaredescribedbyquantum states.Poincar einvarianceofthetheoryimpliesthatelementaryfreeparticlescan beclassi“edinunitaryrepresentationsofthePoincar egroup. ElementsofthePoincar egroupsatisfy, T ( a ) T ( b )= T ( a + b )(3.2) d () T ( a )= T ( a ) d ()(3.3) d () d ( I )= d ( I ) (3.4) Here T ( a )representstheabeliantranslationalgroup,and d ()representsthe Lorentzgroup,where indicates d ()isadoublevaluedrepresentation. Considerthewavefunction ( p, )parameterizedbythemomentumvariables pandthevariable labelsanauxiliaryspace,sothattranslationelementsare T ( a ) ( p, )= eipa ( p, ) (3.5) Nowde“netheoperators P () ( p, )= (Š 1p, ) (3.6) Itiseasytoshow P () T ( a )= T ( a ) p () (3.7)

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21 ConsideringboththeaboveequationandEqn.(3.3),anoperatorde“nedby Q () d () P ()Š 1(3.8) canbeshowntoactontheparameter alone,whichcandepend,however,on p Q () ( p, )= Q ( p, ) ( p, ) (3.9) Q ( p, )isactuallyanunitaryrepresentationofthelittlegroupoftheLorentz group d ().Itsucestoconsider Q ( pfix, )asarepresentationof pfix= pfixfor aparticularvector pfix. RepresentationofPoincar ealgebraisalsocharacterizedbytheparticularvalue ofthecasimiroperators, PPand WW.InEqn.(3.1), M,actingonthewave function ( p, ),isgenericallyrepresentedas M= Š i p pŠ p p + S, (3.10) where Sisassociatedwith .Threeclassesofirreduciblephysicalrepresentations arefound, € PP= m2> 0 Choose P fix=( m, 0 0 0),then WW= m2( Sij)2= m2s ( s +1) (3.11) where Sij( i,j =1 2 3)isanirreduciblerepresentationof SO (3),labeledby spin s .Itiswellknownthat s takesthevalueofzeroorpositiveintegeror halfinteger.Thisclassofrepresentationdescribesmassiveparticleswithspin s € PP=0 ,WW=0 Thisclassofrepresentationisjustthemasslesslimit( m 0)ofthemassive representationanddescribesmasslessparticles.Choose P fix=( E, 0 0 ,E ),or

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22 inlightcone P+= 2 E,PŠ= P1= P2=0 (3.12) Since WW=( W1)2+( W2)2=0 (3.13) W1= W2=0or SŠ 1= SŠ 2=0.Alsoitiseasytoshowthat WŠ=0 and W+= P+S12.Therefore W= S12P,where S12,helicityoperator, isthegeneratorofthelittlegroup U (1).Singleordoublevaluenessofthe Lorentzgroupdemandthevalueoftheh elicitygeneratortobehalfinteger orinteger.Oneparticularvariantofthisclassofrepresentation,obtainedby takingin“nitemomentumlimitofmassiverepresentations[37],canbeused torepresentirreducibledegreesoffreedomofstrings.Itssupersymmetric generalizationwillbeabletorepresen tsuperstringsofvarious”avors. € PP=0 ,WW=2Againchoose P fix=( E, 0 0 ,E ). WW=2 E [( SŠ 1)2+( SŠ 2)2]and WŠ=0,but W1,W2and W+arenonzero.Thelittlegroupwhichleaves Pinvariantis SE (2)withgenerators SŠ 1,SŠ 2and S12.Only,thelengthof W,isneededtolabeltherepresentation.Twotypesofrepresentation,single valuedordoublevalued,belongtothisclass.Thisclassofrepresentationis originallycalledcontinuousspinrepre sentationbyWigner,duetothereason thatforeachrepresentation,thestatescanbelabeledbythevalueofusual helicitygenerator S12whichisalltheintegersorhalfintegers. Theaboverepresentationsshouldbeabletorepresentallphysicalparticlesin 3+1dimension.Therearegoodreasonstodisregardhigherspinrepresentations [38,39]inthe“rstandsecondclass.Naiv equantizationofthecontinuousspin representationleadstononlocalityorbreakdownofcausality[40,41].

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23 SuperPoincar egroupistheextensionofPoincar egroup,includingsupercharge QA,whichsatis“esthecommutationrelations, [ QA,P]=0 (3.14) [ M,QA]= Š 1 2 (Q )A, (3.15) { QA,Q B} =( P0)AB, (3.16) where A,B =1 2 1 2arespinorindices.Intheabove, = Š i 2 [ ,] wherethe matricessatisfytheanticommutationrelation, { ,} =2 . Masslesssupermultipletsareparticula rlyimportant,whichyieldthebasic physicalspectraofthesupersymmetricmodels.SpectraofWess-Zuminomodel canbededucedinthefollowingway[34] .Thesamearguments(Eqn.(3.5)and below)canbeappliedtosuperPoincar egroup.Thelittlegroup Q ()inthiscase includessuperchargegenerators.Since[ W,QA] =0, WWisnotcasimir operatoranymore.Formasslesssupermultiplet,stillchoose P=( E, 0 0 ,E ), thenuserepresentationindependentlight coneprojectorstosplitthesupercharge generators, Q = Q++ QŠ,QPQ. (3.17) Eqn.(3.16),intheWeylrepresentation,showsthatcommutatorsbetweensuperchargegeneratorsgenerallyvanish,except Q+ a,Q + b =2 Eab, (3.18) where a,b =1 2.Generalargumentscanshowthat Q+1and Q+2actually belongtotwodisconnectedalgebras.The refore,consideringjusttheminimal

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24 supersymmetricmodel,thelittlegroupcontainsonlythegenerators, Q+1,Q +1and S12,where S12comesfromthePoincar ealgebra.Thesegeneratorsobeythe commutationrelations, Q+1,Q +1 =2 E (3.19) { S12,Q+1} = Š 1 2 Q+1(3.20) Previously(Eqn.(3.12)andbelow)itisshownthatthemasslessrepresentationof Poincar egroupincludesstateswhicharelabeledby | E, ,where E isproportional to P+ fixand istheeigenvalueofthehelicitygenerator S12.Eqn.(3.20)indicates thatthesupercharge Q+1or Q +1aretheloweringorraisingoperatorsforthe helicity.Thusthestate | E, canbede“nedtobethelowesthelicitystatesince Q2 +1=0.Thereforetheminimumsupermultipletcontainstwostates | E, and Q +1| E, .ThesupersymmetricWess-Zuminomodeldescribesthesupermultiplet withhelicity =0. Othersupermultipletswithmorespinorsuperchargesorwithcentralcharges invariousdimensionshavebeenconstructedexplicitly[34].Inhigher( d +1)dimension( d> 3),littlegroupofsuperPoincar egroupisenlargedtocontain SO ( d Š 1)andcorrespondingspinorsuperchargegenerators.Masslesssupermultipletsin(9+1)-dimensioncorrespondtovarioussuperstringtheories.The littlegroup SO (8)hastrialitysymmetrywhichleadstomarvelouscancelationsin quantumperturbationcalculations([42]).Morerecently,M-theoryin(10+1)dimensionemergesasuni“cationofsupers tringtheories,whoselowenergylimitis suggestedtobe(10+1)-dimensionsupergravitytheory.Thelittlegroup SO (9) isthemaximumsubgroupoftheexceptionalgroup F 4.Asaresult,Eulertriplets arisesassolutionsofKostantequations[43].Thelowestleveltripletissupermultipletandcorrespondsto(10+1)-dimensionsupergravitytheory.Thehigher levelmultipletshaveaccidentalsupersymmetryandmaybeabletodescribethe

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25 zero-tensionlimitsofstringtheory[4]. Chapter5showstheconstructionofthese solutionsatalllevels.Continuousspinrepresentationin( d +1)dimensionisthe representationoflittlegroup SE ( d Š 1).Uponsupersymmetrization,ithasone toonecorrespondencewithordinarymasslesssupermultipletin d dimension[3]. ThepaperalsoshowsthatiflightconetranslationsarerepresentedbyGrassman variables,nilpotentcontinuousspinrepresentationsleadtosupermultipletswith centralcharges.Suchanalogyisalreadys uggestedinpreviousclassi“cationof (3+1)-dimensionrepresentation,whereb oththemasslesssupermultipletandthe continuousspinrepresentationcontainstateswithconnectedhelicities. NC“eldtheory,asalowenergylimitofstringtheory,doesnothaveLorentz symmetry.InthefollowingsectionsdeformedsuperPoincar ealgebraofNC“eld theoriesarestudiedinanintuitiveway,whichisexpectedtogainsomebasic understandingofunderlyingalgebraandre presentationofNC“eldtheories. 3.2 Deformed SuperPoincar e Algebra 3.2.1 Notations and Identities TofacilitatethecalculationsinvolvingNC“eldsstarproduct,weintroducethe followingnotationsandlisttheusefulidentities. De“neanoperator,whichactsnontriviallyonascalarpair-product( f,g ) as, ( f,g ) f g, 2( f,g )= f g, . .= . n( f,g )= nf ng, (3.21) where i 2.

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26 Withourde“nition,astarproductbetweentwoscalar“elds A and B canbe writtenas AB = e( A,B ) = 1++ 2 2! + 3 3! + ( A,B ) = AB + E ()( A, B ) (3.22) wheretheoperator E ()is E ()= eŠ 1 =n =0n ( n +1)! (3.23) Byusingtheabovenotations,weobtainsomeusefulidentities: 1. BA = AB Š E ( Š )( A, B ) 2.[ A,B ] AB Š BA =2 sinh() ( A, B ) 3. { A,B } AB + BA =2 AB +2 cosh() Š 1 ( A, B ) 4.( xA ) B = x( AB )+ A B. 5. B ( xA )= x( BA ) Š BA. 6.[( xA ) ,B ]= x[ A,B ]+ { A, B }. 7.[ B, ( xA )]= x[ B,A ]Š{ A, B }. 8. { ( xA ) ,B }= x{ A,B }+[ A, B ]. Weassume 0 i=0fromnowonforcausalityandunitarityreasons[25].The immediateconsequenceisthatnoncommutativitywillnotintroducehigherorder timederivativesofthe“eldsinLagrangian. 3.2.2 4 Theory NowletuscalculatetheNoethercurrentsoftheNC4theoryfollowing standardtechnique[44].VaryingtheLagrangian(2.7),andusingtheabove

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27 identitiesandalsotheequationofmotion,onegets d4x L = d4x 1 2 { ,0 }+ xL + 12 sinh() ( [ ,0]) (3.24) Underanin“nitesimaltranslation, x= g,0= Š ,oneyieldsthe energy-momentumtensor, T= 1 2 { }Š gL + 12 sinh() ( [ ,]) (3.25) Asexplicitlyseen,theenergy-momentumtensor Tisconservedsinceitsdivergenceiszero. Underthein“nitesimalLorentztransformation, x= x= Š1 2( xg Š xg ) ,0=1 2( x Š x),where isananti-symmetricsecondrank tensor,oneobtainsathree-indexcurrent j = T x+ 1 2 [ ]+ 12 (sinh() / )( ( ) [ ,]) Š 12 sinh() ( g [ ,]+ { }) Š ( ) (3.26) where(sinh() / )=(cosh() Š sinh()) / 2.Thedivergenceofthethreeindexcurrentisnotequaltozerodueto thepresenceofthetermsproportional tothenon-commutativity.However,notethattheNoethercurrentsofthe commutativescalar“eldtheorycanbeobtainedbysettingequaltozero. Inthecaseofthecommutative4theory,oneyieldsthemomentumand Hamiltoniangeneratorsfromtheenergy-momentumtensor,andtheangularmomentumandboostgeneratorsfromthethr ee-indexcurrent[44].Thesegenerators formthePoincar ealgebra.FortheNC4theory,oneobtainsitsgenerators

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28 analogoustothoseofthecommutativeone, Pi d3x Pi= d3x ( i) (3.27) P0 d3x P0= d3x 1 2 ( 2+( )2+ m22)+ 4! 4 (3.28) M0 i= d3x ( x0PiŠ xiP0) (3.29) Mij= d3x ( xiPjŠ xjPi) (3.30) Thesurfacetermsof M0 iand Mijaredroppedout.Thesegeneratorsgeneratethe translational,rotationalandboosttransformationson. Byusingthequantizationcondition,[( x ) ( y )]= i3( x Š y ),onecaneasily obtainthefollowingequal-ti mecommutationrelations: [ P,P]=0 (3.31) [ Mij,Mkl]= i ( ilMjk+ jkMilŠ ikMjlŠ jlMik) (3.32) [ Mij,Pk]= i ( jkPiŠ ikPj) (3.33) [ M0 i,Pj]= iijP0. (3.34) TheabovecommutationrelationsoftheNC4theoryarethesameasthose ofthecommutativeone.Inparticular,(3.31)veri“esthattheNC4Lagrangian hastranslationalinvarianceandthetranslationgenerator Pisconserved.Butthe followingcommutationrelationshavesomeadditionaltermsproportionalto,

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29 duetothesymmetry-breakingterm 4! 4, [ M0 i,P0]= Š i00PiŠ i 4! d3x { [ 2, i]}, (3.35) [ Mij,P0]= Š i 3! d3xxi( j) 3+( i j ) (3.36) [ M0 i,M0 j]= Š i00Mij+ i 4! d3x xj{ [ 2, i]}Š ( i j ) (3.37) [ M0 i,Mjk]= i ( ijM0 kŠ ikM0 j) Š i 4! d3xxi [ k j 3]+ 2k j Š j 2k + k j 2Š j k 2Š ( j k ) (3.38) TheEqn.(3.35)and(3.36)explicitlyshowthattheLorentzgeneratorsare notconservedinthetheory,andallthedef ormationtermsaredirectlyproportional to. 3.2.3 Wess-Zumino Model FortheNCWZmodel,onestartfromanon-shellLagrangiananalogoustothe commutativeone[34], L = 1 2 ( AA Š m2A2)+ 1 2 ( BB Š m2B2)+ 1 2 ( i Š m ) Š mgA ( A 2+ B 2) Š mgB ( AB + BA ) Š g ( A Š B 5) Š 1 2 g2( A Š iB ) 2( A + iB ) 2(3.39) = 1 2 ( Š m2 )+ 1 2 ( i + i Š m Š m ) Š 1 2 mg ( 2+ 2) Š g ( + ) Š 1 2 g2 2 2. (3.40) where A Š iB, A + iB ,and aretheWeylcomponentsoftheMajorana “eld,followingthenotationsandconventionsbyBailinandLove[45]. Followingthesimilarprocedureasdoneinthe4theory,thevariationofthe Lagrangianunderthein“nitesimalPoincar eandsupergaugetransformationsyields

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30 thegeneratorsas, Pi d3x Pi= d3x 1 2 i + 1 2 i + i 0i (3.41) P0 d3x P0= d3x 1 2 ( + ii + m2 )+ 1 2 ( i ii + iii + m + m ) + 1 2 mg ( 2+ 2)+ g ( + )+ 1 2 g2 2 2 (3.42) M0 i= d3x x0PiŠ xiP0 (3.43) Mij= d3x xiPjŠ xjPi (3.44) Q = d3x Š 2 i0 i + im0 + ig 20 (3.45) Q = d3x Š 2 i 0 i + im 0 + ig 2 0 =( Q ), (3.46) where isanarbitraryMajoranaspinorparameter. InthecaseofthecommutativeWess-Zuminomodel,theanalogsofthe abovegeneratorsarethoseofthePoincar ealgebraandsupercharge,whichform the N =1super-Poincar ealgebra.Withtherepresentationsobtainedherein theNCWZmodel,onecancalculatethecommutationrelationsbetweenthose generators, [ P,P]=0 (3.47) [ Mij,Mkl]= i ( ilMjk+ jkMilŠ ikMjlŠ jlMik) (3.48) [ Mij,Pk]= i ( jkPiŠ ikPj) (3.49) [ M0 i,Pj]= iijP0, (3.50) Theabovecommutationrelationsareexactlythesameasthoseobtainedin theNC4theory,whichsuggeststhegeneralityofsuchrelationsforallNCFTs. Inparticular,(3.47)veri“esthetranslationalinvarianceofthetheory.Equation

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31 (3.50)isalittlesurprising.ThecalculationofitinanywayinvolvestheNC interactionterms.NeverthelessitistrueforbothNCFTs. Othercommutationrelationsare [ Q,Q ]=[ Q, Q ]=0 (3.51) [ Q, Q ]=2 P, (3.52) [ P,Q ]=0 (3.53) [ Mij,Q ]= Š iijQ, (3.54) [ Mij, Q ]= Š i ij Q. (3.55) AlltheaboverelationsareexactlythesameasthoseofthecommutativeWessZuminomodel.Inparticular,one“ndsthesuperchargegenerators, Q and Q andthetranslationgenerators Psformaclosealgebra,andthesupercharge generatorsareconserved. Therestcommutationrelationshav eadditionaltermsproportionalto, includingthesimilaronesasappearsintheNC4theory, [ M0 i,P0]= Š i00PiŠ d3x i 2 mg ([ i ] + [ i ])+ mg ([ i ]0 +[ i ] 0 ) Š 2 ig ([ i ]0 ll +[ i ] 0 ll )+ i 2 g2( [ 2, i ]+[ 2, i ] ) + g2([ i ]{ }Š{ }[ i ]) (3.56) [ Mij,P0] = d3x i 2 mg ([ i, j ] +[ i j] )+ ig ([ i j ] +[ i, j ] ) + i 2 g2([ i, j ] 2+[ i j ] 2) Š ( i j ) (3.57)

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32 [ M0 i,M0 j]= Š i00Mij+ d3x mgxi( 0[ j ]+[ j ]0 ) Ši 2mgxi([ j ] + [ j ])+2 igxi([ j ] 0 ll +[ j ]0 ll ) Ši 2g2xi([ 2, j ] + [ 2, j ])+ g2{ xi, }{ ,xj }Š ( i j ) (3.58) [ M0 i,Mjk]= i ( ijM0 kŠ ikM0 j) Š d3x i 2mgxi( [ k, j ]+ [ k j ])+ igxi( [ k j ]+ [ k, j ]) +i 2g2xi([ k, j ] 2+[ k j ]) Š ( j k ) (3.59) andalsothetransformationsofthesup erchargegeneratorsundertheLorentz boosts, [ M0 i,Q ]= Š i0 iQ + d3x g [ ]0 i + g [ il( )] l Š 2 ig + img [ ] i + ig2[ 2, ] i (3.60) [ M0 i, Q ]= Š i 0 i Q + d3x g [ ] 0 i + g [ il ] l Š 2 ig + img [ ] i + ig2[ 2, ] i (3.61) Tosimplifytheexpression,wereordertheconjugate“eldsontherighthand sideoftheaboveequations,whichinducesextrain“niteconstanttermsnot explicitlyshownhere. Insummary,ThecommutationrelationsoftheLorentzrotationandboost generatorsgenerallyhaveadditionalte rmscomparedwiththoseofthePoincar eor super-Poincar ealgebras.Othercommutationrelationsverifycertainsymmetries preservedbyNCFTs,suchasthetranslationalandsupergaugeinvariance.Inthe limitof 0,thePoincar eorSuper-Poincar ealgebraisrecovered.

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33 3.3 Discussions Inthischapterwesuggestarepresentationofthetranslation,Lorentzandsuperchargegenerators.Thecommutationrela tionsofthosequantitiesarecalculated directlybasedonthisrepresentation. TheNCFThasnonlocalinteractionterms ,whichexplicitlybreaktheLorentz invariance,butstillpreservethetranslationalandsupergaugeinvariance.Itis foundthatintheNCFTthetranslationandsuperchargegeneratorsformthe samealgebraasinthecommutativetheory.But,thecommutationrelations oftheLorentzgenerators,orbetweentheLorentzgeneratorsandthetranslationorsuperchargegenerators,genera llyhaveextratermsproportionaltothe non-commutativity.Inadditiontothat,therearealsootherinteresting commutationrelations,suchas[ M0 i,Pj]= iijP0,stillholdtrueintheNCcase. PreservationofsupersymmetryalgebrasuggestsasupersymmetrygeneralizationofNCgeometry,whichisnotclearly understoodyet.Inthatframe,fermions canbetterbede“nedassupersymmetrypartnersofbosons,insteadofbeinga naivegeneralizationofLorentzalgebrarepresentation,sinceLorentzinvariance isbroken.IndeedsupersymmetryalgebrahasbeenrepresentedonNCspace andadivergencefreesupergravitymodelisexpectedtobeconstructedwiththis representation[46]. ThesuperPoincar ealgebrageneratorsarefundame ntalquantitiesŽ,representationofwhichcanbeusedtoconstructatheoryofadynamicalsystem[47].It remainsaquestionwhethertherepresentationweobtained,Eqn.(3.41)to(3.46), couldbeusedtoconstructanewtheoryonNCspaceconsistentwiththeNC theorywestartwith.

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CHAPTER4 QUANTIZATIONOFNONCOMMUTATIVESOLITONS 4.1 Introduction Solitons,knownasextendedobjectsŽ,existin“eldtheorieswithnonlinear interactions[48].Theyarede“nedtobec lassicalsolutionstotheequationof motionofalocal“eldtheorywiththepropertythattheenergydensityis,atall times,localizedwithinagivenregionofspace.Awavepacket,whichspreadsas timeevolves,ingeneralisnotthistype.Theseobjectshavespecialfeaturessince atclassicalleveltheyarealreadyparticle-like,andyettheypossessanextended structure.Suchobjectshavebeenstudiedextensivelyaroundmid-70stoearly 80softhelastcentury.Itwasspeculatedhadronsmaysimplybedescribedas quantizedstatesoftheextendobjects, wherethepossibilityofcon“ningquarks exists,sinceevenclassicallyquarksaretrappedtosomeextent[49,50]. Solitonsdiscussedinthischapterar ealsolocalizedextendedobjects,but existinNC“eldtheories,calledNCsolit ons.NCsolitonswas“rstdiscoveredby Gopakumar,MinwallaandStrominger(G MS)[51]inNCscalar“eldtheorywith thepotentialhavingalocalminimumbesidesaglobalminimumattheorigin. Sincethen,thesolitonsolutionshaveb eenexplicitlyconstructedindierentNC gaugetheorieswithorwithoutmatter[5,6].Monopole-likesolutionsin(3+1) dimensionwithastringattachedturnsouttoberealizationofD3-D1system, whereD1stringendsontheD3brane[52].NCsolitonscanalsobeinterpretedas lowerdimensionalD-branesinstring“eldtheory[53,54]. Asexplainedinchapter1,perturbativedynamicsofNC“eldtheoriesreveals averyintriguingstructure,i.e.UV/IRmixing,whichsuggeststheiranalogyto 34

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35 stringtheories.Thusitbecomesinterestingtoinvestigatethequantumbehaviorof nonperturbativestructures,NCsolitons,inNC“eldtheory. NextsectionisareviewofthetheoryofNCsolitonsandtheirinterpretation asD-branes,andalsoquantumtheoryofordinarycommutativesolitons.In sectionthreeanewtypeoftheNCsolitonsolutions,NC Q -balls,isinvestigated “rstanditsdierencefromNCGMSsolitons,existenceatarbitrarysmall ,is emphasized.NextcanonicalquantizationofNC Q -ballsatverysmallthetais discussedindetail.Quantumcorrectiontotheenergyiscalculatedwithphaseshift summationmethod.ThesamemethodisfurthergeneralizedtoNCGMSsoliton forsmoothenoughsolutions.Inbothcas esUV/IRmixingtermsarepresentinthe energycorrectionofNCsolitons.ThefuturedirectioncouldbequantizationofNC solitonsingaugetheories,whichwoul dincreasetheunderstandingoftheUV/IR mixingterms. 4.2 Noncommutative Solitons and D-branes 4.2.1 Noncommutative Solitons in Scalar Field Theory ClassicalNCsolitonswere“rstdiscoveredinthescalar“eldtheorywithmore thanonespacedimension[22].Startwithanactionofscalar“eldtheoryintwo NCspacedimensions, S = d3x 1 2 ( )2Š V () (4.1) wherethe“eldsaremultipliedbyNCstarproducts(1.1)implicithere.Inchapter 2wediscussedoperatorandfunctionrepresentationsofNCalgebra,andtheisomorphism(Weyltransform)betweenthem.Thetwodescriptionsarecompletely equivalent.Itturnsoutclassicalsolut ionsofNC“eldtheorycanbeeasilyrepresentedinoperatorformalism.Speci“cally,intwoNCspacedimensions,NC“eld theoryareisomorphictoanalgebraofoperatorsde“nedonaoneparticleHilbert space, [ x1, x2]= i. (4.2)

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36 Alsostarproductof“eldsisjustproductofWeyltransformofthe“elds(1.1and 2.5), f ( x ) g ( x ) f ( x ) g ( x ) (4.3) where f ( x )and f ( x )or g ( x )and g ( x )areWeyltransformsofeachother.The integrationoverthespaceisactuallyequivalentofoperatortrace, d2x 2 Tr (4.4) Thederivativeisequivalentto xi ( x ) i ij[ xj, ( x )] (4.5) De“necreationandannihilationoperators a = 1 2 ( x1+ i x2) ,a= 1 2 ( x1Š i x2) (4.6) with[ a,a]=1asusual.The“eld( x )or ( x )canbeexpandedintheorthonormalbasis fnm( x )or | n m | .Asystematicwayofcalculating fnm( x )canbe obtained[5,55].Inparticular,We yltransformofprojectionoperator Pn= | n n | fnn( r ),arejustcentralfunctions,andcanbeexpressedinLaguerrepolynomials, fnn( r )=( Š 1)n2 eŠ r2Ln(2 r2) (4.7) Intheoperatorformalismtheactionintegral(4.1)becomes S [ ]= dt 2 Tr 1 2 2+ 1 ([ a, ][ a, ] Š V ( ) (4.8) Theequationofmotionis 2 0 + 2 [ a, [ a, ]]+ V( )=0 (4.9) Incommutativetheory,itiswellknownthattimeindependentscalarsolitons donotexistindimensionmorethanone[ 56].However,inNCscalar“eldtheory,

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37 suchsolitonsgenerallyexistprovided islargerenough.Forradialsymmetric solution, = nnPn= nn| n n | (4.10) PutintoEqn.(4.9), ( n +1)( n +1Š n) Š n ( nŠ n Š 1)= 2 V( n) (4.11) If V()= c ( Š 1) ( Š l) (4.12) at ,thesolutionsarejust = iPnor i(1 Š Pn) (4.13) 4.2.2 Noncommutative Solitons in Gauge Theory NCsolitonsarealsowidelyfoundingaugeth eories[6],andtheirinterpretation asD-branesisprecise[53,54].Inthefollowingwereviewthetheoryofsolitons foundingaugetheories.Startwiththeaction, S =2 Tr Š 1 4 FF+ 1 2 DD Š V ( ) (4.14) Here Foi= AiŠ iA0, (4.15) and Fij= i [ Di,Dj]= 1 ([ C, C ]+1) ij, (4.16) where Di = iŠ i [ Ai, ] UDi U (4.17) iscovariantderivativeunderU(1)transformation, U U, (4.18)

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38 and C a+ i A Supposethepotentialis V ( Š ),whichhasalocalminimumat = and alocalmaximumat =0,and V (0)=0.Staticsolutionslocallyminimizethe Hamiltonian H =2 Tr 1 4 F2 ij+ 1 [ C, ][ C, ]+ V ( Š ) (4.19) Thelowestenergysolution(vacuum)canbeeasilyfound, = I,C = a. (4.20) Aclassofsolutionswithnonzeroenergycanbeconstructedthroughatransformationwithanonunitaryisometryoperator,shiftoperator S S : | n | n +1 ,S =n =0| n +1 n | (4.21) where SS =1,but S S =1 Š P0 1 Š| 0 0 | .The n th solutionis = SnI Sn= ( I Š Pn) C = Sn a Sn, C = Sna Sn, (4.22) withthe“eldstrength F = 1 2 ijFij= 1 ([ C, C ]+1)= Pn (4.23) andtheenergy E =2 n 1 2 2+ V ( Š ) (4.24) TheNCsolitonsconstructedabovecaninfactbeinterpretedasBosonicDbranes.Asexplainedintheintroductionsection,theeective“eldtheoryofthe tachyonandgauge“elddegreesoffreedomoftheopenstringsontheD-branewill beaNC“eldtheoryintermsoftheeective“eldtheoryat B =0,butwiththe

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39 openstringeectivemetric Gijandthenoncommutativity ijdeterminedby(1.11) and(1.13). ConsideraBosonicopenstringwithanunstablespace-“llingD25-brane.The leadingtermsintheeectiveactionfor constanttachyonandgauge“eldstrength havetheBorn-Infeldform[57…59].Integratingoutthemassivestringdegreesof freedomleadstoaneectiveactionoftheform(for Bij=0), Seff= c gs d26x det g Š 1 4 h ( Š 1) FF+ + 1 2 f ( Š 1) + Š V ( Š 1) (4.25) where c = T25gsisindependentof gswith T25theD25-branetension.Thepotential V ( Š 1)hasalocalmaximumat =0with V ( Š 1)=1representingtheunstable D25branecon“guration,andalocalminimumat =1with V (0)=0representing theclosedstringvacuum,accordingtotheconjectureofSen[60,61]. Nowturnonthebackground B “eldwithonly B24 25= b< 0,thentheaction becomesNCin R2including24and25direction.Inchapter1,explicitformofthe NCgauge“eldtermisgivenin(1.19)analogousto(1.22).ThereforetheNCaction analogousto(4.25)is S = 2 c Gs d24x Lnc, (4.26) with Lnc= det G Tr Š 1 4 h ( Š 1)( F+)( F+) + + 1 2 f ( Š 1) DD + Š V ( Š 1) (4.27) FocusingononlytheNCdirections24 25,aspecialchoice[10] = 1 B = 1 | b | = Š B = | b | (4.28)

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40 inEqn.(1.20)givesthevalueoftheopenstringmetric(1.11)andthestring coupling(1.24)inthezerocouplinglimit, G = Š (2 b )2,Gs= gs(2 | b | ) (4.29) InevaluatingenergyoftheNCsolitonsolution(4.22),rememberthesolitoncanbe obtainedbythetransformationwiththeshiftingoperator Snfromtheclosedstring vacuumcon“guration.Allcovariantderivativesof or Fvanishastheydointhe closedstringvacuum,soarethegauge“elds Averticaltothebrane.Also h ( Š 1)( F +)2= 1 2h ( Š Pn)(1 Š Pn)= 1 2h ( Š 1) Pn(1 Š Pn)=0 (4.30) and V ( Š 1)= V ( Š Pn)= V ( Š 1) Pn= Pn. (4.31) Thereforetheenergyofthesolitonhasonlythepotentialterm,derivedfromEqn. (4.27), E = 2 c det G Gs d24x Tr Pn= (2 )2nc gs, (4.32) whichidenti“esthetensionas T = (2 )2nc gs=(2 )2nT25= nT23. (4.33) Intheaboveconstructioncon“gurationofnD p -branesarisesastheNCsoliton solutionshiftedfromtheclosedstringvacuum.Thisconstructionisexactforany valueof B or“nite ,whichreducestothatof[62]inthelimitoflarge B -“eld. 4.3 Classical Noncommutative Q -ball Solution GMSsolitons,whichexistin(2+1)dimensionalNCscalar“eldtheory,while classicallystable,ceasetoexistatsucientlysmallNCparameter ,duetothe nonexistencetheoremofDerrick[56]inthecommutativelimit( 0).Inthis commutativelimit,however,timedependentnontopologicalsolitons,or Q -balls existinallspacedimensions[48,63].

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41 4.3.1 Hamiltonian and Equation of Motion InthissectionwederivetheequationofmotionforNC Q -ballsolutions, followingbriefintroductionofNCscalar“eldtheory.Theformofthesolutionhas alreadybeengiven[64].Wediscusstheexistenceandstabilityofthesolutions,and showthatinthecommutativelimitNC Q -ballsjustreducetothecommutative Q -balls. ConsideraNCscalar“eldt heoryactionwithglobal U (1)phaseinvariance, S = Š dtd2x + V ( 1 2 { } ) (4.34) wherethespace-timemetricis( Š + +),andthe“eldsaremultipliedbyNCstar product,generallymadeimplicitinthispaper,and { A,B } AB + BA .The potential V hasaglobalminimumattheorigin,withthescalingproperty V ( g, )= gŠ 2V ( g2 ) (4.35) g isthenthecouplingconstantassumedtobesmall.Thecommutativelimitofthis actioniswhereordinary Q -ballshavealreadybeenconstructed[48,63].TheNC starproductisde“nedtobe, ( )( x ) exp( i 2 jk xj yk) ( x ) ( y ) y = x, (4.36) where j,k =1 2. Intheoperatorformalismtheactionintegral(4.34)becomes S [ ]= dt 2 Tr 0 0 + 1 ([ a, ][ a, ]+[ a, ][ a, ]) Š V ( ) (4.37) Theequationofmotionis 2 0 + 2 [ a, [ a, ]]+ V( )=0 (4.38)

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42 Theactionhasglobal U (1)phaseinvariance,whichy ieldsaconservedcharge Q [ ]= d2xj0= i 2 Tr( 0 Š 0 ) (4.39) Q isinterpretedasparticlenumberinthephysicalsystem.Aparticularsystem alwaysexistswith“xedparticlenumber N = Q [ ].To“ndnondissipativesoliton solutions[48,63]underthisconstraint,wewriteHamiltonian H =2 Tr 0 0 Š 1 ([ a, ][ a, ]+[ a, ][ a, ])+ V ( ) + ( N Š Q [ ]) (4.40) withtheconstraintappliedbeforethePoissonbracketisworkedout[47].The minimumenergysolutionoccursat H ( 0 ) N= 0 + i =0 (4.41) whichmeans = 1 2 ( x ) eŠ it. (4.42) Assuminghermitian ( x )orreal ( x ), H becomes H =2 Tr Š 1 2 2 2Š 1 [ a, ][ a, ]+ V ( 1 2 2) + N, (4.43) withtheparticlenumber N =2 Tr( 2) (4.44) andtheequationofmotion(4.38) 2 [ a, [ a, ]] Š 2 + V( 1 2 2)=0 (4.45) Notetheequationofmotion(4.45)alsofollowsfrom( H/ ) |N=0,whichmeans thatthesolution hasthesameformasthestaticGMSsolitonsolutioninthe potential U ( )= V ( 1 2 2) Š 1 2 2 2. (4.46)

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43 Considersphericallysymmetricsolution[ 51]expandedintermsoftheprojection operators, ( x )=n =0nPn, (4.47) where Pn| n>
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44 Thisisnotthecasewith Q -ballsolutions.Theexistenceandstabilityof Q -ballsolutionrelyontheconservationofthecharge Q astheconsequenceof theglobalsymmetry.Thepotentialfor Q -ballsolutionsdoesnothavenontrivial topologicalstructure.ThereforeNC Q -ballsareexpectedtoexistevenforvery small .WewillshowthatsuchNC Q -ballsolutionswouldsmoothlyreducetothe Q -ballsolutioninthecommutativelimit. InthefollowingwediscusstheexistenceofNC Q -ballsolutionsinatypical potentialform, U ( )= V ( 2) Š 1 2 22= a2Š b4+ c6, (4.53) wherethecoecients b and c arelargerthanzero,and a =1 2( m2Š 2). U ( )variesfordierent .If 2>m2or a< 0, U ( )hasalocalmaximum attheorigin.Inthecommutativelimitthereisonlyaplanewavesolution.Here similarplanewavesolutioninNClimitcanalsobeconstructed.Sincefora stablesolitonsolution nwouldhavetotakevaluesbetween s andtheorigin andmonotonicallydecreasein n [65],asimpleargumentcanshowthatsolitons cannotexist.Thereisaconstraintthat K n =0U( n)convergestozeroas K goes toin“nity,whichcannotbesatis“edinthisc ase.Toprovethisconstraint,suppose thatn =0U( n) v =0 (4.54) SumEqn.(4.51)fromaparticular K = q sucientlylargetoapoint p closeto in“nity, pŠ qpK = qv K (4.55) Itistheneasytosee pwillnotconvergetozeroas p goestoin“nity. When 2<2, 2= m2Š b2/ 2 c U ( )hasonlyaglobalminimumat theorigin.Eventhoughinthecommutativetheorynosolitonsolutionsexist,for NCtheoryatsucientlylarge ,thereareGMStypesolitonsexist.Ithasbeen

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45 shownthatthereisacriticallowerboundon fortheexistenceofNCsoliton[66]. SimilarboundswouldbeexpectedtoexistforNC Q -ballfor0 << aswell. As 2<2 .As getssmaller,theeigenvalues n getscloser,andeventuallybecomescontinuous as1 2r2inthecommutativelimit.Thecoecient njustbecomesthe“eld ( r )in 1ThankstoDr.Shabanovforhelpingonthispoint

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46 thislimit.Inthisdescriptionthecommutative Q -ballcanbeconsideredasthe analyticalcontinuationoftheNC Q -ballsin Theformulafortheenergyandparticlenumberinthecommutativelimitcan alsoberecoveredbytakingthecontinuumlimitofEqn.(4.48)and(4.49), H =2 0vdv 1 2 ( d dv )2+ U ( )+ N (4.59) N =2 0du2( u )=2 0vdv2( v ) (4.60) Theexistenceofthecommutative Q -ballsolutionareprovedbyconsidering ananalogousprobleminwhichaclassicalparticlemovesintheone-dimensional potential Š U ( )[48].The“eldcon“guration ( v )ofthe Q -ballstartsfroma uniquevalue = (0)betweenthezero w andtheglobalminimum s ,then monotonicallydecreasesin v ,andapproaches0when v .Thispropertyof ( v )isconsistentwiththoseofthegeneralstableNCsolitonsolutions nat“nite .Itisfound[65]thatthereexistsmooth familiesofsphericallysymmetric solutionsinwhich nismonotonicallydecreasingin n .Inthein“nite limit suchsolutionisjust sPK.As decrease, 0, ,Kdecreasefrom s ,whileother n( n>K )startstomoveawayfromtheorigintowards s ,butthewhole nseriesremainmonotonicallydecreasein n .Sinceinthecommutativelimit njustbecomes ( v ),onecanconcludethatas decreasesfrom tozero, 0will decreasefrom s andeventuallyto atthecommutativelimit. 4.3.3 Virial Relation TheHamiltonian(4.43)inthefunctionformalismis H [ ]=2 rdr 1 2 ( i )2+ U ( )+ N (4.61) wherethepotential U hasexplicitdependenceon throughstarproduct.Suppose ( x )isthe Q -ballsolution, H [ ( x/a )]mustbestationaryat a =1.Achangeofthe

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47 integrationvariablesshowsthat H [ ( x/a )]=2 rdr 1 2 ( i ( x ))2+ a2U ( a2, ( x ))+ N (4.62) and d da H [ ( x/a )] a =1=2 rdr 2 U ( ) Š 2 U ( ) =0 (4.63) UnliketheVirialtheoremfor d =1spacedimension,herethekineticenergyis scaleinvariant.Scalingdependenceoftheenergyincludestwoseparateterms fromthepotentialandfromitsdependenceon throughthestarproducts.The signi“canceofEqn.(4.63)ismoreexplicit inGMSsolitoncase,wherethepotential energy 2 rdrU ( )= n =0U ( n) > 0 (4.64) Thescalingvariable a canbethoughtofasthesizeoftheNCsoliton.Whilethe positivepotentialenergyfavorsshrinkingofthesoliton,buttheNCstarproducts keepitfromdecay. 4.4 Quantization of Noncommutative Q -ball Solitonsareextendedobjectsexistin“eldtheory,thepropertiesofwhich receivequantumcorrectionsasthe“eldsarequantized.Inthissectionwefollow verycloselytothecanonicalquantizationprocedure[49,63].Thenweevaluatethe ultravioletdivergencesinthequantumcorrectionstothesolitonenergyatvery small 4.4.1 Canonical Quantization Thegeneralproceduretoinvestigatethepropertiesofthesolitonsistoexpand the“eldsaroundtheclassicalsolutio n.Becausethemomentumandparticle numberareconservedinthesystem,wewillhavetoimposethecorresponding constraintstoerasethezero-frequencymodesintheexpansion.

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48 Westartbymakingapointcanonicaltransformationof ( x ), = 1 2 eŠ i ( t )[ ( x Š X ( t ))+ ( x Š X ( t ) ,t )] (4.65) ( x Š X ( t ) ,t ) R( x Š X ( t ) ,t )+ iI( x Š X ( t ) ,t ) (4.66) where ( t )and Xi( t )arethecollectivecoordinatesrepresenttheover-allphase andthecenterofmassposition.Imposetheconstraintson toensuretheabove transformationisacanonicaltransformationwithequalnumberofdegreesof freedombeforeandafter, I=0 Ri =0 (4.67) where i =1 2.Theintegralsigndenotestwodimensionalintegrationsover x Thestarproductissuppressed.Unlessindicatedotherwise,fromnowonthe dierential d2x andtheNCstarproductareimpliedwhereverapplicable.The aboveconstraintsalsoremovetheperturbativezeromodesolutionsinthemeson “eld .Let R( x,t )=a =3qRa( t ) fa( x ) (4.68) I( x,t )= a =2qI a( t ) g a( x ) (4.69) where fa( x )and g a( x )aretherealnormalfunctionssatisfy fafb= ab, (4.70) g ag b= a b, (4.71) (4.72)

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49 andundertheconstraints, ifa=0 (4.73) g a=0 (4.74) where a =3 4 ,... and a =2 3 ,... alwaysinthispaper. RewritetheLagrangian(4.34)with(4.65), L = 1 2 qTM q + V ( q ) (4.75) where qT=( X1,X2,,qR 3,...,qI 2,... )and T denotesmatrixtranspose,and V ( q ) ( i i + V ( 1 2 { } ))(4.76) Thematrixelementsofthesymmetric M are Mij= M0ij+ (2 ijR+ iRjR+ iIjI) (4.77) M= I + (2 R+ 2 R+ 2 I) (4.78) Mi= ( Š 2 iI+ RiIŠ IiR) (4.79) Mia= Š faiR, (4.80) Mi a= Š g aiI, (4.81) Ma= faI, (4.82) M a= Š g aR, (4.83) Mab= ab, (4.84) M a b= a b. (4.85) where M01 2 ( i )2and I 2.Theconjugatemomentumof q is p = M q ( P1,P2,N,pR 3,...,pI 2,... )T. (4.86)

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50 Theparticlenumber N andthetotalmomentum Piareconservedsincethe Lagrangian(4.75)isindependentofthecollectivecoordinates and Xi.Quantize thenewcanonicalcoordinates, [ Xi,Pj]= iij, (4.87) [ ,N ]= i, (4.88) [ qRa,pRb]= iab, (4.89) [ qI a,pI b]= i a b. (4.90) TheHamiltonian H = 1 2 JŠ 1pTMŠ 1Jp + V ( q ) (4.91) where J = det M becausetheoperatororderingin H isunambiguouslydeterminedbytheordinaryquantizedHamiltonianwiththecoordinates [49]. Thequantumstatescanbelabeledas | P1,P2,N,qRa,qI a .Onecansolve theSchr¨ odingerEquationperturbativelyaroundtheonesolitongroundstate | P1= P2=0 ,N = I, 0 .Inthisstate Piand N arethemomentumandparticle numberoftheclassicalsolution ,whichcanbeobtainedbyletting = inEqn. (4.49)and Pi= i + i [1].0labelsthelowestenergystatewiththegiven Piand N value. Wecanthentreat Rand Iasperturbativedegreesoffreedom,andexpand theHamiltonianperturbativelyaroundtheonesolitongroundstateorderbyorder intheweakcouplingconstant g ,de“nedinEqn.(4.35). isattheorderof gŠ 1asasolitonsolution. M0and I are gŠ 2order.Then Piand N areatthe gŠ 2order,while pRaand pI aat g0order.Since J commutewith Piand N ,andattheleading gŠ 3order, J = M0 I isaconstantor[ p,J ]=0,one cancheckthat J wouldnotbeafactorintheHamiltonianuptotheorder g0.

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51 TheHamiltoniancanbeexpandedorderbyorder, H = H0+ H1+ H2,withthe expansionrelation, MŠ 1= MŠ 1 0+ MŠ 1 0 MŠ 1 0+ MŠ 1 0 MŠ 1 0 MŠ 1 0+ (4.92) where M = M0+,and M0hasonlynonzerodiagonalelements, Mqq 0= ( M0,M0,I, 1 1). H0,equaltotheenergyoftheclassicalsolution,isattheorderof gŠ 2, H0= M0+ 1 2 I2+ V ( 1 2 2) (4.93) H1,linearin ,vanishesduetothe“xed N and Pi,whichensuresthat Rand Iareattheorderof g0. Thetermquadraticin isatthe g0order, H2= 1 2 ( pRaŠ faI)2+ 1 2 ( pI a+ g aR)2+2 2 M0( iI)2+2 2 I ( R)2+ V2( q ) (4.94) where V2( q )= 1 2 [( iR)2+( iI)2] Š 1 2 2( 2 R+ 2 I) + 1 2 ( 2 R+ 2 I) V(1 / 2 2)+ V 1 2 2, 1 2 { R, } 1 2 { R, } (4.95) where V 1 22,1 2{ R, } ,1 2{ R, } representsthetermsfromtheexpansionofthe potential V quadraticin R. 4.4.2 Energy Corrections at Very Small TheHamiltonianisseparatedintotwoparts,describedbythebaryondegrees offreedom( Pi,N )andmesondegreesoffreedom( R,I)respectively.Thesum ofthefrequenciesofthemesonexcitationsisthezero-pointenergyof H2, P1= P2=0 ,N = I, 0 | H2| P1= P2=0 ,N = I, 0 (4.96)

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52 which,subtractedbythevacuumenergy E vac = d2k/ (2 )2 k2+ m2,givesthe quantumcorrectionstothesolitonenergy. V2( q )istheperturbativeexpansionoftheeectivepotential V ( q ) Š Q [ ], Eqn.(4.76)and(4.39),aroundthesolution .Itiseasytocheckthat R= i and I= aretheeigenmodesof V2( q )witheigenfrequency0,duetothe translationalandrotationalinvarianceofthepotential.Thereforewecande“nethe normalfunctions faand g atobetheeigenmodesof V2,or V2( q )= 1 2 2 Raq2 Ra+ 1 2 2 I aq2 I a, (4.97) withthefrequenciesRaandI a.Thepotential V2arehighlynonlocalsince the“eldsaremultipliedbyNCstarproduct.Inthecommutative Q -ballcase, V2hasbeenshowntohaveonlyones-waveeigenmodein Rsectorwithimaginary frequencyR 3[63].Inlastsection,wehaveshownthatasNCparameter is takentobesmallenough,theNCsolitonsolutionwillreducearbitrarycloseto itscommutativeanalog.Thereforeclosetothecommutativelimit V2isexpected tohavethesimilareigenvaluesandeigenmodesasitscommutativeanalog.NC Q -ballisalsoexpectedtobestableasitscommutativeanalog.Wewillassume ischosentobesuchasmallvalueinevaluatingthequantumeectsofthe noncommutativity. De“ne fi=1 / M0i and g1=1 / I ,RewritetheHamiltonian H2,(4.94), inthematrixform, H2= 1 2 ( PTŠ QTT)( PŠ Q )+2 2QT Q + 1 2 QT2Q (4.98) wherethematricesarede“nedasfollows: PT ( pRa,pI a) QT ( qRa,qI a) (4.99)

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53 0a aŠ T aa0 " " # Fab0 0 G a b! " " # Ra0 0I a! " " # (4.100) where a a fag a, Fab g1fa g1fb, G a b fig a fig b. (4.101) Theequationofmotion, Q = H2 P P = Š H2 Q (4.102) give Q = PŠ Q (4.103) P = T( PŠ Q ) Š 4 2 QŠ 2Q (4.104) Therefore, ¨ Q +2 Q +4 2 Q +2Q =0 (4.105) Lettherealnormaleigenmodesof Q be QA=( A Ra,A I a)T, (4.106) where QA TQB= AB.Replace Q = QAexp( Š i At )(Index A isnotsummedover) intheaboveequation.Since QA T QA=0, A= QA T(4 2+2) QA. (4.107) Introducecreationandannihilationoperators,[ CA, C B]= AB, Q canthenbe quantizedas, Q = AQA 2A( CAeŠ i At+ C Aei At) (4.108)

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54 UsethisequationandEqn.(4.104)and(4.98),onecande“netheonesoliton groundstate, CA| P1= P2=0 ,N = I, 0 =0 (4.109) thenthezero-pointenergyof H2(4.96)is 1 2 AA= 1 2 Tr { } (4.110) wherethematrixisdiagonalwiththeeigenvaluesA. Inthecommutativetheorythezero-po intenergycontainsthedivergences evenaftersubtractionofthevacuumenergy.The“nitenessofthesolitonenergy isrecoveredbystartingfromtherenormalizedformoftheaction(4.34),which inducesthecountertermsalsocontainthedivergences[67]. Workinthespeci“cformofthe 6potential(4.53), V ( 1 2 { } )= m2( 1 2 { } ) Š bm2g2( 1 2 { } )2+ cm2g4( 1 2 { } )3. (4.111) Atthe g0order,ortheone-looporder,thegeneralformulaforthesolitonenergyis E soliton P1= P2=0 ,N = I, 0 | H | P1= P2=0 ,N = I, 0 Š E vac (4.112) = H0+ 1 2 Tr { }Š E vac + 1 2 m2 2Š bm2g2 (4) 4, (4.113) where m2and g2 (4)arethecountertermsforthemassandthe 4coupling respectively.The 6couplingdoesnotreceiveloopcorrections.The 4coupling termsyieldtherightcoecientsandcanberenormalized[32]. TheloopintegrationintheNC“eldtheorygenerallycontainsphasefactors whichyieldtheinterestingUV-IRphenomenonuponrenormalization[22].Inthe followingweevaluatethequantumcorr ectionfromthezero-pointenergyof H2in Eqn.(4.110)andshowthatitcontainsthesamephasefactorsasthoseappearin thecounterterms m2and g2 (4).

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55 Westartbyarguingthatonly1 / 2Tr { } isneededinevaluatingtheleading divergence.InEqn.(4.107),itiseasytosee QA T QAis“nite, QA T QA=( g1faRa)2+( fig aI a)2 [ g2 1+ ( faRa)2]2+[ f2 1+ ( g aI a)2]2+[ f2 2+ ( g aI a)2]2=(1+ 2 Ra)2+2(1+ 2 I a)2 12 (4.114) AswewillseethattheeigenvaluesRaandI abehavelike k2Š 2+ m2atvery large k .TheleadingdivergenceofTr { } willbedeterminedbyTr { } RaandI a,eigenfrequenciesof V2( q )inEqn.(4.95),satisfythelinear equations, ( Š 2 iŠ 2+ m2) IŠ 1 2 bm2g2{ 2,I} + 3 8 cm2g4{ 4,I} =2 I aI, (4.115) ( Š 2 iŠ 2+ m2) RŠ bm2g2( { 2,R} + R )+ 3 4 cm2g4( { 4,R} + 2R2+ { ,R} )=2 RaR. (4.116) TheaboveequationsarejusttimeindependentSchr¨ odingerequations.In particularphaseshiftsfromthecentralpotentialhavebeenusedincalculatingthe solitonenergycorrection[68].Thebasicideaisthatinthecentralpotentialfor eachpartialwave,thedierenceofthedensityofthestatesbetweenthescattered waveandthefreewaveisrelatedtothederivativeofthephaseshift, l( k ) Š 0( k )= 1 dl( k ) dk (4.117) where l goesfrom Š to .The“nitenessoftheparticlenumber, N = 2, determinesthat o as r .ThereforetheNCpotentialinEqn.(4.115)and (4.116)isradialsymmetricandvanishesat .Forthemostgeneralpotentialterm WF( r ) WB( r ), [ WF( r ) WB( r ) ,L ]= WF( r ) [ ,L ] WB( r ) (4.118)

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56 where L = Š iijxijistheangularmomentum.Thestarproductismadeexplicit hereandintherestofthesection.ThisformulacanbeeasilyprovedintheWeyl transformsofthe“elds.Goingtothemomentumspace,onecangeneralizethe resultin[15]andshowthat WF( x ) ( x ) WB( x )= d2pf (2 )2d2pb (2 )2$ WF( pf) $ WB( pb) ei pf( x +i 2 )ei pb( x Ši 2 ) ( x ) (4.119) where i ijj. UsingEqn.(4.117),consideronlytheleadingdivergence,wehave[69], 1 2 Tr { }Š E vac 1 2 Tr { }Š E vac 1 2 d k2+ m2l[ Il( k )+ Rl( k )] (4.120) where Il( k )and Rl( k )arethephaseshiftsfor Iand R.Thesumofthephase shiftscanbeevaluatedthroughBornapprox imation.Inthecommutativecase,this leadstothecancelationofthetadpolediagram[68]. Eqn.(4.115)and(4.116)havetheJostsolutionformforthe l thpartialwave atlarge r h l( kr )+ e2 ilhl( kr ) (4.121) Consideringtheasymptotic( r )behaviorofthesolution,thestandard procedure[70]leadstothescatteringamplitude, f ( k, k )= f ( )= lfl( k ) eil= 1 k leilsin leil, (4.122) where k= k and istheanglebetween kand k .Atlarge l ,or l 0,wecansee ll kf ( =0)(4.123)

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57 f ( k, k )canalsobecalculatedthroughBornapproximation,replacing by eŠ i kxinthepotentialform(4.119), f ( k, k )= Š 1 4 k d2xeŠ i kxiW(i) FeŠ i kx W(i) B= Š 1 4 k d2xeŠ i ( kŠ k ) xiW(i) F( x Š 2 k ) W(i) B( x + 2 k )(4.124) whereilabelsthepotentialtermsinEqn.(4.115)and(4.116),and ki ijkj. Therefore ll= Š 1 4 d2x iW(i) F( x Š 2 k ) W(i) B( x + 2 k ) = Š 1 4 d2p (2 )2i$ W(i) F( p ) $ W(i) B( Š p ) eŠ i p k. (4.125) Therighthandsideonlydependsonthemagnitude k duetothecentralpotential W ( r ). NowwearereadytoevaluateEqn.(4.120), 1 2 d k2+ m2l[ Il( k )+ Rl( k )] = Š 1 8 d k2+ m2 d2p (2 )2i$ W(i) F( p ) $ W(i) B( Š p ) eŠ i p k= Š 1 2 d2p (2 )2 d2k (2 )2eŠ i p k 2 k2+ m2i$ W(i) F( p ) $ W(i) B( Š p ) (4.126) Theintegrationoverkisexactlypartofthetadpolediagrambelongsto m2[32], anditcontainstheUV/IRdivergence( and p 0)evaluatedwiththe cuto[22], d2k (2 )2eŠ i p k 2 k2+ m2= 2 (4 )3 / 2( m eff)1 / 2K1 2 2 m eff = eff 8 + O (1) (4.127) whereeff ( 2p2/ 4+1 / 2)Š 1 / 2.NoticethattheaboveUV/IRdivergencefrom Eqn.(4.126)occursonlywhenboth WFand WBexist.Inotherwords,onlythe terms R,2R2and { ,R} inEqn.(4.115)and(4.116)yieldUV/IR

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58 divergence.AllotherpotentialtermsonlygivethenormalUVdivergencewhere thephasefactorisabsent.Sincethecounterterm m2and g2donotinclude UV/IRdivergence,wearecertainthatthe Q -ballenergycorrectionincludesUV/IR divergence.CancelationoftheUVdivergencesisnotobviousbecausetheexact valueoftheeigenfrequenciesAisunknown. 4.5 Finite and Noncommutative GMS Solitons Theabovecalculationassu mesthattheNCparameter issucientlysmallso thattheNCpotentialwillgeneratetheJostsolutionformasinthecommutative case.LetusconsidertheeectsoftheNCpotential(4.119)inthecasethat is notsmall. Since [ xi i 2 i,xj i 2 j]= iij, (4.128) theeectivescatteringpotentialfortheNCinteraction WF( x ) WB( x )(4.119) isjust % WF( x + i 2 ) % WB( x Š i 2 ) (4.129) multiplicationoftheWeyltransformsof WF( x )and WB( x ).Notice % WFand % WBcommutesince[ xi+ i/ 2 i,xjŠ i/ 2 j]=0. Nowconsidering % WF( x + i 2 )= d2pf (2 )2$ WF( pf) ei pf( x +i 2 ), (4.130) thenoncommutativity, [ p1 f( x1+ i 2 1) ,p2 f( x2+ i 2 2)]= ip1 fp2 f, (4.131) canbesuppressedevenif isnotsmall,aslongas W ( x )issmoothenoughor $ W ( pf) 0atlarge pf.Noticesmall pfisalsotheIRlimitwediscussedinthelast

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59 section.Underthisassumptionwecanwrite % WF( x + i 2 ) WF( x + i 2 )= WF( | x + i 2 |2)= WF( r2Š 2 L + 2 i) (4.132) Actingonthe“eld ( x )= ul( kr ) eil,theeectivepotentialbecomes WF( r2Š l/ 2+ 2 i).Similarcalculationappliedto % WB( x )yields WB( r2+ l/ 2+ 2 i).Thereforeat large k andlarge r ,wecantreatthescatteringpotentialperturbativelyasinthe commutativecase.Thephaseshiftevaluationoftheenergyofthesolitoncould stillapplyprovidedthat W orthesolitonsolution aresmoothenough. NCGMSsolitononlyexistsat“nite .Basedontheabovearguments,we canstillevaluateitsquantumcorrectio nswiththephaseshiftmethodinthelast section. QuantizationofGMSsolitonand Q -ballsharealotofsimilarities.Togetthe GMSsolitontheory,wemakethereplacements( 1 / 2)intheprevious complexscalar“eldtheory(4.34).Withthepotential(4.111),theLagrangian becomes L = Š 1 2 ( )2Š V ()= Š 1 2 ( )2Š 1 2 m22+ 1 4 bm2g24Š 1 8 cm2g46, (4.133) whereismultipliedbythestarproduct.Therenormalizabilityofthetheory hasbeenproved[30].Let =0becausethereisnoconservedcharge Q (4.39)in thetheory.When b2Š 4 c< 0,thepotentialin(4.133)hasalocalminimumat bg/ 2besidestheglobalminimumattheorigin,andtheGMSsolitonsolution exists.Replacetheexpansion(4.65)by= + ,where = Risreal.As aresult,themesondegreesoffreedomareonly Ror .Uponquantization,the solitonenergyisstillgivenbyEqn.(4.113).Since =0and=aretheexact

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60 eigenfrequenciesof H2(4.94),wehave ( Š 2 i+ m2) RŠ bm2g2( { 2,R} + R ) + 3 4 cm2g4( { 4,R} + 2R2+ { ,R} )=2 aR. (4.134) Eqn.(4.120)describestheexactultravioletdivergencesin1 / 2Tr { }Š E vac ThereforeweareabletocheckthecancelationofthedivergencesinEqn.(4.113). Asmentionedintheprevioussection,intheaboveequation,theterms R 2R2and { ,R} yieldUV/IRdivergences,whiletheresttermsyieldUV divergence.AcriticalobservationisthatthosetermsyieldUV/IRdivergence haveonetoonecorrespondencewiththeco ntractionsofthe“eldsyieldNonplanar Feynmandiagrams[22],andthosetermsyieldUVdivergencecorrespondexactly totheplanardiagrams.Wecanjustspareth edetailsofcountingthedivergences. Sincethecounterterms m2and g2 (4)cancelexactlytheUVdivergencepart,we concludethatthesolitonenergy(4.113)isUV“nite,butincludesalltheUV/IR divergences. 4.6 Conclusion and Discussion InthischapterwediscussedthequantizationofNCsolitonsin(2+1) dimensionalscalar“eldtheory.Inparticula r,classicalsolutionsandquantization oftheNC Q -ballsatverysmall areinvestigatedindetail.ClassicallyNC Q ballsreducetothecommutative Q -ballas goestozero.Quantummechanically, becauseloopintegrationsintheNC“eldtheoryhavedierentultravioletstructure fromthoseinthecommutativetheory,i. e.UV/IRmixing,qua ntumcorrectionsto theNCsolitonenergynecessarilyincludetheUV/IRdivergenttermswhichcannot berenormalizedaway.Theexistenceofsuchtermsintheenergyisdemonstrated throughthephaseshiftsummation.ThesamemethodisfurthergeneralizedtoNC GMSsolitonswhichexistonlyat“nite .Inthesmallmomentumlimit,orforthe sucientlysmoothsolitonsolutions,divergencestructureofthesolitonenergycan

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61 becalculatedexactly.InthiscasetheenergyisfoundtocontainnoUVdivergence butalltheUV/IRdivergences.Quantumco rrectionstotheNCsolitonenergyhave alsobeencalculatedbutatverylarge [7],wherenoUV/IRdivergenceisfound. Webelievethatisbecauseatlarge ,thenoncommutativity(4.131)isnotsmall andcannotbeignored,andthepotentialtermisthedominantterminsteadofa perturbativeone.Inthiscasethephasesh iftsumisnotagoodapproximationto theenergycorrection. AninterpretationtotheUV/IRdivergence[23]isthatbecausenewlight degreesoffreedomareintroducedintheWilsonianeectiveaction.UV/IR divergencecanbereproducedbyintegratingoutthosenewdegreesoffreedom, whicharetheninterpretedasclosedstringmodeswithchannelduality.Future researchdirectionistoconsidertheNCsolitonsinthegaugetheory,wherethey areinterpretedasD-branes[53,54]andD-braneactionisproperlyrecovered.One expectstogainbetterunderstandingofin teractionsbetweenD-braneandclosed stringsthroughquantizationofNCsolitons.

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CHAPTER5 SOLUTIONOFKOSTANTEQUATION Inchapter3theoryofsuperPoincar ealgebraanditsrepresentationsis reviewed.Classi“cationofirreducibleunitaryrepresentationsofPoincar eand superPoincar ealgebrarevealsphysicalspectrumofthecorrespondingtheory. M-theory,asuni“cationofstringtheory ,isconjecturedtobe11-dimensional theory[12,13],whoselowenergylimitis11-dimensionalsupergravity.Littlegroup of10dimensionalPoincar egroupis SO (8),whichclassi“esthespectrumof10 dimensionalsuperstringtheory.Trialitysymmetryof SO (8)leadstomarvelous cancelationsbetweenBosonicandFerm ioniccontributions,whichrendersthe theorytobeUV“nite.Littlegroup of11-dimensionalsuperPoincar ealgebra, SO (9)doesnothavesuchsymmetry,and SO (9)isnonrenormalizableinhighloop order[71].Itisfoundthatsomeirrepsof SO (9)naturallygrouptogetherinto anin“nitetoweroftriplets[72],thelowestofwhichincludesthespectrumof11 dimensionalsupergravity.Thissuggeststhatthetoweroftripletsmightbeableto describecertainlimitofM-theory.Apossiblecandidateisthein“niteReggeslope limit,orzerotensionlimitofstringtheory,whereoneexpectsallstatestobecome massless,withanin“nitenumberofstatesforeachspin. Amathematicalunderstandingofthetripletshasbeengiven[73].Theyarise forembeddingswherebothgroupandsubgrouphavethesamerank. SO (9)isa subgroupof F 4withthesamerank,andthequotientspace F 4 /SO (9)hasEuler numberthreegivingatripletof SO (9)toeveryirrepof F 4.Thereexistother casesthetripletsarisethelowestofwhichdescribe N =8supergravity, N =4 Yang-Millsand N =2hypermultiplet[74…76].AlltheseEulertripletsariseas solutionsofKostantsequation[43],whichisaDirac-likeequationonthecoset. 62

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63 ThischapterfocusondierentialformoftherepresentationofKostantsequationanditssolutions,whichisthebeginningstepfortheLagrangianconstruction. AfterareviewofEulertripletinatoycosetspace,coset SU (3) /SU (2) U (1), Eulertripletsolutionsincoset F 4 /SO (9)arediscussedindetail.Inparticular, variablerepresentationsof11-dimensionalsuperPoincar ealgebra,and F 4and SO (9)algebra,areworkedoutexplicitly,aswellasrepresentationsofKostant equationanditstripletsolutions. 5.1 Euler Triplet for SU (3) /SU (2) U (1) Asalearningexamplestartwithadeta iledanalysisoftheEulertriplets associatedwiththecoset SU (3) /SU (2) U (1).Thereisanin“nityofEulertriplets whicharesolutionsofKostantsequationassociatedwiththiscoset.Themost trivialsolutiondescribesthelight-conedegreesoffreedomofthe N =2infour dimensions,whenthe U (1)isinterpretedashelicity. 5.1.1 The N =2 Hypermultiplet in 4 Dimensions Themassless N =2scalarhypermultipletcontainstwoWeylspinorsand twocomplexscalar“elds,onwhichthe N =2SuperPoincar ealgebraisrealized. Introducethelight-coneHamiltonian PŠ= p p p+, (5.1) where p =1 2( p1+ ip2).Thefront-formsupersymmetrygeneratorssatisfythe anticommutationrelations {Qm +, Qn +} = Š 2 mnp+, {Qm Š, Qn Š} = Š 2 mnp p p+,m,n =1 2 (5.2) {Qm +, Qn Š} = Š 2 pmn.

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64 Thekinematicsupersymmetriesareexpressedas Qm += Š mŠ mp+, Qm += m+ mp+, (5.3) whilethekinematicLorentzgeneratorsaregivenby M12= i ( x p Š xp )+ 1 2 m mŠ 1 2 m m, (5.4) M+ Š= Š xŠp+Š i 2 m mŠ i 2 m m, M+ 1 2 ( M+1+ iM+2)= Š xp+, M+= Š xp+, where x =1 2( x1+ ix2),andwherethetwocomplexGrassmanvariablessatisfythe anticommutationrelations { m, n} = { m, n} = mn, { m, n} = { m, n} =0 The(free)Hamiltonian-likesupersymmetrygeneratorsaresimply Qm Š= p p+Qm +, Qm Š= p p+ Qm +, (5.5) andthelight-coneboostsaregivenby MŠ= xŠp Š 1 2 { x,PŠ} + i p p+m m, (5.6) MŠ= xŠ p Š 1 2 { x,PŠ} + i p p+ m m. ThisrepresentationofthesuperPoincar ealgebraisreducible,asitcanbeseento actonreduciblesuper“elds( xŠ,xi,m, m),becausetheoperators Dm += mŠ mp+, (5.7)

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65 anticommutewiththesupersymmetrygenerators.Asaresult,onecanachieve irreducibilitybyactingonsuper“eldsforwhich Dm +=[ mŠ mp+]=0 (5.8) solvedbythechiralsuper“eld ( yŠ,xi,m)= 0( yŠ,xi)+ mm( yŠ,xi)+ 1212( yŠ,xi) (5.9) The“eldentriesofthescalarhypermultipletnowdependonthecombination yŠ= xŠŠ im m, (5.10) andthetransversevariables.Actingont hischiralsuper“eld,theconstraintis equivalenttorequiringthat Qm +Š 2 p+m, Qm + m, (5.11) wherethederivativeismeanttoactonlyonthenaked ms,notonthosehidingin yŠ.Thislight-conerepresentationiswell-known,butwerepeatitheretosetour conventionsandnotations. 5.1.2 Coset Construction Let TA, A =1 2 ,... 8,denotethe SU (3)generators.Its SU (2) U (1) subalgebraisgeneratedby Ti, i =1 2 3,and T8.IntroduceDiracmatricesover thecoset { a,b} =2 ab, for a,b =4 5 6 7,tode“netheKostantequationoverthecoset SU (3) /SU (2) U (1)as K / = a =4 5 6 7aTa=0 .

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66 TheKostantoperatorcommuteswiththe SU (2) U (1)generators Li= Ti+ Si,i =1 2 3; L8= T8+ S8, (5.12) sumsofthe SU (3)generatorsandofthespinŽpart,expressedintermsofthe matricesas Sj= Š i 4 fjabab,S8= Š i 4 f8 abab, (5.13) where ab= ab,a = b, and fjab,f8 abarestructurefunctionsof SU (3). TheKostantequationhasanin“nitenumberofsolutionswhichcomein groupsofthreerepresentationsof SU (2) U (1),calledEulertriplets.Foreach representationof SU (3),thereisauniqueEulertriplet,eachgivenbythree representations { a1,a2} [ a2]Š2 a 1 + a 2 +3 6 [ a1+ a2+1]a 1 Š a 2 6 [ a1]2 a 2 + a 1 +3 6, where a1,a2aretheDynkinlabelsoftheassociated SU (3)representation.Here,[ a ] standsforthe a =2 j representationof SU (2),andthesubscriptdenotesthe U (1) charge.TheEulertripletcorrespondingto a1= a2=0, { 0 0 } =[0]Š1 2 [1]0 [0]1 2, describesthedegreesoffreedomofthe N =2supermultiplet,wheretheproperly normalized U (1)isinterpretedasthehelicityofthefour-dimensionalPoincar e algebra. 5.1.3 Grassmann Numbers and Dirac Matrices Inordertousethesuper“eldtechniquewewillidentifythespinpartofthe U (1)generator S8withthespinpartinEqn.(5.4)takingthecondition(5.8)into account.Thiswillmeanthatwewritealso Siintermsofthe s .Anappropriate representationisthen

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67 4+ i5= i & 2 p+Q1 +,4Š i5= i & 2 p+ Q1 +(5.14) 6+ i7= i & 2 p+Q2 +,6Š i7= i & 2 p+ Q2 +, (5.15) intermsofthekinematic N =2light-conesupersymmetrygeneratorsde“nedin theprevioussection.Wecancheckthat S8indeedagreeswiththespinpartof Eqn.(5.4)(afterpropernormalization) .AstheKostantoperatoranticommutes withtheconstraintoperators {K /, Dm +} =0 (5.16) itssolutionscanbewrittenaschiralsuper“elds,onwhichthe sbecome 4+ i5= Š 2 i 2 p+1,4Š i5= i & 2 p+ 1(5.17) 6+ i7= Š 2 i 2 p+2,6Š i7= i & 2 p+ 2, (5.18) ThecompletespinŽpartsofthe SU (2) U (1)generators,expressedintermsof Grassmannvariables,donotdependon p+, S1= 1 2 ( 1 2+ 2 1) ,S2= Š i 2 ( 1 2Š 2 1) S3= 1 2 ( 1 1Š 2 2) ,S8= 3 2 ( 1 1+ 2 2Š 1) (5.19) UsingGrassmannproperties,the SU (2)Casimiroperatorcanbewrittenas S2= 3 4 ( 1 1Š 2 2)2;(5.20)

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68 ithasonlytwoeigenvalues,3 / 4andzero.These SU (2)generatorsobeyasimple algebra SiSj= 1 3 S Sij+ i 2 ijkSk. (5.21) Thehelicity,identi“edwith S8uptoanormalizingfactorof 3,leadstohalfintegerhelicityvaluesontheGrassmann-oddcomponentsofthe(constant) super“eldrepresentingthehypermultiplet. 5.1.4 Solutions of Kostants Equation ConsidernowKostantsequationover SU (3) /SU (2) U (1).Itisgivenby K / = a =4 5 6 7aTa=0 (5.22) Schwingerscelebratedrepresentationof SU (2)generatorsofintermsof onedoubletofharmonicoscillatorshasb eenextendedtootherLiealgebras[77]. Thegeneralizationinvolvesseveralse tsofharmonicoscillators,eachspanning thefundamentalrepresentations.Thus SU (3)isgeneratedbytwosetsoftriplet harmonicoscillators,onetra nsformingasatriplettheotherasanantitriplet.Its generatorsaregivenby T1+ iT2= z12Š z21,T1Š iT2= z21Š z12, T4+ iT5= z13Š z31,T4Š iT5= z31Š z13, T6+ iT7= z23Š z32,T6Š iT7= z32Š z23, and T3= 1 2 ( z11Š z22Š z11+ z22) T8= 1 2 3 ( z11+ z22Š z11Š z22Š 2 z33+2 z33) wherewehavede“ned 1 z1,1 z1, etc ..

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69 Thehighest-weightstatesofeach SU (3)representationareholomorphicpolynomialsoftheform za11 za23, where a1,a2areitsDynkinindices:allrepresentationsof SU (3)arehomogeneous holomorphicpolynomials. NowexpandtheKostantequation(5.22)withtheDiracmatricesintermsof Grassmannvariablesyieldstw oindependentpairsofequations ( T4+ iT5) 1+( T6+ iT7) 2=0;( T4Š iT5) 2Š ( T6Š iT7) 1=0 and ( T4Š iT5) 0Š ( T6+ iT7) 12=0;( T6Š iT7) 0+( T4+ iT5) 12=0 thatis ( z13Š z31) 1+( z23Š z32) 2=0;( z31Š z13) 2Š ( z32Š z23) 1=0 ( z31Š z13) 0Š ( z23Š z32) 12=0;( z32Š z23) 0+( z13Š z31) 12=0 Thehomogeneityoperators D = z11+ z22+ z33, D = z11+ z22+ z33commutewith K / ,allowingthesolutionsofKostantequationtobearrangedin termsofhomogeneouspolynomials,onwhich a1istheeigenvalueof D and a2that of D .Thesolutionscanalsobelabeledintermsofthe SU (2) U (1)generatedby theoperators Li= Ti+ Si,i =1 2 3; L8= T8+ S8.

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70 Thesolutionsforeachtriplet,areeasilywrittenforthehighestweightstatesof eachrepresentation, = za13 za22labels[ a2]Š2 a 1 + a 2 +3 6, + 1za11 za22labels[ a1+ a2+1]a 1 Š a 2 6, + 12za11 za23, labels[ a1]2 a 2 + a 1 +3 6, (5.23) where[ ... ]arethe SU (2)Dynkinlabels.Allotherstatesareobtainedbyrepeated actionoftheloweringoperator L1Š iL2= 2 1+( z21Š z12) givingusallthestateswithineachtheEulertriplet. 5.2 Supergravity in Eleven Dimensions Theultimate“eldtheorywithoutgravityisthe“nite N =4SuperYangMillstheoryinfourdimensions.Elevendimensional N =1Supergravity[78],the ultimate“eldtheorywithgravity,isnotrenormalizable;itdoesnotstandonits ownasaphysicaltheory.However,theeleven-dimensionaltheoryhasbeenrecently revivedastheinfraredlimitofthepresumably“niteM-theory. 5.2.1 Superalgebra N =1supergravityinelevendimensionisalocal“eldtheorythatcontains threemassless“elds,thefamilia rsymmetricsecond-ranktensor, hwhichrepresentsgravity,athree-form“eld A,andtheRarita-Schwingerspinor.From itsLagrangian,onecanderivetheexpressionforthesuperPoincar ealgebra,which intheunitarytransversegaugeassumestheparticularlysimpleformintermsof thenine(16 16) imatriceswhichformtheCliordalgebra { i,j} =2 ij,i,j =1 ,..., 9 .

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71 Supersymmetryisgeneratedbythesixteenrealsupercharges Qa = Qa , whichsatisfy {Qa +, Qb +} = 2 p+ab, {Qa Š, Qb Š} = p p 2 p+ab, {Qa +, Qb Š} = Š ( i)abpi, andtransformasLorentzspinors [ Mij, Qa ]= i 2 ( ijQ)a, [ M+ Š, Qa ]= i 2 Qa , (5.24) [ M i, Qa ]=0 [ M i, Qa ]= i 2 ( iQ)a. (5.25) Averysimplerepresentationofthe11-dimensionalsuper-Poincar egeneratorscan beconstructed,intermsofsixteenanticommutingreal sandtheirderivatives, whichtransformasthespinorof SO (9),as Qa += a+ 1 2 p+a, Qa Š= Š pi p+ iQ+a, Mij= xipjŠ xjpiŠ i 2 ij, (5.26) M+ Š= Š xŠp+Š i 2 (5.27) M+ i= Š xip+, (5.28) MŠ i= xŠpiŠ 1 2 { xi,PŠ} + ipj 2 p+ij. (5.29) Thelight-conelittlegrouptransformationsaregeneratedby Sij= Š i 2 ij, whichsatisfythe SO (9)Liealgebra.Toconstructitsspectrum,wewritethe superchargesintermsofeightcomplexGrassmannvariables 1 2 + i +8 1 2 Š i +8 ,

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72 and 1 2 Š i +8 1 2 + i +8 where =0 1 2 ,..., 7.Theeightcomplex transformasthe( 4 2 ),and asthe ( 4 2 )ofthe SU (4) SU (2)subgroupof SO (9).Theeightcomplexsupercharges Q + 1 2 Q ++ i Q +8 + = + 1 2 p+, (5.30) Q  + 1 2 Q +Š i Q +8 + = + 1 2 p+ (5.31) satisfy { Q +, Q  +} = 2 p+. Toreducethenumberofthegrassmannvariables,theusualwayistoimpose covariantderivativesasconstraints, D= aŠ 1 2 p+=0 (5.32) since {D, Qa +} =0.Itfollowsthat / acanbereplacedby1 / 2 p+,when actingontheconstraint“eldsdependsonlyon canalsobetakentobezero. Therefore, Q +Š i Q +8 += 2 (5.33) Q ++ i Q +8 +=2 p+. (5.34) Thisgives, Q+= 1 2 + 2 p+i ( Š 2 p+) # (5.35) where = /, and ij=1 2[ i,j].Theyactirreduciblyonchiralsuper“elds whichareannihilatedbythecovariantderivatives Š 1 2 p+ ( yŠ, )=0 ,

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73 where yŠ= xŠŠ i 2 5.2.2 Representations of Grassman Variables SO (2 n +1)representationinDynkinbasis[ HIJ,E( I Š J ),E( I )]canbeconstructed fromitsstandardform[ Mij][79].Here [ Mij,Mkl]= i ( ikMjl+ jlMkiŠ ilMjkŠ jkMil) (5.36) and HI= M2 I, 2 I, (5.37) E( I )= M2 I Š 1 2 n +1+ i( I )M2 I, 2 n +1, (5.38) E( I Š J )= i 2 ( I + J )( M2 I Š 1 2 J Š 1+ i( I )M2 I, 2 J Š 1Š i( J )M2 I Š 1 2 J+ ( I )( J )M2 I, 2 J) (5.39)

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74 Orconversely, M2 I Š 1 2 I= HI, (5.40) M2 I Š 1 2 n +1= 1 2 ( E( I )+ EŠ ( I )) (5.41) M2 I, 2 n +1= Š i 2 ( I )( E( I )Š EŠ ( I )) (5.42) M2 I Š 1 2 J Š 1= Š i 2 ( ( I + J )E( I Š J )Š ( I + J )EŠ ( I Š J )+ ( I Š J )E( I + J )Š ( I Š J )EŠ ( I + J )) (5.43) M2 I, 2 J= Š i 2 ( I )( J )( ( I + J )E( I Š J )Š ( I + J )EŠ ( I Š J )Š ( I Š J )E( I + J )+ ( I Š J )EŠ ( I + J )) (5.44) M2 I, 2 J Š 1= Š 1 2 ( I )( ( I + J )E( I Š J )+ ( I + J )EŠ ( I Š J )+ ( I Š J )E( I + J )+ ( I Š J )EŠ ( I + J )) (5.45) M2 I Š 1 2 J= 1 2 ( J )( ( I + J )E( I Š J )+ ( I + J )EŠ ( I Š J )Š ( I Š J )E( I + J )Š ( I Š J )EŠ ( I + J )) (5.46) where I,J =1 ,n and i =1 2 n +1( n =4)for SO (9). Considerspinorrepresentation( 16 )of SO (9), M( a ) ( b ) ij= Š i 2 f( a ) ( b ) ijŠ i 2 ij. (5.47) Here( a )represent16indices,arelabeledbyfour+or Š signs,and Š ( a )means thatallsignsare”ipped.Theycanallbeswitchedtoindices0 ,..., 15through binarycounting.Forexample,if( a )=(+++ Š )=(0001)=1,then Š ( a )= ( ŠŠŠ +)=(1110)=14. sareantisymmetricrealmatrices.A( 16 )spinor representation(5.47)canbenaturally expressedin16Fermionicoscillators Q+,or 8complex (5.35), Sij= Š i 4 2 p+QT +ijQ+. (5.48)

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75 Aspecialchoiceof s,forthereasonwhichwillbecomeclearsoon,is 1= 3 1 1 3; 2= 1 1 3 1 ; 3= 3 1 3 1; 4= 1 3 1 1; 5= 3 3 1 1 ; 6= 1 1 1 3; 7= 2 2 2 2; 8= 1 1 1 1; 9= Š 1 3 3 3;(5.49) NowexplicitformsofCartangeneratorsandraisingandloweringoperatorscan bederivedfromEqn.(5.48)(Eqn.(5.36)andbelow).Matrixelementsof ijare calculatedwithaC++programdisplayedinappendix. Thecartangeneratorsare S12= Š 1 2 ( Š 00+ 11+ 22Š 33Š 44+ 55+ 66Š 77)(5.50) S34= Š 1 2 ( Š 00Š 11+ 22+ 33+ 44+ 55Š 66Š 77)(5.51) S56= Š 1 2 ( Š 00+ 11Š 22+ 33+ 44Š 55+ 66Š 77)(5.52) S78= Š 1 2 ( Š 00+ 11+ 22Š 33+ 44Š 55Š 66+ 77) (5.53) Theraisingoperatorscorrespondingtothesimplerootsare S(1 Š 2)= 36+ 41,S(2 Š 3)= 12+ 65,S(2+3)= Š ( 03+ 74) S(4)= 07+ 34+ 52+ 61,S(3 Š 4)= i ( 1 2 p+36+ 2 p+27) (5.54) where S12,S34,S56,S(1 Š 2),S(2 Š 3)and S(2+3)belongto SU (4), S78and S(4)belong to SU (2),and S(3 Š 4)mix SU (4)and SU (2)representations. S(2+3),S(1 Š 2),S(2 Š 3)correspondtothesimplerootsof SU (4), S(4)correspondstothesimplerootof SU (2),and S(1 Š 2),S(2 Š 3),S(3 Š 4),S(4)correspondtothesimplerootsof SO (9).Also

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76 theloweringoperatorsare SŠ (1 Š 2)= 63+ 14,SŠ (2 Š 3)= 21+ 56,SŠ (2+3)= Š ( 30+ 47) SŠ (4)= 70+ 43+ 25+ 16,SŠ (3 Š 4)= i ( 1 2 p+27+ 2 p+36) (5.55) Pleasenotethateach isaneigenstateofthecartangenerators S12,S34,S56and S78.Takethehigheststatetobe 0,thenactingtheloweringoperatorsonit, s couldbeeasilyidenti“edwiththestatesof( 4 2 )in SU (4) SU (2)withdynkin labels,representedin( a1,a2,a3) a4,where( a1,a2,a3)and a4aredynkinlabelsfor SU (4)and SU (2)respectively. 0 (1 0 0) 1 ,7 (1 0 0) Š 1 (5.56) 3 ( Š 1 1 0) 1 ,4 ( Š 1 1 0) Š 1 (5.57) 6 (0 Š 1 1) 1 ,1 (0 Š 1 1) Š 1 (5.58) 5 (0 0 Š 1) 1 ,2 (0 0 Š 1) Š 1 (5.59) Alternatively,wecanusetheweightspacerepresentationfor ,andfortheraising andloweringoperators,expressedineigenvaluesof S12,S34,S56,S78. 01 2( e1+ e2+ e3+ e4) ,7 1 2 ( e1+ e2+ e3Š e4) (5.60) 31 2( e1Š e2Š e3+ e4) ,4 1 2 ( e1Š e2Š e3Š e4) (5.61) 61 2( Š e1+ e2Š e3+ e4) ,1 1 2 ( Š e1+ e2Š e3Š e4) (5.62) 51 2( Š e1Š e2+ e3+ e4) ,2 1 2 ( Š e1Š e2+ e3Š e4) (5.63)

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77 S(2+3) (2 Š 1 0) 0 ( e2+ e3) (5.64) S(1 Š 2) ( Š 1 2 Š 1) 0 ( e1Š e2) (5.65) S(2 Š 3) (0 Š 1 2) 0 ( e2Š e3) (5.66) S(4) (0 0 0) 2 ( e4)(5.67) Tocalculatetheaboveformulas,“rstidentify 0withthehighestweightstate, (1 0 0) 1indynkinlabels,i.e.(1 0 0)in SO (6),and1in SO (3).since 0state hasthehighesteigenvaluesintermsof S12,S34,S56and S78,thenbyactingthe loweringoperatorsof SO (6)and SO (3) ( SŠ (2+3),SŠ (1 Š 2),SŠ (2 Š 3)for SO (6)and SŠ (4)for SO (3)),other scanalsobeidenti“edwiththedynkinlabeledstates. Expansionofthesuper“eldinpowersoftheeightcomplex syields256 components,withthefollowing SU (4) SU (2)properties 1 ( 1 1 ) (5.68) ( 4 2 ) (5.69) ( 6 3 ) ( 10 1 ) (5.70) ( 20 2 ) ( 4 4 ) (5.71) ( 15 3 ) ( 1 5 ) ( 20, 1 ) (5.72) andthehigherpowersyieldtheconjugaterepresentationsbyduality.Thesemake upthethree SO (9)representationsof N =1supergravity 44 =( 1 5 ) ( 6 3 ) ( 20, 1 ) ( 1 1 ) (5.73) 84 =( 15 3 ) ( 10 1 ) ( 10 1 ) ( 6 3 ) ( 1 1 ) (5.74) 128 =( 20 2 ) ( 20 2 ) ( 4 4 ) ( 4 4 ) ( 4 2 ) ( 4 2 ) (5.75)

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78 withthehighestweights 44 : 0347=(0 2 0) 0 ( 201 )(5.76) 84 : 07=(2 0 0) 0 ( 10 1 )(5.77) 128 : 037=(1 1 0) 1 ( 20 2 ) (5.78) togetherwiththeir SU (4) SU (2)properties.Allotherstatesaregeneratedby actingonthesehighestweightstateswiththeloweringoperators.Thehighest weightchiralsuper“eldthatdescribes N =1supergravityinelevendimensionsis simply = 07h ( yŠ, x )+ 037 ( yŠ, x )+ 0347A ( yŠ, x ) whichsummarizesthespectrumofthesuper-Poincar ealgebrainelevendimensions ofeitherafree“eldtheoryorafreesuperparticle.Allotherstatesareobtainedby applyingthe SO (9)loweringoperators. 5.2.3 F 4 /SO (9) Oscillator and Dierential form Representations Itturnsoutthatallrepresentationsoftheexceptionalgroup F4aregenerated bythree(notfour[77])setsofoscillatorstransformingas 26 Labeleachcopyof26oscillatorsas A[ ] 0,A[ ] i,i =1 9 ,B[ ] a,a = 0 15,andtheirhermitianconjugates,andwhere =1 2 3.Under SO (9), the A[ ] itransformas 9 B[ ] atransformas 16 ,and A[ ] 0isascalar.Theysatisfythe commutationrelationsofor dinaryharmonicoscillators [ A[ ] i,A[ ]  j]= ij[ ][ ], [ A[ ] 0,A[ ]  0]= [ ]. Notethatthe SO (9)spinoroperatorssatisfyBose-likecommutationrelations [ B[ ] a,B[ ]  b]= ab[ ][ ].

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79 Thegenerators Tijand TaTij= Š i4 =1 A[ ]  iA[ ] jŠ A[ ]  ] jA[ ] i + 1 2 B[ ] ijB[ ] (5.79) Ta= Š i 24 =1 ( i)ab A[ ]  iB[ ] bŠ B[ ]  bA[ ] i Š 3 B[ ]  aA[ ] 0Š A[ ]  0B[ ] a (5.80) satisfythe F4algebra, [ Tij,Tkl]= Š i ( jkTil+ ilTjkŠ ikTjlŠ jlTik) (5.81) [ Tij,Ta]= i 2 ( ij)abTb, (5.82) [ Ta,Tb]= i 2 ( ij)abTij, (5.83) sothatthestructureconstantsaregivenby fijab= fabij= 1 2 ( ij)ab. ThelastcommutatorrequirestheFierz-derivedidentity 1 4 ijij =3 + ii, fromwhichwededuce 3 acdb+( i)ac( i)dbŠ ( a b )= 1 4 ( ij)ab( ij)cd. Tosatisfythesecommutationrel ations,wehaverequiredboth A0and Batoobey Bosecommutationrelatio ns(Curiously,ifbothareanticommuting,the F4algebra isstillsatis“ed).Onecanjustaseasilyuseacoordinaterepresentationofthe oscillatorsbyintroducingrealcoordinates xiwhichtransformastransversespace

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80 vectors, x0asscalars,and yaasspacespinorswhichsatisfyBosecommutationrules Ai= 1 2 ( xi+ xi) ,A i= 1 2 ( xiŠ xi) (5.84) Ba= 1 2 ( ya+ ya) ,B a= 1 2 ( yaŠ ya) (5.85) A0= 1 2 ( x0+ x0) ,A 0= 1 2 ( x0Š x0) (5.86) Fromnowon,letususesquarebrackets[ ]torepresentthedynkinlabelof F 4, androundbrackets( )torepresentthedynkinlabelof SO (9),Intheweight spacesofthecartangenerators,(eigenvaluesof T12,T34,T56,T78),theraising operatorscorrespondtothesimplerootsof F 4are T(2 Š 3) [2 Š 100] ( e2Š e3) (5.87) T(3 Š 4) [ Š 12 Š 20] ( e3Š e4) (5.88) T(4) [0 Š 12 Š 1] ( e4) (5.89) T 1 2 ( T4+ iT12) [00 Š 12] 1 2 ( e1Š e2Š e3Š e4) (5.90) where T(2 Š 3),T(3 Š 4),T(4)arede“nedbyEqn.(5.39)for SO (9)generatorsasusual, and Trepresentsthe F 4simplerootraisingoperatortransformasaspinorunder the SO (9)subgroup,andalso TŠ 1 2( T4Š iT12)willbeusedtorepresentthe loweringoperator.Alsointhesamespace,theraisingoperatorscorrespondtothe simplerootsof SO (9)are T(1 Š 2) (2 Š 100) ( e1Š e2) (5.91) T(2 Š 3) ( Š 12 Š 10) ( e2Š e3) (5.92) T(3 Š 4) (0 Š 12 Š 2) ( e3Š e4) (5.93) T(4) (00 Š 12) ( e4) (5.94)

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81 Theweightstatesare x1+ ix2 [0001] (1000) ( e1) (5.95) x3+ ix4 [100 Š 1] ( Š 1100) ( e2) (5.96) y0+ iy8 [001 Š 1] (0001) 1 2 ( e1+ e2+ e3+ e4) (5.97) y7+ iy15 [01 Š 10] (001 Š 1) 1 2 ( e1+ e2+ e3Š e4) (5.98) y2Š iy10 [1 Š 110] (01 Š 11) 1 2 ( e1+ e2Š e3+ e4) (5.99) y5Š iy13 [10 Š 11] (010 Š 1) 1 2 ( e1+ e2Š e3Š e4) (5.100) 5.2.4 Solution of Kostant Equation in F 4 /SO (9) De“neClifordalgebraover16-dimensionalcoset F 4 /SO (9), { a, b} =2 ab,a,b =0 1 ,..., 15 (5.101) generatedby(256 256)matrices.TheKostantequationisde“nedas K / =16a =1aTa=0 (5.102) where Taare F4generatorsnotin SO (9),withcommutationrelations [ Ta,Tb]= ifabijTij. (5.103) Althoughitistakenoveracompactmanifold,ithasnon-trivialsolutions.Tosee this,werewriteitssquareasthedierenceofpositivede“nitequantities, K / K / = C2 F4Š C2 SO (9)+72 (5.104) where C2 F4= 1 2 TijTij+ TaTa, (5.105)

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82 isthe F4quadraticCasimiroperator,and C2 SO (9)= 1 2 TijŠ ifabij' ab2, (5.106) isthequadraticCasimirforthesum Lij Tij+ Sij, (5.107) where Sij= Š i 8 ( ij)abab, (5.108) is SO (9)generator(5.48),whichactsonthesupergravity“elds.Thequadratic Casimironthespinorrepresentationis 1 2 SijSij=72 (5.109) Kostantsoperatorcommuteswiththesumofthegenerators, [ K /,Lij]=0 (5.110) allowingitssolutionstobelabeledby SO (9)quantumnumbers. Sincethelittlegroupgenerators Sijactona256-dimensionalspace,theycan beexpressedintermsofsixteen(256 256)matrices,a,whichsatisfytheDirac algebra { a, b} =2 ab. Thisleadstoanelegantrepresentationofthe SO (9)generators Sij= Š i 4 ( ij)ababŠ i 2 fijabab. ,whichcanbeidenti“edwithEqn.(5.48),consideringthereplacement a ( 2 p+Qa. (5.111)

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83 Thecoecients fijab 1 2 ( ij)ab, naturallyappearinthecommutatorbetweenthegeneratorsof SO (9)andany spinoroperator Ta,as [ Tij,Ta]= i 2 ijT a= ifijabTb. Butthereismoretoit,the( ij)abcanalsobeviewedasstructureconstantsof aLiealgebra.Manifestlyantisymmetricunder a b ,theycanappearinthe commutatoroftwospinorsintothe SO (9)generators [ Ta,Tb]= i 2 ( ij)abTij= fabijTij, andoneeasilychecksthattheysatisfytheJacobiidentities.Remarkably,the52 operators Tijand TageneratetheexceptionalLiealgebra F4,showingexplicitly howanexceptionalLiealgebraappearsinthelight-coneformulationofsupergravityinelevendimensions. ForKostantsolution=( ) f ( x,y ) aTa=(a( ))( Taf ( x,y ))=0 (5.112) Therefore a( )=0 ,Taf ( x,y )=0 (5.113) since[ Lij, aTa]=0,( )and f ( x,y )areboththehighestweightstatesofthe SO (9)algebra, Sijand Tij,and Sij( )= Tijf ( x,y )=0(5.114) To“ndthesolutionfortheKostanteqn.,wehavetochoosethespeci“c representation,(5.101)and(5.79),(5.80) ,forthegenerators,andthereforethe solutionisformedbythestatesofthisrepresentation.

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84 Letus“rstshowsomeusefulrelationsbasedontheDynkindiagramof26in F 4,shownattheendofthechapter,wherethenumbers1 2 3 4nearthearrow representtheloweringoperators TŠ (2 Š 3),TŠ (3 Š 4),TŠ (4),TŠ respectively.Usingthe explicitformula,Eqn.(5.79),(5.80),(5.39) and(5.90),forthoseloweringoperators, thecalculationshows, TŠ ( x1+ ix2)= i 2 ( 1)4 b+ i ( 2)4 bŠ i ( 1)12 b+( 2)12 b yb= i ( y0+ iy8) TŠ (4)i ( y0+ iy8)= 1 2 ( 79)a 0+ i ( 79)a 8+ i ( 89)a 0Š ( 89)a 8 ya= i ( y7+ iy15) TŠ (3 Š 4)i ( y7+ iy15)= Š i 4 ( 57)a 7+ i ( 57)a 15Š i ( 67)a 7+( 67)a 15+ i ( 58)a 7Š ( 58)a 15+( 68)a 7+ i ( 68)a 15 ya= y2Š iy10TŠ (4)( y2Š iy10)= Š i 2 ( 79)a 2Š i ( 79)a 10Š i ( 89)a 2Š ( 89)a 10 ya= Š ( y5Š iy13) TŠ ( Š y5+ iy13)= i 2 ( i)45Š i ( i)4 13Š i ( i)12 5Š ( i)12 13 xi= i ( x3+ ix4) wherewekeepthecoecientofthestatesforthelaterantisymmetricconstruction ofthehighestweights. ToverifythesolutionsfortheKostantequation,weneedtoidentifythe generators, T4+ iT12= 2 T,T4Š iT12= 2 TŠ T3+ iT11= 2[ T(4),T] ,T3Š iT11= Š 2[ TŠ (4),TŠ ] andalsofromeqn.(5.101),Kostantoperatorcanberewriteas, aTa= i [ ( T+ iT +8) Š ( TŠ iT +8)] (5.115) TheaboveexplicitformshowsthatKostantoperatorsarejustcomposedof raisingandloweringoperatorsconstructedby Ta.Sinceweonlyneedtoverifythe highestweightsolutions,onlytheloweringoperatorsareneededtobetakeninto consideration.ExplicitcalculationshowsthatwhenactingtheKostantoperator

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85 onthehighestweightsolution,mostofthetermswillvanishdueto and ,and onlyfewloweringoperators(1 2( T3Š iT11) ,1 2( T4Š iT12))needtobeconsidered. Thesolutionsofkostantsequationform SO (9)triplets.Foreveryrepresentationof F 4,indynkinlabel,[ a1,a2,a3,a4],thereisa SO (9)tripletsolution associated[72], (2+ a2+ a3+ a4,a1,a2,a3) ( a2,a1, 1+ a2+ a3,a4) (1+ a2+ a3,a1,a2, 1+ a3+ a4)(5.116) Nowletusparameterizetheaby a,and Taby x,y ,withEqn.(5.101) and(5.79).KostantequationaTa=0willhavethesolutionintheform ( ,x,y )=( ) f ( x,y ). Thesolutioninthe“rstleveliswhen a1= a2= a3= a4=0 (2000) (0010) (1001)or(44) (84) (128).Thehighestweightsolutionis 0347,07and 037foundbefore. To“ndsolutionsinthehigherlevel,noticetwothings, 1.Thesolutionsareintheformof( ) f ( x,y ),where( )= 0347,07or 037.( )and f ( x,y )areboththehighestweightsofthe SO (9)subgroup formedbygenerators, Lij= Sij+ Tij,eqn(5.2.4)and(5.79). 2.Forthefundamentalrepresentationof F 4(indynkinlabel, a1,...,a4areall zeroexceptone ai=1),supposetheassociatedsolutionis( ) f ( x,y ),then f ( x,y )willbeformedbythestatesofthe F 4fundamentalrepresentation. Furthermore,since Tijrepresentationsarehomogeneouspolynomialof x and y ,( ) f ( x,y )aiisthesolutionforhigherlevel( ai> 1). Usingthegeneralizedformofthetripletsolutions(5.116),thehighestweight solutionscorrespondtoeachfundamentalrepresentationof F 4areconstructedas follows: 1. a1= a2= a3=0 ,a4=1

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86 F 4statesare =1copyof26states.Thehighestweightsolutionsare 0347( x1+ ix2) (5.117) 07( y0+ iy8) (5.118) 037( y0+ iy8) (5.119) where( x1+ ix2)and( y0+ iy8)arethehighestweightstatesof SO (9) representations(1000)and(0001)respectively,andtheyarealsothestates belongtothe 26 of F 4.DirectcountingoftheDynkinlabelshowstheabove solutionisconsistentwiththegeneralform(5.116).Toverifythesolution, oneneedtousethepropertiesofloweringsimplerootgeneratorstotraverse throughweightstates, aTa 0347( x1+ ix2) = i ( ( T+ iT +8) Š ( TŠ iT +8)) 0347( x1+ ix2) =0 (5.120) aTa 07( y0+ iy8) = Š i 3( T3Š iT11)+ 4( T4Š iT12) 07( y0+ iy8) = Š i 3( Š 2)[ TŠ (4),TŠ ]+ 4 2 TŠ 07( y0+ iy8) =0 (5.121) 2. a2= a3= a4=0 ,a1=1 F 4statesarerepresentedbyantisymmetricproductsof =2copiesof26 states.Fromthegeneralformofthesolution,(5.116),weneedtorepresent the SO (9)highestweightstate(0100)of 36 bytheantisymmetricproducts ofthetwocopiesof 26 of F 4.Thisstateisalsothehighestweightstate of 52 of F 4.Since( 26 26 )a= 52 + 273 ,Toformthisstate,usethe sixthhighestweightstatesof 26 ,antisymmetrizethe“rstandthesixth, thesecondandthefourth,thethirdandthefourth,thenchooseproper

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87 coecientstocombinethem.Thishighestweightstateof SO (9)isfoundis tobe([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10]).Itis annihilatedbyallthesimplerootsraisingoperatorsof SO (9)and F 4.The highestweightsolutionsare 0347([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10]) 07([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10]) 037([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10]) (5.122) where([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10])is thehighestweightofthe SO (9)representation(0100),andherewedenote [ a,b ]= a[1]b[2]Š a[2]b[1],antisymmetricproductof2copiesof a and b states. Theveri“cationofthissolutionissim ilartothepreviouscase.Forexample, aTa 07([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10] = Š i 3( T3Š iT11)+ 4( T4Š iT12) 07([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10] = Š i 3( Š 2)[ TŠ (4),TŠ ]+ 4 2 TŠ 07([ x1+ ix2,x3+ ix4]+[ y0+ iy8,y5Š iy13]+[ y7+ iy15,y2Š iy10] = Š i Š 2 307([ i ( y7+ iy15,x3+ ix4]+[ y7+ iy15, Š i ( x3+ ix4)]) + 2 407([ i ( y0+ iy8) ,x3+ ix4]+[ y0+ iy8, Š i ( x3+ ix4)]) (5.123) 3. a1= a2= a4=0 ,a3=1

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88 F 4states,( 273 ),arealsorepresentedbyantisymmetricproductsof =2 copiesof26states.Thehighestweightsolutionsare 0347[ x1+ ix2,y0+ iy8] (5.124) 07[ y0+ iy8,y7+ iy15] (5.125) 037[ x1+ ix2,y0+ iy8] (5.126) where[ x1+ ix2,y0+ iy8]and[ y0+ iy8,y7+ iy15]arethehighestweightsofthe SO (9)representations(1001) ( 16 9 )and(0010) ( 16 16 )arespectively. [ x1+ ix2,y0+ iy8]isalsothehighestweightof 273 of F 4. 4. a1= a3= a4=0 ,a2=1 F 4representation( 1274 )canberepresentedbyKroneckerproductsof =3copiesof26states.Thehighestweightstateissimplythetotal antisymmetrizationofthehighestth reestatesin26.Thehighestweight solutionsare 0347[ x1+ ix2,y0+ iy8,y7+ iy15] (5.127) 07[ x1+ ix2,y0+ iy8,y7+ iy15] (5.128) 037[ x1+ ix2,y0+ iy8,y7+ iy15] (5.129) where[ x1+ ix2,y0+ iy8,y7+ iy15]isthehighestweightofthe SO (9) representation1010,and[ a,b,c ]istheantisymmetricproductsof3copiesof a,b and c states.Itisalsothehighestweightstateof 1274 of F 4.

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CHAPTER6 SUMMARY Thisdissertationincludestwolooselyconnectedparts.Themainfocusis quantumaspectsofNC“eldtheories,includingbothperturbativeandnonperturbativestructures.Inparticular,perturbativebehaviorofNCsupersymmetric Wess-Zuminomodelisdiscussedindetail.ItisshownthatNCWZmodelhas onlywavefunctionrenormalizationandUV“niteasitscommutativeanalog.It issuggestedsupersymmetricinvarianceofNCWZmodelagainleadstocancelationwhichrendersmassandvertexcorrectionsUV“nite.UV/IRmixingterms, asaresultofphasefactorsinducedinthev ertex,generallyexistinallquantum perturbationcalculations.NCsolitons,nonperturbativestructureinNC“eld theory,areinterpretedaslowenergymani festationoflowerdimensionalD-branes instringtheory.ThroughquantizationofNCsolitons,correctionstotheenergy arecalculatedindetail.EnergyofNCGMSsolitonsisfoundtobeUV“nite,and alsoincludesUV/IRmixingterms,whichneednotbesurprisingconsideringtheir generalexistenceinperturbationtheor y.UV/IRmixingtermsinperturbative theoryaresuggestedasresultsofparticlestravelinginextradimensions,which inthecontextofstringtheory,areinterp retedaslowenergyclosedstringmodes dualtohighenergyopenstringmodeslivingonthebrane.ExistenceofUV/IR mixingtermsinNCsolitonenergysuggeststhesemodesalsointeractwithlower dimensionalD-branes.PropertiesofNCscalarsolitonsalreadymaketheUV/IR termsanintriguingsubject.SomeNCsolitons(GMSsolitons)existonlyatlarge enough ,andquantumcorrectionsofwhichdonotincludeUV/IRtermsnear in“nite limit.Therearemanyquestionsreadytobeanswered.HowdoUV/IR termsaectthestabilityofNCsolitons? Isthereanyinterpretationfromstring 89

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90 theorythatsuchterms,whichrevealthestructureofstringdiagrams,ceasetoexist atverylarge ?WhatnewinsightcanthoseUV/IRtermsgivewithregardtothe interactionbetweenD-braneandclosedstrings? SuperPoincar esymmetryisconsideredasbasicspace-timesymmetryoffundamentaltheory.NC“eldtheories,aslowe nergylimitofstringtheory,explicitly violateLorentzsymmetry.Thusitbeco mesimportanttounderstandthespacetimesymmetriesonwhichNC“eldtheoriesareconstructed.Arepresentationof deformedsuperPoincar ealgebraisobtainedandcommutationrelationsarecalculatedinanintuitiveway.Preservativeofsupersymmetrysupportstheattempt toconstructsupersymmetricNC“eldtheorydirectlyfromsupersymmetricgeneralizationofNCspace.Thepresenceofthe B “eldontheboundaryofD-brane enablesdecouplingoflowenergymodesofstringtheoryincertainlimit,butalso yieldsnoncommutativityexplicitlybrok enLorentzsymmetry.If“eldtheoriesare fundamentallyNC,therewillbeaverysmallupperboundoftheNCparameter, sinceinourspace“eldtheoryseemstobeLorentzinvariantinhighprecision.An alternativeexplanationistotakein toconsiderationtheexistenceofa B “eld. Indeedcovarianceofthetheoryiseasilyjusti“ediftheNCparametersaretaken tobe ,wheretheindicestransformaccordinglyunderLorentzrotation.Symmetry,nonlocality,causalityandunitari tywillcontinuetobeimportantissuesin identifyingNC“eldtheoriesasrealistictheories. SolutionofKostantequation,aswellassupersymmetryalgebrarepresentation in11dimension,isconsideredasana ttempttoconstructzeroslopelimitsof stringtheory,ifwebelievetheyobeysuperPoincar esymmetryandreduceto11 dimensionsupergravityinlowenergylimit.Theappearanceofanin“nitetowerof tripletsisinterestingandexpectedbutconstructionofLagrangianandinteractions forthosemultipletsstillneedsfurtherwork.

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APPENDIX COMPUTERCODE Thisprogramisdesignedtooutputmatrixelementsof iand ij(5.49). //Defineacomplexnumberclassandasigmaclass, //andagammaclassandaF_barclass. #include usingnamespacestd; #ifdef_MSC_VER classF_bar; ostream&operator<<(ostream&,constF_bar&); #endif #ifdef_MSC_VER classsComplex; ostream&operator<<(ostream&,constsComplex&); #endif //defineaclasssComplexwhichrepresentsthecomplexnumberseither //imaginaryorreal. classsComplex { public: intvalue; 91

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92 boolreal; sComplex(){} sComplex(int); sComplex(int,bool); sComplexoperator+(sComplex); sComplexoperator*(sComplex); friendostream&operator<<(ostream&,constsComplex&); }; //defineaSigmaclasstorepresentDiracSigmamatricesorthe //unitmatrix. classSigma { public: intindex; sComplexsign; sComplexele[2][2]; Sigma(){} Sigma(int); Sigma(int,sComplex); //~Sigma(); Sigmaoperator*(Sigma); protected: voidSigmaInit(int); voidEleInit(sComplexa[2][2]);

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93 }; //defineaGammaclassforgammamatrix,whichincludesanoverall //signand4sigmamatrices.Mathematicallyitisthedirect //productoffoursigmamatrices. classGamma { public: Sigmaele[4]; sComplexsign; Gamma(){} Gamma(sComplex,int,int,int,int); Gammaoperator*(Gamma&); }; //afunctionreturnstheantisymmetryorsymmetrypropertyof //sigmamatrices. booltransSigma(Sigma&a) { if(a.index==0||a.index==3||a.index==1)returntrue; if(a.index==2)returnfalse; returntrue; }

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94 //afunctionreturnstheantisymmetryorsymmetrypropertyof //gammamatrices. booltransGamma(Gamma&a) { booly=true; for(inti=0;i<4;i++) { if(y==true)y=transSigma(a.ele[i]); elsey=!transSigma(a.ele[i]); } returny; } //operator==overloadingforgammamatrices booloperator==(Gamma&a,Gamma&b) { for(inti=0;i<4;i++) { if(a.ele[i].index!=b.ele[i].index)returnfalse; } returntrue; } //outputgammamatricesasthedirectproductof4sigmamatrices. ostream&operator<<(ostream&cout,constGamma&gamma) {

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95 cout< //#include usingnamespacestd;

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96 //sComplexclassconstructorwithaninteger. sComplex::sComplex(intvalue) { this->value=value; this->real=true; } //sComplexclassconstructorwithanintegerandabooleanforreal //orimaginary sComplex::sComplex(intvalue,boolreal) { this->value=value; this->real=real; } //operator+overloadingforthesComplexclass sComplexsComplex::operator+(sComplexa) { if(this->real&&a.real)returnsComplex(this->value+a.value); elseif(this->real||a.real)cout<<"mistake"; elsereturnsComplex(this->value+a.value,false); } //operator*overloadingforthesComplexclass.sComplex //objectswillmultiplyeachotherlikecomplexnumbers sComplexsComplex::operator*(sComplexx)

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97 { intval=this->value*x.value; if(this->real==true) { if(x.real==true)returnsComplex(val); elseif(x.real==false)returnsComplex(val,false); } elseif(this->real==false) { if(x.real==true)returnsComplex(val,false); elseif(x.real==false)returnsComplex(-val,true); } } //outputsComplexobjectslikecomplexnumbers. ostream&operator<<(ostream&cout,constsComplex&x) { if(x.real==true||x.value==0)cout<
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98 //Sigmaclassconstructorwithindex.TheconstructedSigma //objectswillbeDiracsigmamatrices,andtheunitmatrix. Sigma::Sigma(intindex) { SigmaInit(index); this->sign=sComplex(1); } //EnabletheSigmaobjectstohavecomplexcoefficients. Sigma::Sigma(intindex,sComplexsign) { SigmaInit(index); this->sign=sign; } voidSigma::SigmaInit(intindex) { this->index=index; if(index==0) { sComplexa[2][2]={{1,0},{0,1}}; EleInit(a); } elseif(index==1) { sComplexa[2][2]={{0,1},{1,0}};

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99 EleInit(a); } elseif(index==2) { sComplexa[2][2]={{0,mAi},{ai,0}}; EleInit(a); } elseif(index==3) { sComplexa[2][2]={{1,0},{0,-1}}; EleInit(a); } } voidSigma::EleInit(sComplexa[2][2]) { for(inti=0;i<2;i++) { for(intj=0;j<2;j++) { ele[i][j]=a[i][j]; } } }

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100 /*Sigma::~Sigma() { for(inti=0;i<2;i++) delete[](ele[i]); }*/ //operator*overloadingforSigmaobjects,whichmultiply //eachotherlikeDiracsigmamatrices SigmaSigma::operator*(Sigmax) { sComplexthisSign=this->sign*x.sign; if(this->index==0)returnSigma(x.index,thisSign); if(x.index==0)returnSigma(this->index,thisSign); if(this->index==1) { if(x.index==1)returnSigma(0,thisSign); if(x.index==2)returnSigma(3,ai*thisSign); if(x.index==3)returnSigma(2,mAi*thisSign); } if(this->index==2) { if(x.index==1)returnSigma(3,mAi*thisSign); if(x.index==2)returnSigma(0,thisSign); if(x.index==3)returnSigma(1,ai*thisSign); } if(this->index==3) {

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101 if(x.index==1)returnSigma(2,ai*thisSign); if(x.index==2)returnSigma(1,mAi*thisSign); if(x.index==3)returnSigma(0,thisSign); } } //defineDiracsigmamatricesandtheunitmatrics. Sigmasigma_0=Sigma(0); Sigmasigma_1=Sigma(1); Sigmasigma_2=Sigma(2); Sigmasigma_3=Sigma(3); //eachgammamatrixisconstructedby4sigmamatricesandasign. //mathematicallyitisthedirectproductofthe4sigmamatrices. Gamma::Gamma(sComplexsign,inta,intb,intc,intd) { this->sign=sign; ele[0]=a; ele[1]=b; ele[2]=c; ele[3]=d; } //operator*overloadingforthegammamatrices GammaGamma::operator*(Gamma&x) { Sigmaa[4];

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102 for(inti=0;i<4;i++) { a[i]=this->ele[i]*x.ele[i]; } sComplexthisSign=this->sign*x.sign; for(intj=0;j<4;j++) { thisSign=thisSign*(a[j].sign); } returnGamma(thisSign,a[0].index,a[1].index,a[2].index, a[3].index); } //calculatethematrixelementsofthemuliplicationof2gamma //matrices,andputtheminthe(16,16)arrayinsideF_barobject. F_bar::F_bar(Gammax,Gammay) { Gammaz=x*y; for(inti1=0;i1<2;i1++) { for(inti2=0;i2<2;i2++) { for(inti3=0;i3<2;i3++) { for(inti4=0;i4<2;i4++) {

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103 for(intj1=0;j1<2;j1++) { for(intj2=0;j2<2;j2++) { for(intj3=0;j3<2;j3++) { for(intj4=0;j4<2;j4++) { ele[i1*8+i2*4+i3*2+i4*1][j1*8+j2*4+j3*2+j4*1]= z.sign*z.ele[0].ele[i1][j1]*z.ele[1].ele[i2][j2]* z.ele[2].ele[i3][j3]*z.ele[3].ele[i4][j4]; } } } } } } } } } //outputeachnonzeroelementintheF_bararray. ostream&operator<<(ostream&cout,constF_bar&f_bar) { /*offdiagonalelements,halfofthematrixelements*/ for(inta=0;a<16;a++)

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104 { for(intb=a+1;b<16;b++) { if(f_bar.ele[a][b].value!=0) { cout<
PAGE 114

105 } /*Thepurposeofthisprogramistocalculatethestructurefunctions fortheExceptionalLiealgebraF4.F4has36generatorsforminga subalgebraSO(9),and16generatorstransforminginaspinor representationoftheSO(9).ThespinorrepresentationoftheSO(9) iswellknowntobeconstructedby9gammamatrices.Thisprogram willoutputallthenonzeromatrixelementsofthespinor representation,giventheinputofthegammamatrices.*/ //#include"F_bar.h" #include"F_bar.cpp"//includeclassfiles. intmain() { Gamma*gamma=newGamma[10];//initialize gamma[0]=Gamma(1,0,0,0,0);//aunitmatrixforspecialpurpose gamma[9]=Gamma(-1,0,3,3,3);//construct9gammamatrices gamma[1]=Gamma(1,3,1,0,3); gamma[2]=Gamma(1,1,1,3,0); gamma[3]=Gamma(1,3,0,3,1); gamma[4]=Gamma(1,1,3,0,1); gamma[5]=Gamma(1,3,3,1,0); gamma[6]=Gamma(1,1,0,1,3); gamma[7]=Gamma(1,2,2,2,2); gamma[8]=Gamma(1,0,1,1,1);

PAGE 115

106 //outputtheF_barobjectsforeachi,j.Theoutputisthenonzeroelements //ofthematrixelements. for(inti=1;i<10;i++) { for(intj=i+1;j<10;j++) { cout<
PAGE 116

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111 [72]T.PengpanandP.Ramond,Phys.Rep. 315 ,137(1999). [73]B.Gross,B.KostantP.RamondandS.Sternberg, Proc.Natl.Acad.Scien. 8441(1998). [74]LarsBrinkandP.Ramond,DiracEquations,Light-ConeSupersymmetry, andSuperconformalAlgebras,ŽinShifman,M.A.(ed.):TheManyFacesof theSuperworld398-416;hep-th/9908208, http://www.arxiv.org [75]LarsBrink,EulerMultiplets,Light-ConeSupersymmetryandSuperconformalAlgebras,ŽinProceedingsoftheInternationalConferenceon Quantization,GaugeTheory,andStrings:ConferenceDedicatedtothe MemoryofProfessorE“mFradkin,Moscow2000. [76]P.Ramond,BosonFermionConfusion:TheStringPathtoSupersymmetry,ŽNucl.Phys.Proc.Suppl. 101 ,45(2001);hep-th/0102012, http://www.arxiv.org [77]T.Fulton,J.Phys.A 18 ,2863(1985). [78]E.Cremmer,B.JuliaandJ.Scherk,Phys.Lett. B76 ,409(1978) [79]H.Georgi, LieAlgebrasinParticlePhysics (Reading,Mass.:PerseusPublishing,2ndedition,1999).

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BIOGRAPHICALSKETCH Xiao-ZhenXiongwasbornonJuly30th,1972,inXiu-Shui,avillagetownin Jian-Xi,aprovinceinthesouthofChina.HeistheonlysonofYa-teXiongand Qi-fangZhou,andthelittlebrotherofXiao-linXiong. Xiao-ZhenstayedinXiu-ShuiforroughlytwoyearsbeforehemovedtoNanChang,capitalcityofJiang-Xi,accord ingtohismother.Xiao-Zhenonlyhas ”ashesofmemoriesbeforehisfourthyear.Hestartedtoshowtalentinmathat “veyearsofage,whenhewasabletomultiplynumberswithoutusingapencil. Alsohewaseagertolearnmoremathandotherthingsandenteredtheelementary schoolatsixyearsold.Hismother,ateacherinphysics,deservesmostofthecredit forXiao-Zhensearlyeducationandinterestinmathandphysics. Xiao-ZhenstayedinNan-Changfor15yearsuntilhe“nishedhighschool,just likemostpeopleathisageinChinadid.Forsomereason,althoughXiao-zhenhas alwaysbeenregardedasapersonwithverygoodpotential,hehasneverbeenable todisplaythattruly.Accordingtohissi ster,althoughheisaveryhappyperson livinginanicefamily,heseemstoliveinsomesortofunconsciousway.Somehow heisalittleseparatefromthesurroundingworld.Itiscertainthathewouldfacea lotofchangeswillinglyorunwillinglylaterinhislife. Thesituationstartedtochangealittlebitafterhelefthisparentsforacollege educationthousandsofmilesawayint henorthofChina.Itwasasmallcollege inChang-Chun,Ji-Lin,averycoldplacecomparedtoNan-Chang.Therehehad verygoodclassmatesandfriends.Hestartedtogrowstrongerandplaybasketball andsoccer.Thephysicallifewasnotsoeasy.Herememberssittinginthecrowded trainforthousandsofmilesgoingtoandfromtheschooltwiceayear.Healso 112

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113 remembersthehardnessofstudyinthecoldclassroomandsleepinthecolddorm forthepoorheatingheatingsystem.Butitwasnotallbad.Duringthattime,he learnedhowtotryhisbesttoachievehisgoal,althoughheattemptedbutfailedto entergraduateschooltostudyhighenergytheory. AfterXiao-Zhensgraduationfromcollege,hewentontoworkinasmall companyinXia-Men,Fu-Jian,asanengineer.Afterafewmonths,herealizedthe simpleworklifewasnothisdestiny.Inhi shearthestillwantedmorechallenges andopportunities. ThedreamallcametruewhenhewasacceptedbythePhysicsDepartment aththeUniversityofFloridainfall1996.Hestartedtoenjoyeverything,fromthe lecturesbythebestprofessorstofantasticfootballandNBAbythebestplayers. ThefreedomintheUSAalwaysbroughtavarietyofchoices,andhewasnotreally surewhichtochoose.Aftersometries,he“nallysettledtostudywithProf.Pierre Ramondonhighenergytheorypartlybecausehehadalwaysdreamedofbeinga puretheoreticalphysicist. Sincethenforhimithasbeenanotherlife-changing4years.Xiao-Zhenhad metalotoffrustrationsfromacademicr esearchtopersonalemotions.Hewas veryfortunatetohavePierreashisadvisor,fromwhomhelearnedtheessenceof researchprogress,aswellasmentallybeingreadyforthechallenges.Hisresearch areahascoveredsuperPoincar ealgebraandnoncommutative“eldtheoriesand solitons,fromwhichhepublished4papers.Hehasalsotaughthundredsof studentsinthephysicslab.Bythetimeofhisgraduation,hewouldsayhehad masteredtheabilityoflearningandrese archandestablishedcon“dencetodeal withthechallengesinanytheoreticalsciencearea. Afterhisgraduation,hehopestoswitchtoaresearchareawhichhasmore practicalapplications.Currentlyheisswit chingtothecomputationalbiologyarea,

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114 andhopefullyinthisareahecanexploithistheoreticalbackgroundandresearch ability.


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Title: Noncommutative field theories, solitons and superalgebra
Physical Description: Mixed Material
Creator: Xiong, Xiaozhen ( Author, Primary )
Publication Date: 2002
Copyright Date: 2002

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NONCOMMUTATIVE SCALAR FIELD THEORIES,
SOLITONS AND SUPERALGEBRA















By

XIAOZHEN XIONG


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2002


































Copyright 2002

by

Xiaozhen Xiong















I dedicate this work to my family.















ACKNOWLEDGMENTS

When one thinks about his or her success, one would find that an individual's

success alv--l- relies on the kindness and help from many other people. It is no

exception for the PhD degree that I will be receiving. Listed in the following are

the persons towards whom I feel deepest appreciation. Certainly they are not the

only ones that deserve the credit for my success. With this opportunity I would

also like to acknowledge my thanks to all other people who have helped me.

I would like to begin by thanking my parents, Yate Xiong and Qifang Zhou.

When I was little and naughty, they alv--,v- led me through hardship to happiness.

When older and ambitious, they were ah v--l very supportive of me. They ah--,v-

leave me the best, but never ask me to do .nivthing in return. Every time I

succeed, it is ah--bv- been a great joy to them as well. This time certainly I would

like to share the wonderful moment with them again.

A special thank you also goes to my sister Xiaolin Xiong. As an elder sister,

she is ah--bv- ready to look after me. She ahv--, walks one step ahead of me, and

shares with me her success and failure. I feel very lucky to have such a nice sister.

If it was not because of her, I would have experienced much more failure while I

was growing up. I am sure she will feel very proud to see her brother to get a PhD

one di-.

Of course I owe a great debt of gratitude to my advisor, Prof. Pierre Ramond.

It has been a great honor and pleasure to have him as my supervisor, and work

with him through the PhD. He not only led me through the academic research

work, but also from time to time gave me -ii. -1 i.- ll- on mental strength for

being a dedicated scientist. Some of his criticisms were even blunt, but I am sure









I will remember all of them because they have contributed a lot to my success in

obtaining my PhD and will have a tremendous amount of influence in my future

work and life as well.

A lot of thanks go to Prof. Richard Woodard. He is the very first one who

introduced me to the field theory, and was ah--lv- willing to answer my questions

in detail. The knowledge I have learned in the field theory class has ahlv- i been a

great support to my PhD research. His principles for physicist and for nerd have

aliv-- been an encouragement for me to move forwards. I admire his courage and

determination in holding his principles, because often I found they are far more

easy to understand than to stick to.

Very special thanks go to my very good friend Vivian Guo. She has encour-

aged me a lot during the hardest time in my PhD work, when I felt exhausted

physically and mentally. The time we spent together is .i.-- li-, a precious and

beautiful memory to me. At this moment of my success I would like to i- thanks

to her and wish her the best.

Many thanks go to my girlfriend, Qiang Mei. She gave me a lot of encour-

agements when I met difficulties in the research, and felt that I was not able to

finish the research work. It is a great pleasure to talk to her every iv. I am looking

forward to continuing to develop our relationship after I finish the PhD.

I would also like to thank Susan Rizzo and Darlene Latimer. Their work has

.il.-- i,- been, and will ah--xi-s be very helpful to graduate students and faculty in

the department. Certainly I appreciate the convenience they bring to me during

the long time I have spent here.

Finally a lot of thanks would have to go to the faculty, postdoctorals and

fellow students in IFT. They are ah--xi-i very friendly and ready to help and discuss

the questions. The research environment here is wonderful. It would be a great

memory of mine spending the best time in my life with the people here. Particular









thanks go to my other PhD committee members, Dr. Pierre Sikivie, Dr. C'!i ,i

Thorn, Dr. Zongan Qiu and Dr. Paul Ehrlich. I appreciate their time in guiding

my process towards the PhD.















TABLE OF CONTENTS
page

ACKNOWLEDGMENTS ................... ...... iv

ABSTRACT ....................... ........... ix

CHAPTER

1 INTRODUCTION ........................... 1

2 NONCOMMUTATIVE PERTURBATIVE DYNAMICS ........... 8

2.1 Noncommutative Perturbation Theory ...... ......... 8
2.2 Noncommutative 04 Theory ....... ........ ... 10
2.3 Renormalization in Wess-Zumino Model ............. 14

3 DEFORMED SUPERPOINCARE ALGEBRA .............. 19

3.1 Unitary Representations of SuperPoincar6 Algebra ........ 19
3.2 Deformed SuperPoincar6 Algebra ................. 25
3.2.1 Notations and Identities ......... ........... 25
3.2.2 K4 Theory ........ ........... ...... 26
3.2.3 Wess-Zumino Model .................. ..... 29
3.3 Discussions .................. ........... 33

4 QUANTIZATION OF NONCOMMUTATIVE SOLITONS . ... 34

4.1 Introduction .................. ........... .. 34
4.2 Noncommutative Solitons and D-branes . . ..... 35
4.2.1 Noncommutative Solitons in Scalar Field Theory ..... 35
4.2.2 Noncommutative Solitons in Gauge Theory . ... 37
4.3 Classical Noncommutative Q-ball Solution . . ..... 40
4.3.1 Hamiltonian and Equation of Motion . . 41
4.3.2 Q-ball Solutions .................. ..... 43
4.3.3 Virial Relation .................. ..... .. 46
4.4 Quantization of Noncommutative Q-ball . . ...... 47
4.4.1 Canonical Quantization ............ .. .. .. 47
4.4.2 Energy Corrections at Very Small 0 . . ..... 51
4.5 Finite 0 and Noncommutative GMS Solitons . . 58
4.6 Conclusion and Discussion ........... ...... 60









5 SOLUTION OF KOSTANT EQUATION ....... ......... 62

5.1 Euler Triplet for SU(3)/SU(2) x U(1) ........ ........ 63
5.1.1 The N = 2 Hypermultiplet in 4 Dimensions . . 63
5.1.2 Coset Construction .................. ..... 65
5.1.3 Grassmann Numbers and Dirac Matrices . ... 66
5.1.4 Solutions of Kostant's Equation . . ..... 68
5.2 Supergravity in Eleven Dimensions ............... .. 70
5.2.1 Superalgebra .......... . . .... 70
5.2.2 Representations of Grassman Variables . . ... 73
5.2.3 F4/SO(9) Oscillator and Differential form Representations 78
5.2.4 Solution of Kostant Equation in F4/SO(9) . ... 81

6 SUMMARY .................. ............... .. 89

APPENDIX COMPUTER CODE .................. .. 91

REFERENCES ................... ............. 107

BIOGRAPHICAL SKETCH ............. . . .. 112















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

NONCOMMUTATIVE SCALAR FIELD THEORIES,
SOLITONS AND SUPERALGEBRA

By

Xiaozhen Xiong

December 2002

C'!I i': Pierre Ramond
Major Department: Physics

This dissertation presents perturbative and nonperturbative aspects of

noncommutative (NC) field theories, as well as superalgebras in NC field theory

and higher dimensional theories. In particular, the perturbative structures of

the NC Wess-Zumino model are investigated in detail, as well as the deformed

superalgebra relations of the model. NC solitons in scalar field theory are quantized

and quantum corrections to the energy are calculated, where UV-IR divergences

are found similar to those in the perturbative theory. Kostant equations in higher

dimensions are constructed with differential form representations, in which the

solutions are also expressed.















CHAPTER 1
INTRODUCTION

This dissertation is based on the following papers on noncommutative(NC)

field theory and superalgebra [1-4].

The dissertation research focuses on the quantum behavior of NC field theory,

including renormalization of the perturbative and nonperturbative structures in the

theory. The chapters are organized as follows.

Following an introduction to NC geometry as certain limit of string theory in

the beginning chapter, chapter 2 and 3 cover the work on NC perturbative field

theory [1]. In chapter 2, we discuss the perturbative dynamics of NC field theory.

Renormalization of the Wess-Zumino model is studied in detail. In chapter 3, a

slightly digressed topic, the deformed superPoincar4 algebra in NC field theory,

is discussed. The representation of the conserved generators is also useful for the

soliton theory later.

There are comprehensive discussions in literature about NC solitons and their

interpretations as D-branes in NC scalar or gauge field theories [5, 6]. However,

quantization of NC solitons has only been discussed in limited places [7, 8], where

only large NC limit or 1/0 corrections are considered. An important feature of

NC field theories, UV/IR mixing, is omitted in the large NC limit. (! Ilpter 4 is

dedicated to quantization of NC solitons in the small enough 0 limit. In particular

quantization of NC Q-ball solitons is discussed in detail. Quantum corrections

to the soliton energy are then calculated and UV/IR mixing structures similar to

those in perturbation theory are recovered. Energy of NC GMS solitons is found to

be UV renormalizable at one-loop with UV/IR terms included in the corrections.









The structure is i -l-, -I-- 1 as the consequence of interactions between D-branes and

strings, which could be a future direction to pursue.

C'! ipter 5 summarizes the work [4] on 11 dimensional superPoincar4 algebra,

where an alternative representation of the solutions of Kostant equations in

coset space F4/SO(9) is given. The final chapter summarizes the results in this

dissertation and discusses future research directions.

Recently there has been a revival of interest in the study of noncommutative

(NC) geometry, due to the discovery that NC field theory appears to be certain

limit of the effective action of the open string modes living on branes [9, 10]. NC

geometry has been formulated in strict mathematical fashion [11]. The idea can

be captured as follows: On commutative manifold M there exists an algebra

A = C"(M) of commutative smooth functions, with the product being function

multiplication. NC algebra is a deformation of commutative algebra with deformed

product, or more specifically, star product for the concerns of this dissertation,


f *g(x) exp(' )f(x)g(y) (1.1)
2 a&x qai

where 0' = -0j is a non-degenerate constant antisymmetric matrix. NC geometry

is then defined in terms of NC generalizations of the algebraic constructs defined in

the ordinary commutative geometry.

The motivation for studying NC geometry has been manyfold. The idea can

come from the fundamental principle of quantum mechanics, where the phase

variables, position and momentum do not commute. One can just conjecture that

the positions themselves might not commute, leading to


[x xj]= .. (1.2)

where 0ij is an antisymmetric parameter measuring noncommutativity.









Quantum field theory can be well formulated with NC geometry concept,

called NC field theories. One simple reason to investigate NC field theory is that

noncommutativity would introduce phase factors that could better regularize

ultraviolet (UV) divergence present in ordinary field theories. However, as we will

see, often part of UV divergence associated with planar diagrams are still present

and other UV divergences associated with nonplanar diagrams become UV/IR

divergence, which still needs further interpretation.

Another motivation is from the uncertainty principle in quantum gravity,

where position is not expected to be measured accurately at the Planck scale.

People also believe quantum gravity should be nonlocal in general. One expects

by investigating NC field theory as nonlocal theory that a better understanding of

nonlocality can be achieved conceptually and practically.

String theory actually provides stronger motivation for studying NC field

theory. Yang-Mills theory is proved to arise in a natural limit in the context of

the matrix model of M-theory [9, 12, 13], with the noncommutativity arising from

the expectation value of a background field. NC geometry has also been used as

a framework for open string field theory [14]. Later Seiberg and Witten studied

open strings in the presence of a constant Neveu-Schwarz B field nonzero on the

Dp-brane [10]. In the zero slope limit (a' -- 0), NC geometry arises as a limit of

string theory. The effective action of the open string modes on the brane becomes

NC field theory due to the presence of B field. The same paper actually shows the

equivalence between NC Yang-Mills theory and ordinary Yang-Mills theory.

NC field theory has also appeared naturally in condensed matter theory. A

simple example has been shown [15] in which noncommutativity arises when the

theory of electrons moving in a magnetic field is kept in the lowest Landau level

in certain limit. The idea is generalized in the theory of the quantum Hall effect

[16]. Basically an observable algebra, which is well defined in periodic case, can be









generalized to non-periodical background (presence of magnetic field) and actually

becomes a NC manifold called NC Brillouin zone.

Despite the reasons above, experimental support for NC field theory as

realistic low energy theory is limited, due to nonlocality and violation of Lorentz

symmetry introduced by uncertainty relations between the coordinates. An upper

bound of NC parameter, (10Tev)-2, is obtained in the Lorentz violating extension

of standard model [17].

In this dissertation we mainly consider NC field theory from string theory

perspective. In the following we briefly illustrate the idea [10, 15] that NC Yang-

Mills theory arises from certain limit of open string theory.

The worldsheet action for an open string with nonzero B field on Dp-brane

boundary in the Euclidean signature is

S = gij iaxJ Bijxitx (1.3)


where at is a tangential derivative along the worldsheet boundary a0. For xi along

the brane, we have the equation of motion at the boundary zE,


gijOnxj + 27ia'Bijatxj la = 0 (1.4)


The above boundary conditions actually interpolate from Neumann boundary

conditions (B = 0) to Dirichlet boundary conditions (B oc or gij 0). With a

special boundary condition when the world sheet is a disc conformally mapped to

upper half plane, the propagator becomes [18-20]

1 z- '
(x(z)x (z)) = -a'[g' In Iz-z' -g' In z-z' +G' In Iz-z' + In Z- +D'
27aT z Z'
(1.5)









where


( 1 Y 1 1
g
g + 2a'B s g + 2a'B g 2-a'B

ij (27a')2(Bg-1B) ,
(^r 1 2" 1 1
+a'B A (2a )2 2B 'B
G + 2?7a'B A g + 27a'B g 27a'B


(1.6)

(1.7)

(1.8)


The propagator of open string vertex operators inserted on the boundary of E is


-a'G' ln(r r') + e(T T ') ,
2


(1.9)


where G" the coefficient that determines the anomalous dimensions of open string

vertex operators, is referred to as open string metric.

To focus on the low energy behavior while decoupling the string behavior, take

the zero slope limit (a' -+ 0) of the open string system,


S~ e1
Oa C2 -- 0


gij ~ C -- 0 ,


(1.10)


where i,j refer to the directions along the brane. Then G and 0 become


1 1 I 1 ( j )

g'
Gij = g(22a')2 -18 g i




{ g(27a )2(B-1B
[9ij


for i,j along the brane

otherwise


for i, j along the brane

otherwise


(1.11)


(1.12)


(1.13)


S (1)3 for i,j along the brane

0 otherwise

In this limit, the action has only the topological term,


BijxOtxJ ,
2 ,


(1.14)


(xT(r>x (''))









and the propagator (1.9) becomes


(xt(r)x(0r')) = O'" '). (1.15)
2

Interpreting r as time, NC geometry arises by evaluating the commutator,


[x(7) x'()] T(x(Tr)x(T-) x)(T) x(T+r)) .- ,"' (1.16)

With the above equation, one can then argue that for general operator products

O(Tr)O'(T' the leading terms would be independent of r 7' for r r' and

would have to give star products (1.1) of operators, because of the associativity and

translation invariance. Explicitly, normal ordered operators satisfy

Seipxi(r) eiqxiz(0) : e-~Piqjc() eipx(r)+iqx(0) (1.17)


or more generally,


: f(x(r)) :: (0)) : eO2t x O)f(x(T))g(x(0)) (1.18)

where the right hand side is exactly the star product (1.1) of the functions on the

NC space.

Through the general procedure for reduction of open string field theory with

nonzero B field along the brane [10], taking a' -- 0 and keeping G and the effective

open string coupling GC fixed, it is -ii-.-. -1. 1 that the effective action is gauge

invariant NC Yang-Mills theory with field F ,

(F) = det(G + 27a'(F + 4)) (1.19)

where c = Tp/g8 is independent of gs and Tp is the Dp-brane tension for B = 0. In

this form 0 appears only in the star product affected only by B, and certain degrees









of freedom exist in the choice of the parameter 4, which is given by


1
G + 27ra'4


0 1
27a' g + 2ra'B


To determine the effective open string coupling Gs, take F


0 to find the constant


term,


[(F = 0)= -c/det(G +2-a'o) .
z;(P o G,


Also for the Dirac-Born-Infeld Lagrangian (see [21] for a review) for slowing varying

fields,


take F = 0,


LCDBI = cdet(g+2 a'(F + B)),
9s


C(F =0) C-V/det(g T+2WB) .
9s


The equivalence of the above two terms gives

S g( det(G + 27ao)
Cdet(g + 27a'B)


(1.20)


(1.21)


(1.22)


(1.23)


(1.24)















CHAPTER 2
NONCOMMUTATIVE PERTURBATIVE DYNAMICS

NC field theory, after quantization, shows different ultraviolet structures

from the ordinary field theory [22]. Basically noncommutativity introduces phase

factors in the vertices, which in the loop integration become convergent factors

that regularize the UV divergence. However, some UV divergences are still left,

and additional UV/IR divergences are introduced. The intriguing UV/IR mixing

terms can be reproduced by integrating out some new light degrees of freedom with

special propagators and interactions. These new light degrees of freedom can be

interpreted as closed string modes with channel duality [23]. Even for field theory

concern, renormalization of NC field theory needs to be reexamined because of the

change in the divergence structure.

In this chapter following a basic introduction to the NC field theory, perturba-

tion dynamics in NC 04 theory [22], as well as the implications from string theory,

are reviewed. Then we discuss renormalization of supersymmetric NC Wess-Zumino

model.

The commutative Wess-Zumino model is the simplest supersymmetrical field

theory model in (3 + 1) dimension. It includes a scalar and a fermion field with

supersymmetry between them. Because of the supersymmetry, cancelation of the

divergence occurs generally. The only mass renormalization is due to wave function

renormalization, and the vertex and mass corrections are absent [24].

2.1 Noncommutative Perturbation Theory

As explained in the introduction chapter, a specific NC algebra can be defined

by the star product of the commutative functions, Eqn. (1.1). The underlining








noncommutative space has phase space quantization structure,

[. ,x.3] .= .,' (2.1)

where i,j is assumed to label the space dimensions only in this dissertation.
Noncommutativity associated with time coordinate brings the problems of causality
and unitarity when the theory is canonically quantized [25]. Weyl proposed Eqn.
(2.1) as Lie algebra of a group with group elements,

U(p) = exp(ipx) (2.2)

In the function representation of the NC algebra with Weyl ordering,

f () = (2r)" dxf = x)e- ) jp)U(p) (2.3)

where n is the number of the space dimensions, the operator products

((2)" (2 )" (q) (p)U(q)

S(2)" (2) (p q)g(q)e-' U(p) (2.4)
NC star product (1.1) is just the function representation of the above operator
products, for
dnp dnq 'fp e (2.5)
f(x g() i ( (2r)n (2)n e- '' ) f(p)(g (2.5)

The Weyl representation is initially used in phase space quantization [26]. NC
field theory is first proposed by Filk [27], replacing the ordinary products of the
fields by NC star products. The propagator remains the same since

I f(x)g(x)= 0. (2.6)

The interaction vertices now depend on the external momentum through phase fac-
tors, which is induced by the star products. The phase factors, while independent









of the overall permutation of the momentum, distinguish the Feynman diagrams to

be planar and nonplanar ones. The phase factors for planar diagram depend only

on external moment, and do not affect the UV divergence in the loop integration.

But for nonplanar diagram, the dependence of the internal moment by the phase

factor introduces regularization in the momentum integral, and UV divergence is

generally converted into UV/IR divergence. Renormalization of the theory needs

to be reexamined case by case. It is interesting that renormalization of the NC

field theories differs significantly from their commutative analog. For example,

NC QED, as a simple extension of NC U(1) YM theory, is renormalizable at one

loop due to Slavnov-Taylor identity for SU(2) like symmetry, but the 3 functions

include contributions from electrons in U(1) facet [28]. General consideration of

convergence theorem and renormalization in NC field theory has been discussed

[29, 30]. The rest of the chapter discusses renormalization of NC K4 theory and

Wess-Zumino model in detail.

2.2 Noncommutative 04 Theory

The NC K4 theory in the four-dimensional space-time, is described by

1 1 A
L = -2 0I^ -m2I _A I ". (2.7)
2 2 4!

It is well known [10, 22, 27] that under the integration the star product of the

fields does not affect the quadratic parts of the Lagrangians, whereas it makes the

interaction Lagrangian become nonlocal. Hence, the Feynman rules in momentum

space of NC field theory are similar to those of commutative theory except that the

vertices of the NC theory are modified by a phase factor. For the Lagrangian (2.7),









the Feynman rule for the deformed vertex is

A(cos -(p x p2 + x p3 2 xp)
3 2
1
+ cos -(P x p2 + P x P3 P2 x P3)
2
1
+ cos -(pi X p2 p X 3 2 X p3)), (2.8)
2

where pi's, i = 1... 4, are moment coming out of the vertex and pi x pj

PipO/"pj,. When O/ -- 0, the deformed vertex becomes the non-deformed one.
By using the above vertex, one yields a wave function renormalization of the scalar

field Q at one-loop order that has only one diagram as follows:

F( )) (p2)
A J d4k (2 + cos(p x k))
6 i(27)4 (k2 + m2)

e -1j 9 -eIam-+ )A2a
487r2 0 a2 2
--2 (2 2ln )) 9- (A 2 m21n(/ ) + (2.9)
4872 M 2 967 2 C M 2

The Schwinger parametrization technique to deal with the above integration can

be found in Itzykson and Zuber [31] and Hayakawa [28]. In the second line, the

term is proportional to exp (-ip2/4p), where p = p,9", is due to the nonplanar

contribution and the exp (i/pA2) factor is introduced to regulate the small p

divergence in the planar contribution. Note that the nonplanar contribution is one-

half of the planar one. In the third line, we keep only the divergent terms and the

effective cutoff, A f = 1/(1/A2 + (p2)/4), shows the mixing of UV divergence and

IR singularity [22]. The above integration can also be done by using dimensional

regularization method [32]. In the case that Q is a complex scalar field, there are

two v-,v of ordering the fields Q and Q* in the quartic interaction (Q*0)2. So, the









most general potential of the NC complex scalar field action is

AO ** 0* + BO ** 0.

The potential is invariant under global transformation since the star product

has nothing to do with the constant phase transformation. It was shown by
Aref'eva, Belov and Koshelev [33] that the theory is not generally renormalizable

for arbitrary values of A and B and is renormalizable at one-loop level only when
B = 0 or A = B.

The one loop 1PI quadratic effective action is

S(2) 4 2 + M2 A
1PI d2 96w22(Jp2 +
AM2 1
96 rlnp2 ) + p)(-p) (2.10)

where
n2 AA2 Am2 A2
M+2 2- In( ) + (2.11)
4872 4872 M2
is the renormalized mass.

The appearance of UV/IR mixing terms -i-i- -1; the presence of new degrees

of the freedom. Indeed the correct IR singularity in the effective action can be

systematically reproduced by the introduction of new light degrees of freedom

[22, 23]. A brief review of the idea is as follows.

Consider the modified effective action


S'f (A) ( Se[ + d4p2 1() 1 (P)X( -P
2 p24 -p2 727-

+ I (l P2n( ) ( ) X2(P)X2(-)

+ 1 A(i + M2X2) (2.12)
V967w2 1








Upon integrating out X in the above action, the correct quadratic and logarithmic
divergences can be reproduced, and the UV/IR mixing terms are cancelled by the X
exchange diagrams.
X's have special propagators,


(xI(p)xi(-p))

(X2(p)X2(-p))


4 1
p2 1p2 + I
SIn P2 + )- ln(-p2)
4 A 2 4


(2.13)

(2.14)


A possible interpretation to the presence of those new degrees of freedom X is that
they are actually transverse modes of particles i's which propagate freely in more
dimensions. For example, a particle i propagates in two extra dimensions and
couples linearly to Q on the brane will produce the logarithmic propagator. Define
X(x) = (x, x = 0) and write the action of i with a Lagrange multiplier A(x),

exp d4x(x)( xx = 0)- d4xd 2xr (q2)

J f[dA][d exp (- Jdx (x)xx)+\ iA(x)[x(x)- x,xi o0)}

Jd 4xd2x1j5a02

= [dA][dx]exp(- Jd4[(P)X(p) + i(-p)X(

4 pd2 q -p, -q)(p + 2)(p, q) i- A(-p)(p, q)] (2.15)
2 4


Integrate out first, leaving

p1 f dq
exp p 4 p i)(pP) )+ A(p)-p) + t7A(p)IA(-p) dq2 2
2 j Jt /4p2 +q 2
Then integrate out the Lagrange multiplier A, giving the desired action,

exp (-/ p X-P) (-p)t(p)n I 1/4 A .


). (2.16)


(2.17)









The duality between the high momentum degrees of freedom in Q and propaga-

tion of # in the extra dimensions -,-.- -i; that O's are associated with open strings

modes and X's or K's are associated with closed string modes, since in the string

theory the low energy closed string modes are related to the high energy modes

of the open strings by channel duality. A nonplanar loop diagram is topologically

equivalent to a string diagram in which a number of open strings becomes a closed

string that freely propagates in the bulk and turns back into open strings. Con-

nections between NC field theories and string theories -ii--.- -1. I by those analogies

have not been clearly understood yet.

2.3 Renormalization in Wess-Zumino Model

In this section, we investigate renormalization at one loop in the NCWZ

theory. NCWZ Lagrangian is given by introducing star products in the interaction

terms and permutating those terms to preserve supersymmetry transformations.

Here we follow the conventions by Sohnius [34]. The NCWZ model is described by

the sum of the free off-shell Lagrangian and of the two invariants,


Ltot = Co + Cm + Lg, (2.18)


where

1
Lo -= (/,,AQ^'A + Q,,B^A'B + iTf TI + F2+G2), (2.19)
1
L -n(FA +GB + 2I), (2.20)
2
Lg = -g [A*4_F-B B F+ AB*G+T*(A-75B)*
3


+permutation terms] .


(2.21)









The off-shell Lagrangians Lo, Im and g are separately invariant under the

supersymmetry transformations:


6A = a4, 6B = a5, 6F = i-a OT, 6G = iaM5 O,

-6 = -(F + 5G)a- i (A + 5B)a, (2.22)


where a and a are the global infinitesimal Majorana spinor parameters.

The Feynman rules in momentum space can be extracted out directly from the

Lagrangians (2.18). One gets as follows:

1. Propagators

The propagators of the fields and the mixed fields on the NC space are the

same as those on the commutative one.

2. Deformed vertices

(A A F + permutation terms)

1
-2ig cos(2PA, X PAf).
2 A1).

(B B F + permutation terms)

1
ig cos(2pB x pBf).


(A B G + permutation terms)

1
-2ig cos(-pA X PB).


g- (9 AA T + permutation terms)


-igIcos(pi x pf).


3 (T 7B + permutation terms)


2ig75 cos( pi x pf).
2









The deformed vertices we obtain differ from the non-deformed ones by a factor
cos( pi xpf). By using the above Feynman rules, one can study the renormalization

of the NC Wess-Zumino model. The results are summarized as follows:

1. Wave function renormalization
M, i ..i i, field T

For Majorana field, at one loop there are two diagrams. The sum of

them gives a contribution

Sd4k 1 V
rF(ff) = 8g2 COS(-p x k)
) 8g (27)4 co 2 (k2 -_ 2)((k + p)2 2)
da(- a) -d (m2 2) + e p2
47 2 JoA Jo P C

2 ( )+ln( )) +.... (2.23)

Scalar fields A, B

For each scalar field, at one loop there are five diagrams. The sum of

them gives a contribution

F(AA)(2) p(BB) (2)
f d4k 1 k p
8g2 Cjos 2 x px k)p
Si(2)4 cos (k2 m2)((k + p)2 n2)

-p2 In( )+n( a)) +.... (2.24)

Auxiliary fields F, G

For F field, at one loop there are two diagrams While, for G field,

at one loop there is only one diagram. However, they have the same









contribution

(FF) (p2) p(GG)(p2)
d4k 21 1
-4g2J 2 cos2( p x k)2-
i(2)4 2 (k2 m2)((k + p)2_ m2)
82 2 ff
8- 2 l(L' -)+ln( )+ .... (2.25)

Mixed fields
F(FA)(p2) = (GB)(p2)= 0. (2.26)

Again, all the integration can be done directly by using the Schwinger

parametrization technique [28, 31]. The divergent terms of the wave function

renormalizations of all fields are the same, whereas the finite terms of F(FF)

and F(GG) are different from those of the others. Note that in the NC

Wess-Zumino model the planar and nonplanar contributions have the same

multiplicative factor. Renormalization is cut by half compared to that of the

ordinary case.

2. Mass renormalizations

Since, at one-loop F('1)(/) is proportional to only / and both FFA and
F0B are zero, there is no mass renormalization.

3. Vertex corrections

FA2, FB2, ABG

For each vertex, at one loop there are two diagrams and they add up to

zero. So, there is no correction for each vertex.

"* \1A< .y*75 B-

Similarly, there is no correction for each of these two vertices. Since, at

one loop there are two diagrams and they add up to finite values.

Just as in the )4 theory, the UV/IR mixing also appears in the NCWZ

theory, which is the general consequence of the uncertainty relations among







18

NC coordinates [10]. Renormalization in the NCWZ theory is very similar to

the commutative one. Compared with the ordinary Wess-Zumino theory, the

counter term for the wave function renormalization reduces by one-half, but the

cancellations, in particular the absence of mass and vertex corrections, persist

due to supergauge invariance. The renormalization of the wave function of the

commutative theory can be recovered by setting 0P" equal to zero. Supergauge

invariance sustains generally in NC field theories. In the next chapter we will

study superPoincar4 algebra in NC field theories and again verify the supergauge

invariance from an algebraic point of view.














CHAPTER 3
DEFORMED SUPERPOINCARE ALGEBRA

In this chapter following an introduction on classification of representations

of Poincar4 and superPoincar4 algebra, algebra of NCFT's are studied. Conserved

currents are derived by Noether's procedure, then a representation of the genera-

tors of deformed Poincar4 or superPoincar4 algebra is -ir -.-. -1 I. and commutation

relations are calculated explicitly. NC 04 and NCWZ theory are studied as the

examples.

3.1 Unitary Representations of SuperPoincar4 Algebra

Poincar4 invariance is considered as a fundamental property of modern theory

since the discovery of special relativity. In recent decades superPoincar4 invariance,

as an enlarged invariance, is also considered as a property of fundamental theory

for theoretical concerns (see [35] for a review), although there is no experimental

evidence clearly supporting the conjecture yet. The importance of the unitary rep-

resentations of the Poincar4 algebra and their classification is originally recognized

by Wigner [36]. The following is a review about the theory of unitary Poincar4 and

superPoincar4 representation in 3 + 1 dimensional space.

Poincar4 algebra includes Lorentz generators .1,,.. and translation generators

P,, satisfying commutation relations,

[P, P"] 0 ,

[M^, P"] i(r"P" P') ,

[Mv, _I I j(qTl. + rMlav + ,i M"p + ) ,









where l, = (-1,1,1, 1) and p = 0, 1, 2, 3. The representations are characterized by

the values of the Casimir operators, P,PP, and the squares of the Pauli-Lubanski

forms built out of the Levi-Civit4 symbols. In d = 4 space-time dimensions, the

Pauli-Lubanski vector is
1
W" P"l 3.,, P, (3.1)
2
Modern physics theory, built in the framework of quantum mechanics, assumes

existence of a Hilbert space in which physical particles are described by quantum

states. Poincar4 invariance of the theory implies that elementary free particles can

be classified in unitary representations of the Poincare group.

Elements of the Poincar4 group satisfy,


T(a)T(b) T(a + b) (3.2)

d(A)T(a) = T(Aa)d(A) (3.3)

d(A)d(I) = d(AI) (3.4)

Here T(a) represents the abelian translational group, and d(A) represents the

Lorentz group, where indicates d(A) is a double valued representation.

Consider the wave function Q(p, i) parameterized by the momentum variables

p, and the variable labels an auxiliary space, so that translation elements are


T(a)O(p, ) =e~" (p, ) (3.5)

Now define the operators


P(A)Q(p, ) Q0(A-lp, ) (3.6)

It is easy to show


P(A)T(a) = T(Aa)p(A) .


(3.7)









Considering both the above equation and Eqn. (3.3), an operator defined by


Q(A) = d(A)P(A)-1 (3.8)

can be shown to act on the parameter alone, which can depend, however, on p,


Q(A) (p, () Q(p, A) ...(p, I) (3.9)


Q(p, A) is actually an unitary representation of the little group of the Lorentz

group d(A). It suffices to consider Q(pfix, A) as a representation of Apfix = Pfi for

a particular vector Pfix.

Representation of Poincar6 algebra is also characterized by the particular value

of the casimir operators, PPP and WW, In Eqn. (3.1), MP", acting on the wave

function 0(p, ), is generically represented as

MP = -i (p -- p- V + (3.10)


where SP" is associated with (. Three classes of irreducible physical representations

are found,

SPPP" = m2 > 0

C'!,-.... P-, (m, 0, 0, 0), then


WW = m2(S')2 M2s(s + ) (3.11)

where Sj(i,j = 1, 2, 3) is an irreducible representation of SO(3), labeled by

spin s. It is well known that s takes the value of zero or positive integer or

half integer. This class of representation describes massive particles with spin

S.

P/P = 0, W/WP" 0

This class of representation is just the massless limit (m -+ 0) of the massive

representation and describes massless particles. ('! .... Pf, = (E, 0, 0, E), or









in light cone

P+ 2E, P- = P = P2 0 (3.12)

Since

WW = (Wi)2 + (W2)2 0 (3.13)

W1 = W2 0 or S-1 = S-2 = 0. Also it is easy to show that W- = 0

and W+ = P+S12. Therefore W" = S12p, where S12 helicity operator,

is the generator of the little group U(1). Single or double valueness of the

Lorentz group demand the value of the helicity generator to be half integer

or integer. One particular variant of this class of representation, obtained by

taking infinite momentum limit of massive representations [37], can be used

to represent irreducible degrees of freedom of strings. Its supersymmetric

generalization will be able to represent superstrings of various flavors.

PP 0, W=,W B= 2

Again choose P, (E, 00,0E). W,W" = 2E[(S-1)2 + (S-2)2] and

W- = 0, but W1, W2 and W+ are nonzero. The little group which leaves P"

invariant is SE(2) with generators S-1, S-2 and S12. Only E, the length of

W", is needed to label the representation. Two types of representation, single

valued or double valued, belong to this class. This class of representation is

originally called continuous spin representation by Wigner, due to the reason

that for each representation, the states can be labeled by the value of usual

helicity generator S12 which is all the integers or half integers.

The above representations should be able to represent all physical particles in

3 + 1 dimension. There are good reasons to disregard higher spin representations

[38, 39] in the first and second class. Naive quantization of the continuous spin

representation leads to nonlocality or breakdown of causality [40,41].









SuperPoincar4 group is the extension of Poincare group, including supercharge

QA, which satisfies the commutation relations,

[QA P" 0 (3.14)
1
[M1" QA] -2 (ZQ)A (3.15)

{QA Q} = (PP70)AB (3.16)

where A, B = 1, 2, i, 2 are spinor indices. In the above,


2

where the 7 matrices satisfy the anticommutation relation,


{Q7 7'y} 2TV .

Massless supermultiplets are particularly important, which yield the basic

physical spectra of the supersymmetric models. Spectra of Wess-Zumino model

can be deduced in the following way [34]. The same arguments (Eqn. (3.5) and

below) can be applied to superPoincar6 group. The little group Q(A) in this case

includes supercharge generators. Since [W" QA] / 0, WWM is not casimir

operator any more. For massless supermultiplet, still choose P = (E, 0, 0, E),

then use representation independent light cone projectors to split the supercharge

generators,

Q Q + Q_ Q PQ. (3.17)

Eqn. (3.16), in the Weyl representation, shows that commutators between super-

charge generators generally vanish, except

{Q+a Qtb} 2Eb (3.18)

where a, b = 1, 2. General arguments can show that Q+i and Q+2 actually

belong to two disconnected algebras. Therefore, considering just the minimal









supersymmetric model, the little group contains only the generators, Q+1, Qt

and S12, where S12 comes from the Poincar4 algebra. These generators obey the

commutation relations,


Q+i Qt} 2E (3.19)
1
{S12 Q+} -Q+ (3.20)

Previously (Eqn. (3.12) and below) it is shown that the massless representation of

Poincar4 group includes states which are labeled by IE, A), where E is proportional

to P+ and A is the eigenvalue of the helicity generator S12. Eqn. (3.20) indicates

that the supercharge Q+I or Qt are the lowering or raising operators for the

helicity. Thus the state |E, A) can be defined to be the lowest helicity state since

Q2 = 0. Therefore the minimum supermultiplet contains two states |E, A) and
Qt I\E, A). The supersymmetric Wess-Zumino model describes the supermultiplet

with helicity A = 0.

Other supermultiplets with more spinor supercharges or with central charges

in various dimensions have been constructed explicitly [34]. In higher (d + 1)-

dimension (d > 3), little group of superPoincar4 group is enlarged to contain

SO(d 1) and corresponding spinor supercharge generators. Massless super-

multiplets in (9 + 1)- dimension correspond to various superstring theories. The

little group SO(8) has triality symmetry which leads to marvelous cancellations in

quantum perturbation calculations ([42]). More recently, M-theory in (10 + 1)-

dimension emerges as unification of superstring theories, whose low energy limit is
-Ii.., -1. .1 to be (10 + 1)- dimension supergravity theory. The little group SO(9)

is the maximum subgroup of the exceptional group F4. As a result, Euler triplets

arises as solutions of Kostant equations [43]. The lowest level triplet is supermul-

tiplet and corresponds to (10 + 1)-dimension supergravity theory. The higher

level multiplets have accidental supersymmetry and maybe able to describe the









zero-tension limits of string theory [4]. C'!i pter 5 shows the construction of these

solutions at all levels. Continuous spin representation in (d + 1) dimension is the

representation of little group SE(d 1). Upon supersymmetrization, it has one

to one correspondence with ordinary massless supermultiplet in d dimension [3].

The paper also shows that if light cone translations are represented by Grassman

variables, nilpotent continuous spin representations lead to supermultiplets with

central charges. Such analogy is already ii-.- -1. I1 in previous classification of

(3 + 1)-dimension representation, where both the massless supermultiplet and the

continuous spin representation contain states with connected helicities.

NC field theory, as a low energy limit of string theory, does not have Lorentz

symmetry. In the following sections deformed superPoincard algebra of NC field

theories are studied in an intuitive way, which is expected to gain some basic

understanding of underlying algebra and representation of NC field theories.

3.2 Deformed SuperPoincar6 Algebra

3.2.1 Notations and Identities

To facilitate the calculations involving NC fields star product, we introduce the

following notations and list the useful identities.

Define an operator A, which acts nontrivially on a scalar pair-product (f, g)

as,


A(f, g) = 9,fa'g,

A2(f, 9) 0/, f 6 9,



A(f, g) a 9a ... f 0" ... Pg, (3.21)
nT T


where 0" = -1"A,.








With our definition, a star product between two scalar fields A and B can be
written as

A*B = e (A,B)

S(1++ + -++. (A,B)
2! 3!
AB + a, (E(A)(A, B)) (3.22)

where the operator E(A) is

t 1 An
E(A) (n + ) (3.23)
n=0
By using the above notations, we obtain some useful identities:
1. B A AB (E(-A)(A, B)) .
2. [A, ],= A B A= 2(, ( (A, B)).
3. {A, B}, = A B + B A 2AB + 2A (cosh()-(A, 6 )) .
4. (x,A) B = x,(A B) + A pB.
5. B (xA) xp,(B A) 3,,B A.

6. [(xpA), B], x,[A, B], + {A, aB},.
7. [B, (xpA)], xp[B, A], {A, 8pB},.
8. {(xpA), B}.* xp{A, B},+ [A, dB] .
We assume 0i = 0 from now on for causality and unitarity reasons [25]. The
immediate consequence is that noncommutativity will not introduce higher order
time derivatives of the fields in Lagrangian.
3.2.2 #4 Theory
Now let us calculate the Noether currents of the NC #4 theory following
standard technique [44] Varying the Lagrangian (2.7), and using the above









identities and also the equation of motion, one gets

(//1 A sinh(A)
Sd4x d4( ~x 2 + 2 + 12 A
(3.24)

Under an infinitesimal translation, 6x" = gpv"c, 60 = -C"~v one yields the

energy-momentum tensor,
T 1 A sinh(A)
TP= -{a9, aV+} g L+ A( i [4, 4 9 ,)]. (3.25)
2 12 A

As explicitly seen, the energy-momentum tensor TP" is conserved since its diver-

gence is zero.

Under the infinitesimal Lorentz transformation, 6x" = c "x, = ~c(xpg -

xg"), 00@ = e"C(Xp,4@ xZp4), where eP" is an anti-symmetric second rank

tensor, one obtains a three-index current

1 A
j =- TP'x,+ [ ta8 i 1+-(sinh(A)/A)'(,(4(K ) 61[4,8,0],)
A sinh(A)
A sinh(A (4 i'l ., a,] + a"{a,], a,},) -(p a), (3.26)
12 A

where (sinh(A)/A)' (A cosh(A) sinh(A))/A2. The divergence of the three-

index current is not equal to zero due to the presence of the terms proportional

to the non-commutativity 0"". However, note that the Noether currents of the

commutative scalar field theory can be obtained by setting "" equal to zero.

In the case of the commutative 44 theory, one yields the momentum and

Hamiltonian generators from the energy-momentum tensor, and the angular mo-

mentum and boost generators from the three-index current [44]. These generators

form the Poincar4 algebra. For the NC #4 theory, one obtains its generators








analogous to those of the commutative one,


Pi 3= idp

jd


J 3X + (j(2 + rn2+2) + X 4 ,
( \2' 4! 1


(3.27)

(3.28)

(3.29)

(3.30)


Sd3x(xOpi Xip 0),

Sd3X(xipj -_ X i).


The surface terms of 1/" and IM [ are dropped out. These generators generate the
translational, rotational and boost transformations on K.
By using the quantization condition, [0)(j), (y)] = i63(x y), one can easily
obtain the following equal-time commutation relations:


[P", P"]



LI_ Pk]

'I'/", P]l


i(IilMjk + jkj

i(Ijkpi ikpj

ij P0.


Iik lMik),


(3.31)

(3.32)

(3.33)

(3.34)


The above commutation relations of the NC )4 theory are the same as those
of the commutative one. In particular, (3.31) verifies that the NC j4 Lagrangian
has translational invariance and the translation generator P" is conserved. But the
following commutation relations have some additional terms proportional to 0"",









due to the symmetry-breaking term (*


-io 00Pi i d3 [42

- d3 Xi u )3 + (i j),

00 /' jd (+X i3 [)2 ,ai]

i( j ;' _/ 1") I' d3X i

x ([9k^, aI j 3]. + *2 ,k. 6ja 6j3.

+ k 4j*2 6aj 9 k (*2 (j


(3.36)

({ i)) (3.37)


2 O *

k)) .


(3.38)


The Eqn. (3.35) and (3.36) explicitly show that the Lorentz generators are
not conserved in the theory, and all the deformation terms are directly proportional
to 09".
3.2.3 Wess-Zumino Model

For the NCWZ model, one start from an on-shell Lagrangian analogous to the
commutative one [34],

L (= ((AOaA m2A2) + t(Bd"B m2B2) + mw )
2 2 2
-mgA(A*2 + B2) mgB(A B + B A)


1 1
-g(AT. T BT. *75T) -9 (A iB*(A + iB).2 (3.39)

2 2

Smg(,, + ,-. g( +- g2 .2 .2. (3.40)

where = A iB, -_ A + iB, and b, b are the Weyl components of the Majorana
field T, following the notations and conventions by Bailin and Love [45].
Following the similar procedure as done in the #4 theory, the variation of the
Lagrangian under the infinitesimal Poincar6 and supergauge transformations yields


L/ PO]

I Po]



I/" Mjk]









the generators as,

P o3 i 3_ + Oii
P J d^x7 J d^x + + i2uadi+ ), (3.41)

PO 3 0
d J + 0 2 0


1 1

/" = d3 (xi


/ / d3X (xiPJ j) (3.44)

xQ = J d3x -( 2 0o0 +o. i im oo0 i + jgo*2,0 (3.45)

XQ X d3x ( 2 oia9470 + onab0 + g*2 0(XQ), (3.46)

where X is an arbitrary Majorana spinor parameter.
In the case of the commutative Wess-Zumino model, the analogs of the
above generators are those of the Poincar6 algebra and supercharge, which form

the N = 1 super-Poincar6 algebra. With the representations obtained here in

the NCWZ model, one can calculate the commutation relations between those
generators,

[P" PV] = 0, (3.47)

1_/, 1/=" i(i, "'Mjk +jk Tik '- ~IMi) (3.48)

L I -,Pk j (TjkPi Tikpj), (3.49)

Lr ,Pj] =- ijpo, (3.50)

The above commutation relations are exactly the same as those obtained in

the NC )4 theory, which .-.i--- -r the generality of such relations for all NCFT's.
In particular, (3.47) verifies the translational invariance of the theory. Equation








(3.50) is a little surprising. The calculation of it in any way involves the NC
interaction terms. Nevertheless it is true for both NCFT's.
Other commutation relations are

[XQ, (Q] [xQ, (Q] 0, (3.51)

[XQ, (Q] 2X"(P,, (3.52)

[P", XQ 0, (3.53)

[- ',XQ] -ix1"Q, (3.54)

[\ ', Q] = -ixa1"Q. (3.55)

All the above relations are exactly the same as those of the commutative Wess-
Zumino model. In particular, one finds the supercharge generators, Q and Q,
and the translation generators PT's form a close algebra, and the supercharge
generators are conserved.
The rest commutation relations have additional terms proportional to 0",
including the similar ones as appears in the NC K4 theory,

L1/" p0o] = -ioopi

Sd3x ( ng([, 6o] + [0, &ai]) +mng([a, t,'].,O+ [, ,' 1]..-T)

-2ig([0, a V].,TO, + [Q, ',' ]..T01, ,',) + g2([*2, 6l + [2,aI )

+g2([, ].{ }-{ ,].), (3.56)




1dP2
Sdx (-mg([aiQ,aJQ.Q+ [YQ&,Y].) + ig([9YQ,tW ]. + [at,a ,].








I" ,'] = -i +jd3x (mgx i(pOv[,J, +a [ '], T..To)

~gx([Q, a ,io + [Q, 0],) + 2igx ([Q, o, ].. ,1 + [, j,].To ,a.)

g2 ([*2,j + '.', ) + g2{x }{J } x(i )r) ,( (3.58)


_p" ,Mjk] I(Tij 1" -_ ikM/')

f3 dx mgx(0akQ, 6i] + ,ak,, 60]) +igxij 0 .p + [9',, p' .)

+jg2x k[9kQ, j + [a J]) ( k)) (3.59)

and also the transformations of the supercharge generators under the Lorentz
boosts,

Li", xQ] = -ix Q+ dX (gt 0 Q + gOi3Qii1
-21ig* + irng[, + f+f ig2[ 2 ,Q2 i) (3.60)


I" xQ] = -ix70iQ x (g 0]QX0 + gi [ i, ai.- .1 V' Q

-2igr Q + im + g[ Q] + i g2 /W [2,]X) (3.61)

To simplify the expression, we reorder the conjugate fields on the right hand
side of the above equations, which induces extra infinite constant terms not
explicitly shown here.
In summary, The commutation relations of the Lorentz rotation and boost
generators generally have additional terms compared with those of the Poincar4 or
super-Poincar4 algebras. Other commutation relations verify certain symmetries
preserved by NCFT's, such as the translational and supergauge invariance. In the
limit of 0[" 0, the Poincar4 or Super-Poincar4 algebra is recovered.









3.3 Discussions

In this chapter we -ii-.- -1 a representation of the translation, Lorentz and su-

percharge generators. The commutation relations of those quantities are calculated

directly based on this representation.

The NCFT has nonlocal interaction terms, which explicitly break the Lorentz

invariance, but still preserve the translational and supergauge invariance. It is

found that in the NCFT the translation and supercharge generators form the

same algebra as in the commutative theory. But, the commutation relations

of the Lorentz generators, or between the Lorentz generators and the transla-

tion or supercharge generators, generally have extra terms proportional to the

non-commutativity 0"". In addition to that, there are also other interesting

commutation relations, such as /I" pj] iriJ P, still hold true in the NC case.

Preservation of supersymmetry algebra '--:-- I a supersymmetry generaliza-

tion of NC geometry, which is not clearly understood yet. In that frame, fermions

can better be defined as supersymmetry partners of bosons, instead of being a

naive generalization of Lorentz algebra representation, since Lorentz invariance

is broken. Indeed supersymmetry algebra has been represented on NC space

and a divergence free supergravity model is expected to be constructed with this

representation [46].

The superPoincard algebra generators are "fundamental qu ,il ii represen-

tation of which can be used to construct a theory of a dynamical system [47]. It

remains a question whether the representation we obtained, Eqn. (3.41) to (3.46),

could be used to construct a new theory on NC space consistent with the NC

theory we start with.















CHAPTER 4
QUANTIZATION OF NONCOMMUTATIVE SOLITONS

4.1 Introduction

Solitons, known as "extended obj, I exist in field theories with nonlinear

interactions [48]. They are defined to be classical solutions to the equation of

motion of a local field theory with the property that the energy density is, at all

times, localized within a given region of space. A wave packet, which spreads as

time evolves, in general is not this type. These objects have special features since

at classical level they are already particle-like, and yet they possess an extended

structure. Such objects have been studied extensively around mid-70's to early

80's of the last century. It was speculated hadrons may simply be described as

quantized states of the extend objects, where the possibility of confining quarks

exists, since even classically quarks are trapped to some extent [49, 50].

Solitons discussed in this chapter are also localized extended objects, but

exist in NC field theories, called NC solitons. NC solitons was first discovered by

Gopakumar, Minwalla and Strominger (GMS) [51] in NC scalar field theory with

the potential having a local minimum besides a global minimum at the origin.

Since then, the soliton solutions have been explicitly constructed in different NC

gauge theories with or without matter [5, 6]. Monopole-like solutions in (3 + 1)

dimension with a string attached turns out to be realization of D3-D1 system,

where D1 string ends on the D3 brane [52]. NC solitons can also be interpreted as

lower dimensional D-branes in string field theory [53, 54].

As explained in chapter 1, perturbative dynamics of NC field theories reveals

a very intriguing structure, i. e. UV/IR mixing, which -ii.:: -1- their analogy to









string theories. Thus it becomes interesting to investigate the quantum behavior of

nonperturbative structures, NC solitons, in NC field theory.

Next section is a review of the theory of NC solitons and their interpretation

as D-branes, and also quantum theory of ordinary commutative solitons. In

section three a new type of the NC soliton solutions, NC Q-balls, is investigated

first and its difference from NC GMS solitons, existence at arbitrary small 8, is

emphasized. Next canonical quantization of NC Q-balls at very small theta is

discussed in detail. Quantum correction to the energy is calculated with phase shift

summation method. The same method is further generalized to NC GMS soliton

for smooth enough solutions. In both cases UV/IR mixing terms are present in the

energy correction of NC solitons. The future direction could be quantization of NC

solitons in gauge theories, which would increase the understanding of the UV/IR

mixing terms.

4.2 Noncommutative Solitons and D-branes

4.2.1 Noncommutative Solitons in Scalar Field Theory

Classical NC solitons were first discovered in the scalar field theory with more

than one space dimension [22]. Start with an action of scalar field theory in two

NC space dimensions,

S J d3x 2 V(0) (4.1)

where the fields are multiplied by NC star products (1.1) implicit here. In chapter

2 we discussed operator and function representations of NC algebra, and the iso-

morphism (Weyl transform) between them. The two descriptions are completely

equivalent. It turns out classical solutions of NC field theory can be easily repre-

sented in operator formalism. Specifically, in two NC space dimensions, NC field

theory are isomorphic to an algebra of operators defined on a one particle Hilbert

space,


[x1 ,92] = i(.


(4.2)








Also star product of fields is just product of Weyl transform of the fields (1.1 and
2.5),

f(x) g(x) f(O)g9() (4.3)

where f(x) and f(x) or g(x) and g(x) are Weyl transforms of each other. The
integration over the space is actually equivalent of operator trace,

Sd2x ~+ 27Tr (4.4)

The derivative is equivalent to

Ci(j.) -,[ .)]. (4.5)

Define creation and annihilation operators

1 1
a = (- + ix2), t 1(x1 i2) (4.6)
\20 V20

with [a, at] = 1 as usual. The field 4(x) or 4(x) can be expanded in the or-
thonormal basis f,,(x) or In) (m. A systematic way of calculating f,,(x) can be
obtained [5, 55]. In particular, Weyl transform of projection operator P, = In)(nl,

f,,(r) are just central functions, and can be expressed in Laguerre polynomials,

f,(r) (-1)'2- L,(2r2) (4.7)

In the operator formalism the action integral (4.1) becomes

S[] = /dt2T0Tr- D+ -([a, ] [at, () (4.8)

The equation of motion is

9o2 + [a, [at, W]] + V'(=) 0 (4.9)

In commutative theory, it is well known that time independent scalar solitons
do not exist in dimension more than one [56]. However, in NC scalar field theory,









such solitons generally exist provided 0 is larger enough. For radial symmetric

solution,

S= nPnp,, = n ,,n)(n (4.10)
n n
Put into Eqn. (4.9),


(n + 1)(A,+1 A,) n(A, A,_1) -V'(A,) (4.11)
2

If

V'(I) = c4(4) ac) ( al) (4.12)

at 0 -- o, the solutions are just


1 = aP, or ai(1 P,) (4.13)


4.2.2 Noncommutative Solitons in Gauge Theory

NC solitons are also widely found in gauge theories [6], and their interpretation

as D-branes is precise [53,54]. In the following we review the theory of solitons

found in gauge theories. Start with the action,


S= 270Tr (- ,F'+ -DOD, -V()) (4.14)


Here

Foi = A aAo (4.15)

and
1
Fi = i[Di, Dj] ([C, C] + 1)e (4.16)

where

Di = a, i[Ai, 0] UDioU (4.17)

is covariant derivative under U(1) transformation,


(4.18)


S-U UOU ,









and C -at + i vA.

Suppose the potential is V(O 0,) which has a local minimum at = and

a local maximum at = 0 and V(0) = 0 Static solutions locally minimize the

Hamiltonian

H = 270Tr -F + [CC,]+ V(- )) (4.19)

The lowest energy solution (vacuum) can be easily found,


= IJ C =at (4.20)

A class of solutions with nonzero energy can be constructed through a transforma-

tion with a nonunitary isometry operator shift operator S ,



n=o

where SS 1 but SS = 1 Po 1 10)(01. The nth solution is


S= s"IS" = 0,(I P,) ,

C = SaS C S"aS" (4.22)


with the field strength

11 P
F = -'F = -([C, C] + 1) = (4.23)
2 00

and the energy

E = 270n ( + V(- )) .(4.24)

The NC solitons constructed above can in fact be interpreted as Bosonic D-

branes. As explained in the introduction section, the effective field theory of the

, 1i ,on and gauge field degrees of freedom of the open strings on the D-brane will

be a NC field theory in terms of the effective field theory at B = 0, but with the









open string effective metric Gi and the noncommutativity Oij determined by (1.11)

and (1.13).

Consider a Bosonic open string with an unstable space-filling D25-brane. The

leading terms in the effective action for constant tachyon and gauge field strength

have the Born-Infeld form [57-59]. Integrating out the massive string degrees of

freedom leads to an effective action of the form (for Bj = 0),


Seff = 46 F
2 11
Ss Jd26x detg r 4 h(b 1)FLVF~
+ + f 1)at- 1 8 + V -V 1) (4.25)

where c = T25g is independent of g8 with T25 the D25-brane tension. The potential

V(p 1) has a local maximum at 0 = 0 with V(-1) = 1 representing the unstable

D25 brane configuration, and a local minimum at = 1 with V(0) = 0 representing

the closed string vacuum, according to the conjecture of Sen [60, 61].

Now turn on the background B field with only B24,25 = b < 0 then the action

becomes NC in R2 including 24 and 25 direction. In chapter 1, explicit form of the

NC gauge field term is given in (1.19) analogous to (1.22). Therefore the NC action

analogous to (4.25) is

S=2c d24xc (4.26)
G, J
with


c = vetGTr h( 1)(F"v + Pf")(F, + 4,1)

+ + ,(+ 1)DD + .. V( ) (4.27)


Focusing on only the NC directions 24, 25, a special choice [10]

1 1
0 = -B = |bl (4.28)
B Ibl









in Eqn. (1.20) gives the value of the open string metric (1.11) and the string

coupling (1.24) in the zero coupling limit,


G = -(27a'b)2 Gs = gs(2a' b) (4.29)


In evaluating energy of the NC soliton solution (4.22), remember the soliton can be

obtained by the transformation with the shifting operator S" from the closed string

vacuum configuration. All covariant derivatives of Q or F,, vanish as they do in the

closed string vacuum, so are the gauge fields A, vertical to the brane. Also


h( 1)(F + h)2 (- P ( Pn) h(- )P(1 P) 0 (4.30)


and

V( 1) = V(-P,) V(-1)P, P, (4.31)

Therefore the energy of the soliton has only the potential term, derived from Eqn.

(4.27),

E d f 2TrP, 2(4.32)

which identifies the tension as


T (2=)2 nc (2r)2 'nT25 nT23 (4.33)
98

In the above construction configuration of n Dp-branes arises as the NC soliton

solution shifted from the closed string vacuum. This construction is exact for any

value of B or finite 0, which reduces to that of [62] in the limit of large B-field.

4.3 Classical Noncommutative Q-ball Solution

GMS solitons, which exist in (2 + 1) dimensional NC scalar field theory, while

classically stable, cease to exist at sufficiently small NC parameter 0, due to the

nonexistence theorem of Derrick [56] in the commutative limit(0 -- 0). In this

commutative limit, however, time dependent nontopological solitons, or Q-balls

exist in all space dimensions [48, 63].









4.3.1 Hamiltonian and Equation of Motion

In this section we derive the equation of motion for NC Q-ball solutions,
following brief introduction of NC scalar field theory. The form of the solution has

already been given [64]. We discuss the existence and stability of the solutions, and
show that in the commutative limit NC Q-balls just reduce to the commutative
Q-balls.

Consider a NC scalar field theory action with global U(1) phase invariance,

S =- dtd2x OOpoQ + V({ }) (4.34)

where the space-time metric is (-, +, +), and the fields are multiplied by NC star
product, generally made implicit in this paper, and {A B} A B + B A. The
potential V has a global minimum at the origin, with the scaling property

V(g, ) g-2V(g2 ) (4.35)

g is then the coupling constant assumed to be small. The commutative limit of this
action is where ordinary Q-balls have already been constructed [48, 63]. The NC
star product is defined to be,


(Q x)(x) exp(i--Ejk a )aJ ) (W ) (4.36)
2 Oxj Oyk y^

where j, k = 1, 2.
In the operator formalism the action integral (4.34) becomes

S[V, ] dt270Tr (aoa + (a,-([a, ] + [at,] [a,) V(P) (4.37)

The equation of motion is

a2 + 2 [a, [at, ]] + V' ) = 0 (4.38)









The action has global U(1) phase invariance, which yields a conserved charge


Q[0, 1 = I d2xj = i2-r0Tr(00o o0) (4.39)

Q is interpreted as particle number in the physical system. A particular system
.1.- ,-; exists with fixed particle number N Q[O, 0]. To find nondissipative soliton
solutions [48, 63] under this constraint, we write Hamiltonian

H = 270Tr (00000 [a, ] [a, + [a ] [a, ]) + V( ) + w(N Q[, ]) ,
(4.40)
with the constraint applied before the Poisson bracket is worked out [47]. The
minimum energy solution occurs at

H = 0 + iw = 0 (4.41)
6(00) N
which means

2u= ()e-it (4.42)

Assuming hermitian u(x) or real o(x), H becomes

H 2w0Tr w2-2 t a,] [at. ] + V( Q 2) + N, (4.43)

with the particle number
N 2= 27rTr((2) (4.44)

and the equation of motion (4.38)

2 1
2 [a, [at, _] w2 + 6V'(2 2) = 0 (4.45)
o 2

Note the equation of motion (4.45) also follows from (6H/6) IN = 0, which means
that the solution a has the same form as the static GMS soliton solution in the
potential
U(-)= V(2i(2)_ L22 (4.46)
2 2









Consider spherically symmetric solution [51] expanded in terms of the projection

operators,
00
Sx) = (4.47)
n=0
where P, n >< nI. Replace a in (4.43), (4.44), and the equation of motion

(4.45),

H = 27 [(n + 1)(AX l -A)2 + OU(A,)] + wN (4.48)

N 2wOw A (4.49)


(n + 1)(A,+1 A,) n(A, An,1) = -U(A,) (4.50)
2

Sum the equation of motion from n = 0 to n = K

0 K
AK+1 K =2(K' ) Y U(An) (4.51)
2 ( n=

where K > 0 is an arbitrary integer. A particular set of A,'s defines a solution.

TM in: properties of the solution can still be derived from Eqn. (4.48-4.51) though

a closed form has not been constructed. For example, because of the finiteness of

both the energy H and the particle number N, we have,


A+ = An An = 0 for n -oo (4.52)

4.3.2 Q-ball Solutions

In static GMS soliton theory, the global minimum of the potential is generally

assumed to be at the origin, and the core of the soliton is localized at the local

minimum of the potential (false bubble solution). It is the noncommutativity

that forbids the classical decay of the solitons. The corresponding commutative

potential does not have nontrivial topological structures, and hence yields no

soliton solutions. Therefore NC GMS solitons are genuine NC effects and they

disappear at small enough 0, where the commutative limit is approached.









This is not the case with Q-ball solutions. The existence and stability of

Q-ball solution rely on the conservation of the charge Q as the consequence of

the global symmetry. The potential for Q-ball solutions does not have nontrivial

topological structure. Therefore NC Q-balls are expected to exist even for very

small 0 We will show that such NC Q-ball solutions would smoothly reduce to the

Q-ball solution in the commutative limit.

In the following we discuss the existence of NC Q-ball solutions in a typical

potential form,

U(a) = V(a2) 2 2 2 b4 + ca6 (4.53)
2

where the coefficients b and c are larger than zero, and a = {(2 a2).

U(a) varies for different w. If a2 > m2 or a < 0, U(a) has a local maximum

at the origin. In the commutative limit there is only a plane wave solution. Here

similar plane wave solution in NC limit can also be constructed. Since for a

stable soliton solution AX would have to take values between s and the origin

and monotonically decrease in n [65], a simple argument can show that solitons

cannot exist. There is a constraint that EPK U'(An) converges to zero as K goes

to infinity, which cannot be satisfied in this case. To prove this constraint, suppose

that
OO
U'(A,) v / 0, (4.54)
n=0
Sum Eqn. (4.51) from a particular K = q sufficiently large to a point p close to

infinity,

A, A-Y (4.55)
K=q
It is then easy to see Ap will not converge to zero as p goes to infinity.

When a 2 < V2 V2 = 2 b2/2c U(a) has only a global minimum at

the origin. Even though in the commutative theory no soliton solutions exist, for

NC theory at sufficiently large 0, there are GMS type solitons exist. It has been









shown that there is a critical lower bound on 0 for the existence of NC soliton [66].

Similar bounds would be expected to exist for NC Q-ball for 0 < c < v as well.

As v2 < U2 < m2, U(a) has a local minimum at the origin, a global minimum

at s (U(s) < 0) and a zero w = [(b-Vb2 4ac)/2a]1/2 between 0 and s (0 < w < s).

In the commutative case such potential form enables the existence of Q- ball

solutions. In the NC case it is expected that NC-Q ball solutions exist even for

small 0. In the following we take the continuum limit of Eqn. (4.51) for very

small 0 and show that all the solitons sPK exist at 0 oo converges to the

commutative Q-ball solution as 0 0.

For very small 0, all An's can be considered as sufficiently close. Therefore

A, can be approximated by a continuous function A(u) 1 .Let u = KO and

AK = A(KO) = A(u) Eqn. (4.51) becomes

A'(u) t U'[A(s)]ds + 0(0) (4.56)
2(u + 0) [

Ignore 0(0) term, we have
dA d2A 1 dU
u+ n (4.57)
du du2 2 dA
Let u = v2 ,
d2A 1 dX dU(A)
+ (4.58)
dv2 v dv dA
This is exactly the equation of motion for the commutative Q-ball solution A(v)

, with v identified as radius r This can be explained as follows: The Weyl
transform of r2 is ata and ata has the eigenvalue nO on the state In > As 0

gets smaller, the eigenvalues nO gets closer, and eventually becomes continuous

as 2 in the commutative limit. The coefficient An just becomes the field A(r) in
2s7


1 Thanks to Dr. Shabanov for helping on this point









this limit. In this description the commutative Q-ball can be considered as the

analytical continuation of the NC Q-balls in 0 .

The formula for the energy and particle number in the commutative limit can

also be recovered by taking the continuum limit of Eqn. (4.48) and (4.49),

H = 27 vdv ( )2+ U(A)+ N (4.59)
Jo [2 dv I I
N = 27uw duA2(u) = 27w vdvA2(v) (4.60)
JO Jo

The existence of the commutative Q-ball solution are proved by considering

an analogous problem in which a classical particle moves in the one-dimensional

potential -U(A) [48]. The field configuration A(v) of the Q-ball starts from a

unique value p = A(0) between the zero w and the global minimum s, then

monotonically decreases in v and approaches 0 when v -- oc. This property of

A(v) is consistent with those of the general stable NC soliton solutions AX at finite

0 It is found [65] that there exist smooth 0 families of spherically symmetric

solutions in which An is monotonically decreasing in n In the infinite 0 limit

such solution is just sPK As 0 decrease, Ao, AK decrease from s while other

A,(n > K) starts to move away from the origin towards s but the whole AX

series remain monotonically decrease in n Since in the commutative limit An

just becomes A(v) one can conclude that as 0 decreases from oo to zero, Ao will

decrease from s and eventually to p at the commutative limit.

4.3.3 Virial Relation

The Hamiltonian (4.43) in the function formalism is

H[al] = 27 rdr (5ia)2 + U( ,a) +UN (4.61)

where the potential U has explicit dependence on 0 through star product. Suppose

a(x) is the Q-ball solution, H[a(x/a)] must be stationary at a = 1. A change of the









integration variables shows that


H[a(x/a)] = 27 rdr (L ,(x))2 + a2U(4 a(x)) + N (4.62)
2 a

and

H[a(x/a)] = 2 rdr U( a) 2 U )= 0 (4.63)
da a d 0

Unlike the Virial theorem for d = 1 space dimension, here the kinetic energy is

scale invariant. Scaling dependence of the energy includes two separate terms

from the potential and from its dependence on 0 through the star products. The

significance of Eqn. (4.63) is more explicit in GMS soliton case, where the potential

energy

27 rdrU(a) = 0 U(A,) > 0 (4.64)
n=O
The scaling variable a can be thought of as the size of the NC soliton. While the

positive potential energy favors shrinking of the soliton, but the NC star products

keep it from decay.

4.4 Quantization of Noncommutative Q-ball

Solitons are extended objects exist in field theory, the properties of which

receive quantum corrections as the fields are quantized. In this section we follow

very closely to the canonical quantization procedure [49, 63]. Then we evaluate the

ultraviolet divergences in the quantum corrections to the soliton energy at very

small 0 .

4.4.1 Canonical Quantization

The general procedure to investigate the properties of the solitons is to expand

the fields around the classical solution. Because the momentum and particle

number are conserved in the system, we will have to impose the corresponding

constraints to erase the zero-frequency modes in the expansion.









We start by making a point canonical transformation of (x) ,

1 te-_/i [a7(x- X(t)) + (x X(t),t)] (4.65)


x(x X(t), t) XR(X X(t), t) + ixi(x X(t), t) (4.66)

where 0(t) and X'(t) are the collective coordinates represent the over-all phase

and the center of mass position. Impose the constraints on X to ensure the above

transformation is a canonical transformation with equal number of degrees of

freedom before and after,


Jaxi 0, J'XROa =0, (4.67)

where i = 1, 2 The integral sign denotes two dimensional integration over x.

The star product is suppressed. Unless indicated otherwise, from now on the

differential d2x and the NC star product are implied wherever applicable. The

above constraints also remove the perturbative zero mode solutions in the meson

field X. Let
OO
XR(x, t) = qa (ta(x) (4.68)
a=3
oo
Xi(x,t) = qja(t)g(x) (4.69)
&=2

where fa(x) and g&(x) are the real normal functions satisfy


Sffb 6ab (4.70)

g g99 6g (4.71)

(4.72)









and under the constraints,


Soaf = 0,

Jrga 0,

where a= 3, 4,... and a 2, 3,... ahv--o in this paper.

Rewrite the Lagrangian (4.34) with (4.65),

L =q Mq + V(q),
2

where qT = (X X2 ,3 ,.. q12 ,.. ) and T denotes matrix transpose,


V(q) (00 + V( {t }))
./ 1


The matrix elements


Mi =










Where i, =
M2p =p

Mi. =



M3a =



M ab =

May =

where .1, = f ( oi7)


P


of the symmetric M are

MI ,+ / (29aojXR + 9iXRjXR + OixajX)
I+ (2x + x2 + X2) ,


I (-28(o- + (,i7I XI+iXR) ,

faziXR ,R

fga iXI ,

I faXI ,

gx 9 ,


2


6ab



and I

- Mqc-


(4.77)

(4.78)

(4.79)

(4.80)

(4.81)

(4.82)

(4.83)

(4.84)

(4.85)




(4.86)


f= 2. The conjugate momentum of q is


(PI ,P2 ,N ,pR3 ,... ,p 2 ,...) .


(4.73)

(4.74)






(4.75)

and

(4.76)









The particle number N and the total momentum Pi are conserved since the

Lagrangian (4.75) is independent of the collective coordinates 3 and Xi. Quantize

the new canonical coordinates,


[X1 P] = iij (4.87)

[ N] = (4.88)

[qa pRb] 6ab (4.89)

[qia pi] = i6b (4.90)

The Hamiltonian

H = J-lpTA-MJp + V(q) (4.91)
2
where J = Mdet because the operator ordering in H is unambiguously deter-

mined by the ordinary quantized Hamiltonian with the coordinates 0 [49].

The quantum states can be labeled as IP1 P2 N q qi&). One can solve

the Schr6dinger Equation perturbatively around the one soliton ground state

|P1 = P2 = 0 N 1= I 0). In this state Pi and N are the momentum and particle

number of the classical solution a, which can be obtained by letting = a in Eqn.

(4.49) and Pi =f 0Q' + Qia [1]. 0 labels the lowest energy state with the given

P' and N value.

We can then treat XR and XI as perturbative degrees of freedom, and expand

the Hamiltonian perturbatively around the one soliton ground state order by order

in the weak coupling constant g defined in Eqn. (4.35).

a is at the order of g-1 as a soliton solution. .11, and I are g-2 order. Then Pi

and N are at the g-2 order, while pRa and pia at gO order. Since J commute with

Pi and N and at the leading g-3 order, J= 11,,I/I is a constant or [p J] 0, one

can check that J would not be a factor in the Hamiltonian up to the order gO








The Hamiltonian can be expanded order by order, H = Ho + H1 + H2, with the
expansion relation,

M-1 = M1- + M1AM- + Mo1AM1-AM0o1 + (4.92)

where M = M0 + A, and M0 has only nonzero diagonal elements, M~f
(.p 1,, 1 ,1 ,1).
Ho, equal to the energy of the classical solution, is at the order of g-2

Ho = 3l,, + 2I2 + V( a2) (4.93)
2 2

H1, linear in X, vanishes due to the fixed N and Pi, which ensures that XR and
X, are at the order of g.
The term quadratic in X is at the gO order,

1 fpXI)2+-1a t gXR LL aIX,)2 ,9&2{f IXR)+V2q)
H2 2R a /2 wJ L ah PI& IW lR 2J2 ( )2+2

(4.94)
where

2(q) J [(iXR)2 + (iXI,)2] t 2(x2 + X )

+2(X2 + X2)V'(1/22) + V (a2 1{ 2 } 2{XR, a} ,(4.95)

where V (172 {XR, "} {XR, o7}) represents the terms from the expansion of the
potential V quadratic in XR.
4.4.2 Energy Corrections at Very Small 0
The Hamiltonian is separated into two parts, described by the baryon degrees
of freedom (Pi N) and meson degrees of freedom (XR XI) respectively. The sum
of the frequencies of the meson excitations is the zero-point energy of H2,


p2 = 0 ,N = I 0) (4.96)


(p =P2 =0, N I,O0 H2 IP1









which, subtracted by the vacuum energy Evac = f d2k/(2)2k2 + m2, gives the

quantum corrections to the soliton energy.

V2(q) is the perturbative expansion of the effective potential V(q) wQ[Q, ] ,
Eqn. (4.76) and (4.39), around the solution a It is easy to check that XR = aia

and XI = a are the eigenmodes of V2(q) with eigenfrequency 0, due to the

translational and rotational invariance of the potential. Therefore we can define the

normal functions fa and g& to be the eigenmodes of V2 or


2(q) -1Q 2 q2 + I2 q2
2 'a 'a 2 1 l&


(4.97)


with the frequencies pRa and Qa6 The potential V2 are highly nonlocal since

the fields are multiplied by NC star product. In the commutative Q-ball case, V2

has been shown to have only one s-wave eigenmode in XR sector with imaginary

frequency QR3 [63]. In last section, we have shown that as NC parameter 0 is

taken to be small enough, the NC soliton solution will reduce arbitrary close to

its commutative analog. Therefore close to the commutative limit V2 is expected

to have the similar eigenvalues and eigenmodes as its commutative analog. NC

Q-ball is also expected to be stable as its commutative analog. We will assume

0 is chosen to be such a small value in evaluating the quantum effects of the

noncommutativity.

Define fi = 1/I,,. .T and gi = 1/V-a Rewrite the Hamiltonian H2 (4.94),

in the matrix form,


1 1
H2 (T LQTTT)(P TQ)+ 2L2 QT-QQ T+ 2 Q
2 2


(4.98)


where the matrices are defined as follows:


T (PRa ,Pl) p (qRa qi) ,


(4.99)









o Pa

0 La

-0Fa 0


Tab 0


0 ab)


QRa


0


where


Fad J fad ,


Fab /9lfa /lfb ,


Gb -


The equation of motion,


SaH2
P '


OH2
7 OQ


P UTQ ,


7 = T(P TQ)


4w2 Q 2Q .


Therefore,


Q+2,oTQ+4W2Q + Q22 Q


Let the real normal eigenmodes of Q be

QA ( A ,)T


where QATQB


AB. Replace Q


QAexp(-iAAt) (Index A is not summed over)


in the above equation. Since QATTQA = 0 ,


Introduce creation and annihilation operators, [CA B


6AB, Q can then be


quantized as,


A
Q (CAC-iAAt + CtCeiAAt)
A V2AA


0


(4.100)


fi9a fig .


(4.101)


give


(4.102)


(4.103)

(4.104)


(4.105)


(4.106)


AA -/QAT(4w2 + Q2)QA


(4.107)


(4.108)









Use this equation and Eqn. (4.104) and (4.98), one can define the one soliton

ground state,

CAl1 p2 0 ,N = I ,0) =0 (4.109)

then the zero-point energy of H2 (4.96) is

EAA Tr1{ } (4.110)
A

where the matrix A is diagonal with the eigenvalues AA.

In the commutative theory the zero-point energy contains the divergences

even after subtraction of the vacuum energy. The finiteness of the soliton energy

is recovered by starting from the renormalized form of the action (4.34), which

induces the counter terms also contain the divergences [67].

Work in the specific form of the 06 potential (4.53),

1 1 241
V( {, }) = m2(- {, ) bm2g2(t{, )2 + cm2g4( ({, })3 (4.111)
2 2 2 2

At the gO order, or the one-loop order, the general formula for the soliton energy is


Esoliton (p P2 0 ,N = I ,0| H P1 = P2 =0 ,N =Iw ,0)- ,I 2)

Ho + -Tr{A} Evac + 2 b2 ) (4.113)
2 2 m4

where 6m2 and 6g4) are the counter terms for the mass and the 4 coupling

respectively. The 06 coupling does not receive loop corrections. The 04 coupling

terms yield the right coefficients and can be renormalized [32].

The loop integration in the NC field theory generally contains phase factors

which yield the interesting UV-IR phenomenon upon renormalization [22]. In the

following we evaluate the quantum correction from the zero-point energy of H2 in

Eqn. (4.110) and show that it contains the same phase factors as those appear in

the counter terms 6m2 and 6g24)'








We start by arguing that only 1/2Tr{Q} is needed in evaluating the leading
divergence. In Eqn. (4.107), it is easy to see QAT Q is finite,

QAT glf.(.)2+ (/ f ig& a)2

< [Jgf J(f.a )2]2 [f+ t J(g6a)2]2+ [J + f J(g6)2]2
(< ([ 2 + 2 2

( + <)2 + + 2 )2 < 12. (4.114)

As we will see that the eigenvalues Ra and Qi6 behave like k2 2 + m2 at very
large k. The leading divergence of Tr{A} will be determined by Tr{Q}.
Ra, and Q16, eigenfrequencies of V2(q) in Eqn. (4.95), satisfy the linear
equations,

(_a? 02 m2)Xy -bm2g2a2,2 } + -cm2 44, } = x ,(4.115)
2 8
(-o? L2 + m2)XR bTm22({(2, XR} + AXR) +
Scm2g4({4, XR} + XR 2 + a, XR 2 QXR (4.116)

The above equations are just time independent Schr6dinger equations. In
particular phase shifts from the central potential have been used in calculating the
soliton energy correction [68]. The basic idea is that in the central potential for
each partial wave, the difference of the density of the states between the scattered
wave and the free wave is related to the derivative of the phase shift,

1 d61(k)
pi(k) po(k) (4.117)
x dk

where I goes from -oo to oo. The finiteness of the particle number, N = f a2 ,
determines that a -- o as r -- oo Therefore the NC potential in Eqn. (4.115) and
(4.116) is radial symmetric and vanishes at oo. For the most general potential term

WV(r) X VV (r) ,


[W (r) x* B (r) ,L] = WF (r) [ ,L] W (r) ,


(4.118)









where L = -ie3x'03 is the angular momentum. The star product is made explicit

here and in the rest of the section. This formula can be easily proved in the Weyl

transforms of the fields. Going to the momentum space, one can generalize the

result in [15] and show that

W )(x)n) x d2pf d2pb p(p ) (ei(x+a)eib(x- ~)X(X)
wVF (X) (X) B w (X) (2 )2 (2 )2
(4.119)

where 'i = c0i1.

Using Eqn. (4.117), consider only the leading divergence, we have [69],

1 1
-Tr{A}- Evac -~ tTr} Evac
2 2
2~ kdVk2 2[jI6(k)+6m(k)] (4.120)

where 1n(k) and j6m(k) are the phase shifts for Xj and XR. The sum of the phase

shifts can be evaluated through Born approximation. In the commutative case, this

leads to the cancelation of the tadpole diagram [68].

Eqn. (4.115) and (4.116) have the Jost solution form for the Ith partial wave

at large r ,

X ~ h*(kr) + t- hi(kr) (4.121)

Considering the ..i-mptotic (r oo) behavior of the solution, the standard

procedure [70] leads to the scattering amplitude,

f(k',k) f() = efl(k)e"i =t eS6 sin 61eiz (4.122)

where k' = k and 0 is the angle between k' and k. At large 1, or 61 w 0, we can see

i v/kf ( 0) (4.123)
1








f(k', k) can also be calculated through Born approximation, replacing X by
-ikx in the potential form (4.119),

f(k', k) d2Xik'x w(i) eikx ()
0 0
d1 ( d2-ei(k,-k)x () (_ -k) V(x+ -k) (4.124)
4 v/k F 2 2

where i labels the potential terms in Eqn. (4.115) and (4.116), and k' = e .kj.
Therefore

> -4Jd2x ) (x-2 k)B + -k2
1 i
2p (P)W (-p)e-i8pk (4.125)

The right hand side only depends on the magnitude k due to the central potential
W(r).
Now we are ready to evaluate Eqn. (4.120),

-J d k2 +m2 1l(k) + 6w(k)]

-- d k2 d22) -p-iopk
1 t dv n2 { d2P -"(i),O ,i)C
87T j (271)2
/1 d2pfd2k e-iOpk
d d2 k2 pk 2 )(p) )(-p) (4.126)
2 (2 7e)2 (2 7c)2 2 / -k2 -+ 2 Y -(4 )

The integration over k is exactly part of the tadpole diagram belongs to 6m2 [32],
and it contains the UV/IR divergence (A oo and p 0) evaluated with the
cutoff A [22],
I d2k e -ipk 2 /29m Ae
f w2 d k + (mAn) / 2K ( ) A-f + o(1) (4.127)
j d(2k7)2 2/k2 + 2 (47) 3/2 ( nA )If 87-
where Aeff (02p2/4 + 1/A2)-1/2. Notice that the above UV/IR divergence from
Eqn. (4.126) occurs only when both WF and WB exist. In other words, only the
terms rXRO 2XRo-2 and a{o, XRj} in Eqn. (4.115) and (4.116) yield UV/IR









divergence. All other potential terms only give the normal UV divergence where

the phase factor is absent. Since the counter term 6m2 and 6g2 do not include

UV/IR divergence, we are certain that the Q-ball energy correction includes UV/IR

divergence. Cancelation of the UV divergences is not obvious because the exact

value of the eigenfrequencies AA is unknown.

4.5 Finite 0 and Noncommutative GMS Solitons

The above calculation assumes that the NC parameter 0 is sufficiently small so

that the NC potential will generate the Jost solution form as in the commutative

case. Let us consider the effects of the NC potential (4.119) in the case that 0 is

not small.

Since

[x + i x3 + 3] = iei (4.128)
2 2
the effective scattering potential for the NC interaction 4Wp(x) X WB(x) (4.119)

is just

W (x + -)W (X a) (4.129)
2 2
multiplication of the Weyl transforms of Wp(x) and WB(x) Notice WVp and WB

commute since [xi + i0/2', xj i0/29j] = 0.

Now considering

W(X +i )(p )eip(x+ ) (4.130)
2 (27)2

the noncommutativity,


[p(x + ) ,p6x2 2 + i 62 p}p} (4.131)

can be suppressed even if 0 is not small, as long as W(x) is smooth enough or

W(pf) -- 0 at large pf. Notice small pf is also the IR limit we discussed in the last









section. Under this assumption we can write


W^(x + -) WF(x + 5-) W (|x + |2) F 2- L + 9?) (4.132)
2 2 2 2

Acting on the field X(x) = uz(kr)ei, the effective potential becomes WF)(r2-01/2+

a?). Similar calculation applied to WB(x) yields WB(r2 + 01/2 + a?). Therefore at

large k and large r, we can treat the scattering potential perturbatively as in the

commutative case. The phase shift evaluation of the energy of the soliton could

still apply provided that W or the soliton solution a are smooth enough.

NC GMS soliton only exists at finite 0. Based on the above arguments, we

can still evaluate its quantum corrections with the phase shift method in the last

section.

Quantization of GMS soliton and Q-ball share a lot of similarities. To get the

GMS soliton theory, we make the replacements (0 -- 1/V2)) in the previous

complex scalar field theory (4.34). With the potential (4.111), the Lagrangian

becomes


2= (0) V() = -(0 )2 m2 2 + bm2+ 4 cg 46 (4.133)
2 2 2 4 8

where ) is multiplied by the star product. The renormalizability of the theory

has been proved [30]. Let u = 0 because there is no conserved charge Q (4.39) in

the theory. When b2 4c < 0, the potential in (4.133) has a local minimum at

bg/2 besides the global minimum at the origin, and the GMS soliton solution a

exists. Replace the expansion (4.65) by ) = a + X where X = XR is real. As

a result, the meson degrees of freedom are only XR or X. Upon quantization, the

soliton energy is still given by Eqn. (4.113). Since w = 0 and A = 2 are the exact









eigenfrequencies of H2 (4.94), we have


(-a; + m2)XR bm2g2({a2, XR} + aXR,)

+cm 2g4(4, XR+ 72XR2 + XR}a) -A R (4.134)

Eqn. (4.120) describes the exact ultraviolet divergences in 1/2Tr{A} Evac.

Therefore we are able to check the cancelation of the divergences in Eqn. (4.113).

As mentioned in the previous section, in the above equation, the terms UXRj ,

a2XR-2 and a({a, XR} yield UV/IR divergences, while the rest terms yield UV

divergence. A critical observation is that those terms yield UV/IR divergence

have one to one correspondence with the contractions of the fields yield Nonplanar

Feynman diagrams [22], and those terms yield UV divergence correspond exactly

to the planar diagrams. We can just spare the details of counting the divergences.

Since the counter terms 6m2 and 6g4) cancel exactly the UV divergence part, we

conclude that the soliton energy (4.113) is UV finite, but includes all the UV/IR

divergences.

4.6 Conclusion and Discussion

In this chapter we discussed the quantization of NC solitons in (2 + 1)

dimensional scalar field theory. In particular, classical solutions and quantization

of the NC Q-balls at very small 0 are investigated in detail. Classically NC Q-

balls reduce to the commutative Q-ball as 0 goes to zero. Quantum mechanically,

because loop integration in the NC field theory have different ultraviolet structure

from those in the commutative theory, i. e. UV/IR mixing, quantum corrections to

the NC soliton energy necessarily include the UV/IR divergent terms which cannot

be renormalized away. The existence of such terms in the energy is demonstrated

through the phase shift summation. The same method is further generalized to NC

GMS solitons which exist only at finite 0. In the small momentum limit, or for the

sufficiently smooth soliton solutions, divergence structure of the soliton energy can









be calculated exactly. In this case the energy is found to contain no UV divergence

but all the UV/IR divergences. Quantum corrections to the NC soliton energy have

also been calculated but at very large 0 [7], where no UV/IR divergence is found.

We believe that is because at large 0, the noncommutativity (4.131) is not small

and cannot be ignored, and the potential term is the dominant term instead of a

perturbative one. In this case the phase shift sum is not a good approximation to

the energy correction.

An interpretation to the UV/IR divergence [23] is that because new light

degrees of freedom are introduced in the Wilsonian effective action. UV/IR

divergence can be reproduced by integrating out those new degrees of freedom,

which are then interpreted as closed string modes with channel duality. Future

research direction is to consider the NC solitons in the gauge theory, where they

are interpreted as D-branes [53, 54] and D-brane action is properly recovered. One

expects to gain better understanding of interactions between D-brane and closed

strings through quantization of NC solitons.














CHAPTER 5
SOLUTION OF KOSTANT EQUATION

In chapter 3 theory of superPoincar4 algebra and its representations is

reviewed. Classification of irreducible unitary representations of Poincar6 and

superPoincar6 algebra reveals physical spectrum of the corresponding theory.

M-theory, as unification of string theory, is conjectured to be 11-dimensional

theory [12, 13], whose low energy limit is 11-dimensional supergravity. Little group

of 10 dimensional Poincar6 group is SO(8), which classifies the spectrum of 10

dimensional superstring theory. Triality symmetry of SO(8) leads to marvelous

cancellations between Bosonic and Fermionic contributions, which renders the

theory to be UV finite. Little group of 11-dimensional superPoincar4 algebra,

SO(9) does not have such symmetry, and SO(9) is nonrenormalizable in high loop

order [71]. It is found that some irreps of SO(9) naturally group together into

an infinite tower of triplets [72], the lowest of which includes the spectrum of 11

dimensional supergravity. This --. -:- -I -; that the tower of triplets might be able to

describe certain limit of M-theory. A possible candidate is the infinite R.--i slope

limit, or zero tension limit of string theory, where one expects all states to become

massless, with an infinite number of states for each spin.

A mathematical understanding of the triplets has been given [73]. They arise

for embeddings where both group and subgroup have the same rank. SO(9) is a

subgroup of F4 with the same rank, and the quotient space F4/SO(9) has Euler

number three giving a triplet of SO(9) to every irrep of F4. There exist other

cases the triplets arise the lowest of which describe N = 8 supergravity, N = 4

Yang-Mills and N = 2 hypermultiplet [74-76]. All these Euler triplets arise as

solutions of Kostant's equation [43], which is a Dirac-like equation on the coset.









This chapter focus on differential form of the representation of Kostant's equa-

tion and its solutions, which is the beginning step for the Lagrangian construction.

After a review of Euler triplet in a toy coset space, coset SU(3)/SU(2) x U(1) ,

Euler triplet solutions in coset F4/SO(9) are discussed in detail. In particular,

variable representations of 11-dimensional superPoincar6 algebra, and F4 and

SO(9) algebra, are worked out explicitly, as well as representations of Kostant

equation and its triplet solutions.

5.1 Euler Triplet for SU(3)/SU(2) x U(1)

As a learning example start with a detailed analysis of the Euler triplets

associated with the coset SU(3)/SU(2) x U(1). There is an infinity of Euler triplets

which are solutions of Kostant's equation associated with this coset. The most

trivial solution describes the light-cone degrees of freedom of the N = 2 in four

dimensions, when the U(1) is interpreted as helicity.

5.1.1 The N = 2 Hypermultiplet in 4 Dimensions

The massless N = 2 scalar hypermultiplet contains two Weyl spinors and

two complex scalar fields, on which the N = 2 SuperPoincar4 algebra is realized.

Introduce the light-cone Hamiltonian


P- = (5.1)
p+
where p (pl + ip2). The front-form supersymmetry generators satisfy the

anticommutation relations


{QOn,} = -J26'mnp+

{Qn+} ,m j m,n= 1,2, (5.2)

{ n } = -2p6 "n










The kinematic supersymmetries are expressed as


a0
( r
+m = _-


Omp+


while the kinematic Lorentz generators are given by


1 a
i(;p Xp) + 0

-x-P+ i- 2

(M+ + iM+2)


t a
2 jr'


-p ,
+ ,M
xp- N


where x = -(x' + i2), and where the two complex Grassman variables satisfy the

anticommutation relations


{ Om, a
{-}
('0 j


---0}

{ m }


0.n
0 .


The (free) Hamiltonian-like supersymmetry generators are simply


PQrn p
+


+ -+c-Z,
p+


and the light-cone boosts are given by

M 1 p p}+- + O
M- x p {x, P~}+ i ,
2 p+


1
x-p x, P-
2


(5.5)


(5.6)


S+ -- .
p+


This representation of the superPoincar6 algebra is reducible, as it can be seen to

act on reducible superfields 4(x-, x, Om, m), because the operators



~D a, p+ (5.7)
0(1


-m a
+ __ + 0nP


M12


(5.3)


(5.4)


-p ,


t









anticommute with the supersymmetry generators. As a result, one can achieve

irreducibility by acting on superfields for which


a [) Op+] (5.8)


solved by the chiral superfield


((Y-,X, xOm) = o(y-, x) + e', O( y-, x) + 0 ,' ?(y-, x) (5.9)

The field entries of the scalar hypermultiplet now depend on the combination


y- x- i0mtm, (5.10)


and the transverse variables. Acting on this chiral superfield, the constraint is

equivalent to requiring that


Q -2pOm, (5.11)


where the derivative is meant to act only on the naked Om's, not on those hiding in

y-. This light-cone representation is well-known, but we repeat it here to set our

conventions and notations.

5.1.2 Coset Construction

Let TA A = 1, 2,...8, denote the SU(3) generators. Its SU(2) x U(1)

subalgebra is generated by Ti i = 1, 2, 3, and Ts. Introduce Dirac matrices over

the coset

{7(, qb} 26ab

for a, b = 4, 5, 6, 7, to define the Kostant equation over the coset SU(3)/SU(2) x

U(1) as

a = 4,,7 0.
a-4,5,6,7









The Kostant operator commutes with the SU(2) x U(1) generators


L= T,+S, i 1,2,3; L,= Ts+S, (5.12)


sums of the SU(3) generators and of the "spin" part, expressed in terms of the 7

matrices as

S = fabab S= -S f8ab7ab (5.13)
4q 4

where y^ b = ab a / b and fjab I f8b are structure functions of SU(3).

The Kostant equation has an infinite number of solutions which come in

groups of three representations of SU(2) x U(1), called Euler triplets. For each

representation of SU(3), there is a unique Euler triplet, each given by three

representations



{al,a2} [a2] 2a +a2+3 ( [a1 + a2 + 1] ai 2 E [al 2a2+al+3
6 6 6

where al, a2 are the Dynkin labels of the associated SU(3) representation. Here, [a]

stands for the a = 2j representation of SU(2), and the subscript denotes the U(1)

charge. The Euler triplet corresponding to al = a2 = 0,



{0,0} = [0]_- E [l]o [0] ,

describes the degrees of freedom of the N = 2 supermultiplet, where the properly

normalized U(1) is interpreted as the helicity of the four-dimensional Poincar4

algebra.

5.1.3 Grassmann Numbers and Dirac Matrices

In order to use the superfield technique we will identify the spin part of the

U(1) generator S, with the spin part in Eqn. (5.4) taking the condition (5.8) into

account. This will mean that we write also Si in terms of the 0's. An appropriate

representation is then












74 + 75

76 + i7


.2
. 2+ +
P2 +


4 i 5 -1

Vp+
6 *7 F2-
<7 P< ++


(5.14)

(5.15)


in terms of the kinematic N = 2 light-cone supersymmetry generators defined in

the previous section. We can check that Ss indeed agrees with the spin part of

Eqn. (5.4) (after proper normalization). As the Kostant operator anticommutes

with the constraint operators


{ C Dl} = 0 ,


(5.16)


its solutions can be written as chiral superfields, on which the 7's become


-2i 2 01 ,

-2i 2 2 ,


74 i5 = i2

6 7 = i
<7 <7 P+ OH


The complete "spin" parts of the SU(2) x U(1) generators, expressed in terms of

Grassmann variables, do not depend on p+,


1 a a 0 a a
S, (01 + S2 -( -- 02
2 002 S2 2 802 10
S3 1 ) S ( 02 -1).
2 00, 802 2 +0, 02a

Using Grassmann properties, the SU(2) Casimir operator can be written as


2 3(1 0 2
4 a01, 02


(5.19)


(5.20)


74 + i75

76 + 1i7


(5.17)

(5.18)









it has only two eigenvalues, 3/4 and zero. These SU(2) generators obey a simple

algebra
1 2i
7S, S=-S- +2i kSk. (5.21)

The helicity, identified with Ss up to a normalizing factor of v3, leads to half-

integer helicity values on the Grassmann-odd components of the (constant)

superfield representing the hypermultiplet.

5.1.4 Solutions of Kostant's Equation

Consider now Kostant's equation over SU(3)/SU(2) x U(1) It is given by


p 7aTap = 0. (5.22)
a=4,5,6,7

Schwinger's celebrated representation of SU(2) generators of in terms of

one doublet of harmonic oscillators has been extended to other Lie algebras [77].

The generalization involves several sets of harmonic oscillators, each spanning

the fundamental representations. Thus SU(3) is generated by two sets of triplet

harmonic oscillators, one transforming as a triplet the other as an antitriplet. Its

generators are given by


T1 + iT2 102 2i T1 iT2 = Z20 1 0


T4 + iT5 z13 30 T4 iT5 z301 T03

T6 + iT z2-3 32 T6 iT7 3 -2 3 ,2

and
1
T3 -= 1(Z1 Z22- T101 + T22) ,
2

T8= (z101 + z2 2- 1 1- 2 2Z303 + 2303) ,
23
where we have defined

a a
1 -- 1 etc..
zz\ 8z\









The highest-weight states of each SU(3) representation are holomorphic polynomi-

als of the form
01-al02
z1 3

where al, a2 are its Dynkin indices: all representations of SU(3) are homogeneous

holomorphic polynomials.

Now expand the Kostant equation (5.22) with the Dirac matrices in terms of

Grassmann variables yields two independent pairs of equations


(T4 + iT5)1 + (T6 + iT7'2

and


(T4 T5)0 (T6 + T7)12


0 (T4 iT5)2 (T6 iT7)Q




0 ; (T6 iT7' + (T4 + iT5) 12


that is


(-,13 :3a1) + (203 :3 2)Q2

(-3a1 -:1a3)b0 (Z2a3 -3 0)212

The homogeneity operators


0 ; (3 1 103) 2 (3 2 2 3) 1

0 ; (z3a2 -:2a3)Q0 + 1 ( 3 -:3a1)Q12


D = z011 + z22 + z3a3 ,


D = 11 + 20 + :303


commute with 1, allowing the solutions of Kostant equation to be arranged in

terms of homogeneous polynomials, on which al is the eigenvalue of D and a2 that

of D. The solutions can also be labeled in terms of the SU(2) x U(1) generated by

the operators


L = T + S, i 1,2,3 ; L T +S .









The solutions for each triplet, are easily written for the highest weight states of

each representation,




S= l -2 labels [a2] 2.+02+3
6
+ 01 4z1 2 labels [ai + a2 + 1 -2
6
+ 0102 z~1 72 labels [ail2a,+a+3 (5.23)
6

where [...] are the SU(2) Dynkin labels. All other states are obtained by repeated

action of the lowering operator


Li iL2 02 + (z21 02) ,
80,

giving us all the states within each the Euler triplet.

5.2 Supergravity in Eleven Dimensions

The ultimate field theory without gravity is the finite N = 4 Super Yang-

Mills theory in four dimensions. Eleven dimensional N = 1 Supergravity [78], the

ultimate field theory with gravity, is not renormalizable; it does not stand on its

own as a physical theory. However, the eleven-dimensional theory has been recently

revived as the infrared limit of the presumably finite M-theory.

5.2.1 Superalgebra

N = 1 supergravity in eleven dimension is a local field theory that contains

three massless fields, the familiar symmetric second-rank tensor, h~, which repre-

sents gravity, a three-form field A P, and the Rarita-Schwinger spinor I ,. From

its Lagrangian, one can derive the expression for the super Poincard algebra, which

in the unitary transverse gauge assumes the particularly simple form in terms of

the nine (16 x 16) 7y matrices which form the Clifford algebra


{7', 7 } 26" i,j 1,...,9 .








Supersymmetry is generated by the sixteen real supercharges

Q0, = or*,


which satisfy


{ Qa } PP a 6ab
/2 p+


{Q;, Qb}


(i)abi ,


and transform as Lorentz spinors


[MAi, Q ]


Q [M+-,Q] Q
2 2
0, [MAi,] (yi )a
2


(5.24)

(5.25)


A very simple representation of the 11-dimensional super-Poincar6 generators can
be constructed, in terms of sixteen anticommuting real X's and their derivatives,
which transform as the spinor of SO(9), as


Qa + a ,a
Q+ o vo+ 2


$- ('<'Q^Ta


x'iP x'JP -X 7Jx ,

-x- p -X Op ,

-xip+

x-p {x, P-}+ i 7p i J
x-- {, -7 "


The light-cone little group transformations are generated by


i


which satisfy the SO(9) Lie algebra. To construct its spectrum, we write the
supercharges in terms of eight complex Grassmann variables


Oa 1 ]
Oa t ( ,a + ixa+8)
v2-


-a 1 7 -
O t (x iXa 8)
-2


(5.26)

(5.27)

(5.28)

(5.29)








and


a 1 a 9 a a 1 a
~0-2 9,a O7^ V,2 a .s a 0 X a+8

where a = 0, 1, 2,..., 7. The eight complex 0 transform as the (4, 2), and 0 as the
(4, 2) of the SU(4) x SU(2) subgroup of SO(9). The eight complex supercharges


(Qa + iQa+s8)

1 +
2 -2 i G oj8 )


a 1
_ + _p+O
a 1
+ p+e
0" v/2


{Q, Q }t 2p+ 6a3

To reduce the number of the grassmann variables, the usual way is to impose
covariant derivatives as constraints,

a- t + 0, (5.32)

since {D", Q~} =0. It follows that 0/00a can be replaced by l/2p+O", when
acting on the constraint fields depends only on 0 0 can also be taken to be zero.
Therefore,

Qa -Qa+ = 2 (5.33)

QI + iQ+s = 2p+ (5.34)


This gives,


where Oa = /cO0', and r-
which are annihilated by t


(


1 ( =) (5.
V/2 O V/20ap+)

S[7, 7j]. They act irreducibly on chiral superfields
he covariant derivatives


a


satisfy


(5.30)

(5.31)


Qat


1- poa) (y,o)









where
ie0
y =x 2


5.2.2 Representations of Grassman Variables

SO(2n+ 1) representation in Dynkin basis[Hij, E(I_j), E(I)] can be constructed

from its standard f, ii [ _11 ;] [79]. Here


., [ }=[ i(iMj + J -, iMjk 6jk 3.), (5.36)

and


Hi = i.1,21, (5.37)

E(I) = If.- 1,2n+l + il(I)- 11,2n+1, (5.38)

E(I-J) = 21(I+J) -[-I-1,2J-1 + i(I)-1-,2J-1

-iT(J) 1-I-1,2J + TI(I)TI(J )-V1 ,2J), (5.39)









Or conversely,


/-1-11,21


-l,2n+l
-1 1-,2n+l

- I-1,2J-1



-I_1,2J



3 _1,2J-1



_1-1,2J


HI,

S(E(1) + E_(1)),
2
2 (T) (E(1) E_(1)),

2(T(I+J)E(I-J) (I+j)E-(I-j)

+](I- J)E(I+j) 7](i_j)E_(i+j)),
i
2- 1(I)1 (J))(1(l+J)E(I_-j) T(1+J)E-(I-J)

--](I-J)E(I+j) + r](I_-)E_(I+J)),
1
27 I(I)(T1(I+J)E(I-J) + 17(I+J)E-(I-J)

+T1(I-J)E(I+J) + 7(I-J)E- (I+J)),
1
27T(J) (T1(I+j)E(I-_) + 17(I+j)E-(IJ)

-rl(-J)E(I +j) rl(-_j)E_ (i+J)) ,


(5.40)

(5.41)

(5.42)



(5.43)



(5.44)



(5.45)



(5.46)


where I, J =1, t n and i =1, t 2n + 1 (n 4) for SO(9).

Consider spinor representation (16) of SO(9),


2-777j .


(5.47)


Here (a) represent 16 indices, are labeled by four + or signs, and -(a) means

that all signs are flipped. They can all be switched to indices 0,..., 15 through

binary counting. For example, if (a) = (+ + +-) = (0001) = 1, then -(a)

(- -+) = (1110) = 14. 7s are antisymmetric real matrices. A (16) spinor

representation (5.47) can be naturally expressed in 16 Fermionic oscillators + or

8 complex 0 (5.35),

Sij = Q Q T (5.48)
4ZZ, p*I


3 -/, (b) i (f(),(b)
2 fjl









A special choice of 7's, for the reason which will become clear soon, is


l = 3 X O X l1 3; 72 1 1= 9i X i X 3 X 1;

73 a3 X 1 X 3 X ; 74 9i x 3X 1 X ai;

75 = 3 X 3 X X 1; 76 1 X I X 3;

77 =2 X 02 X 02 x 02; 78 1 x -X1 X1 1;

79 -1 x 03 x 03 x 03; (5.49)

Now explicit forms of Cartan generators and raising and lowering operators can

be derived from Eqn. (5.48) (Eqn. (5.36) and below). Matrix elements of r are
calculated with a C++ program di- p1 i1 in appendix.
The cartan generators are

S( = O-0Oo+O1ol+O22- o 3 o-o4+o 55+o6a6- o77) (5.50)
S34 -0000 o 11 + 022 + 0303 3+ 044 + 0505 66 6- 077) (5.51)
2

S56 = -0000 11 02 2 + 033 + 0404 -55 66 6- 0707) (5.52)
2


S78 -(-0000o + 8101 + 0202 0303 + 0404 050 -5 6 + 0707) (5.53)

The raising operators corresponding to the simple roots are

S(1-2) 03 6 + 041, S(2-3) =012 + 065, S(2+3) -( 0003 +074),

S(4) -007 + 004 + 052 + 060, S(3-4) I --/ 36 + 2Vp+0O27). (5.54)
2p+
where S12, S34, S56, S(1-2), S(2-3) and S(2+3) belong to SU(4), STs and S(4) belong
to SU(2), and S(3-4) mix SU(4) and SU(2) representations. S(2+3), S(1-2), S(2-3)
correspond to the simple roots of SU(4), S(4) corresponds to the simple root of
SU(2), and S(I_2), S(2-3), S(3-4), S(4) correspond to the simple roots of SO(9). Also








the lowering operators are

5-(1-2)- 06 3 + 01'4, S-(2-3) 0291 + 056, 5-(2+3) -(030 + 04a7),

1
-_(4) 0700 + 403 + 0205 + 0l06, S_(3-4) a7 -- 9+2 + Vp+0306). (5.55)
V2p+
Please note that each 0" is an eigenstate of the cartan generators S12, S34, S56 and
S78. Take the highest state to be 0, then acting the lowering operators on it, 0a's
could be easily identified with the states of (4, 2) in SU(4) x SU(2) with dynkin
labels, represented in (al, a2, a3) x a4, where (al, a2, a3) and a4 are dynkin labels for
SU(4) and SU(2) respectively.

00 (,0, 0) x 07 ~ (1,0, 0) x -1, (5.56)

03 ~ (-1,1,0) x l, 4 ~ ( -1,1,0) x -1, (5.57)

06 ~ (0, -1,1) x 1, 01 (0, 1,1) x -1, (5.58)

05 ~(0,0,-1) x 1, 02 (0,0, -1) x -1, (5.59)

Alternatively, we can use the weight space representation for 0", and for the raising
and lowering operators, expressed in eigenvalues of S12, 534, 556, 578.



00 ~- (eie2 + e3 + e4), 7 (e1 + 62 + 3 4), (5.60)
2
03 ~ (~ 2 3 + 4), (1 62 C3 4), (5.61)

06 (-I + 2 -3+4), (-1 + C62 63 4), (5.62)

S (-i 62+63 4), 0264), (5.63)
05 I -+C+(--)l -- C2 + C3 e4), (5.63)
2











S(2+3) (2,-1, 0) x 0 (e2 + 3), (5.64)

S(1-2) (-, 2,-) x 0 (e- 62), (5.65)

S(2-3) (0, -1, 2) x 0 (62 e3), (5.66)

S(4) (0,0, 0) x 2 (64) (5.67)

To calculate the above formulas, first identify 0 with the highest weight state,

(1, 0, 0) x 1 in dynkin labels, i.e. (1, 0, 0) in SO(6), and 1 in SO(3). since 80 state
has the highest eigenvalues in terms of S12 34 S56 and S7s, then by acting the

lowering operators of SO(6) and SO(3) ,(S-(2+3), S-(1-2), S-(2-3) for SO(6) and

S-(4) for SO(3)) other 0's can also be identified with the dynkin labeled states.
Expansion of the superfield in powers of the eight complex O's yields 256

components, with the following SU(4) x SU(2) properties

1 ~ (1,1), (5.68)

0 ~ (4,2) (5.69)

00 ~ (6,3) (10,1), (5.70)

000 ~ (20, 2) (4, 4) (5.71)

0000 ~ (15,3) @ (1,5) (20', 1) (5.72)

and the higher powers yield the conjugate representations by duality. These make

up the three SO(9) representations of N = 1 supergravity

44 (1,5) (6,3) (20',1) (1,1), (5.73)

84 (15, 3) (10, 1) (10, 1) (6, 3) (1,1), (5.74)

128 = (20, 2) (20, 2) (4, 4) (4, 4) (4, 2) (4, 2) (5.75)









with the highest weights

44 00030407 (0,2,0) x 0 (20' 1) (5.76)

84 o007 =(2,0,0) x0 (10, 1) (5.77)

128 : 00307 = (1,1,0) x 1 ~ (20,2) (5.78)

together with their SU(4) x SU(2) properties. All other states are generated by

acting on these highest weight states with the lowering operators. The highest

weight chiral superfield that describes N = 1 supergravity in eleven dimensions is

simply

S- 0007 h(y-,7 ) + 000307Q(y,7) + 00030407A(y-,7X ,

which summarizes the spectrum of the super-Poincar6 algebra in eleven dimensions

of either a free field theory or a free superparticle. All other states are obtained by

applying the SO(9) lowering operators.

5.2.3 F4/SO(9) Oscillator and Differential form Representations

It turns out that all representations of the exceptional group F4 are generated

by three (not four [77]) sets of oscillators transforming as 26.

Label each copy of 26 oscillators as A ], A [], i = 1,- ,9, BRI a =

0, ,15, and their hermitian conjugates, and where = 1, 2, 3. Under SO(9),

the A [] transform as 9, Bi] transform as 16, and Ah] is a scalar. They satisfy the

commutation relations of ordinary harmonic oscillators



[A1-], A[('] t= b6 o l['] [A["] A[-] t = 6 l "']

Note that the SO(9) spinor operators satisfy Bose-like commutation relations


[ bo Bji t] = bo6 [b









The generators Ti and Ta

4
T { (-AtA A-]t]A +] 1B[lt iB } (5.79)
4
To] ( )ab A IMtBD Bl tA
K1=l
-3 B(l]tA] A]tB']) } (5.80)


satisfy the F4 algebra,


[Tij,Tk i(jk Til + il Tj 6ik Tj jl Tk) (5.81)

[ Ti, T] = (j)ab Tb (5.82)

[Ta,Tbl 2~ (7j)b ij, (5.83)

so that the structure constants are given by

1
fijab abij 2 (7ij)ab

The last commutator requires the Fierz-derived identity

1
t07iJOXYX = 30X XOO+07X XO ,
4

from which we deduce

1
3 6c6db + (yi)ac (7i)db (a +- b) -= (7j)ab (ij )cd
4

To satisfy these commutation relations, we have required both Ao and Ba to obey

Bose commutation relations (Curiously, if both are anticommuting, the F4 algebra

is still satisfied). One can just as easily use a coordinate representation of the

oscillators by introducing real coordinates xi which transform as transverse space









vectors, xo as scalars, and y, as space spinors which satisfy Bose commutation rules

1 1
A, (x+ ), A 1 (xa- ) (5.84)
1 1
Ba (a + ) Bt (y a) (5.85)
1 1
Ao (xo + o) A (X ao) (5.86)

From now on, let us use square brackets [ .. ] to represent the dynkin label of F4,
and round brackets ( ... ) to represent the dynkin label of SO(9), In the weight
spaces of the cartan generators, (eigenvalues of T12, T34, T56, T78), the raising
operators correspond to the simple roots of F4 are


T(2-3) ~ [2 100] ~ (e2 63), (5.87)

T(3-4) [-12- 20] (e3- 64), (5.88)

T(4) [0- 12 1] (e4), (5.89)
1 1
T,- (T + iT12) [00- 12] -(ei e2- 3 e 64). (5.90)
V2 2

where T(2-3), T(3-4),T(4) are defined by Eqn.(5.39) for SO(9) generators as usual,
and T, represents the F4 simple root raising operator transform as a spinor under
the 5'O(9) subgroup, and also T_, -(T4 iT12) will be used to represent the
lowering operator. Also in the same space, the raising operators correspond to the
simple roots of SO(9) are


T(1-2) (2 100) (ei 2), (5.91)

T(2-3) (-12- 10) (e2 63), (5.92)

T(3-4) (0- 12 2) (e3 4), (5.93)

T(4) (00 12) (e4). (5.94)









The weight states are


xI + ix2 ~ [0001] (1000)

X3 + ix [100- 1] (-1100)

yo + :,I. ~ [001 1] ~ (0001)

Y7 + .:',i-. ~ [01 10] (001 1)

2 .,,, [1 110] (01- 11)

Y5 .:,i. [10- 11] ~ (010- 1)


~ (e1),

~ (e2),
1
S-(e + + e 3 + 4),
2
1
(eI + e+2 + 3 e4),
2


~ (ei + 62
2
1
S-(e + e2
2


63 + 64),

63 64),


5.2.4 Solution of Kostant Equation in F4/SO(9)

Define Cliford algebra over 16-dimensional coset F4/SO(9),


{Fa b} 2 ab a,b 0,1,...,15 ,


generated by (256 x 256) matrices. The Kostant equation is defined as


16

a-1


where Ta are F4 generators not in SO(9), with commutation relations


[T, Tb] ifbijTij (5.1(


Although it is taken over a compact manifold, it has non-trivial solutions. To see

this, we rewrite its square as the difference of positive definite quantities,


S- C Co(9) + 72 (5.1(


where


12
C 2- Tu" TiJ + Ta Ta
F4 2


(5.95)

(5.96)

(5.97)

(5.98)

(5.99)

(5.100)


(5.101)


(5.102)


)3)


)4)


(5.105)









is the F4 quadratic Casimir operator, and


0o(9) -2

is the quadratic Casimir for the sum



Where
where


i f bij ab)


(5.106)


7'3 + Si


(5.107)


Sc"ij = abpapb8
8


(5.108)


is SO(9) generator (5.48), which acts on the supergravity fields. The quadratic

Casimir on the spinor representation is


1
2 Si" Si"
2


72 ,


(5.109)


Kostant's operator commutes with the sum of the generators,


[}, L" ] = 0,


(5.110)


allowing its solutions to be labeled by SO(9) quantum numbers.

Since the little group generators Sij act on a 256-dimensional space, they can

be expressed in terms of sixteen (256 x 256) matrices, F", which satisfy the Dirac

algebra


{ p b }


26ab


This leads to an elegant representation of the SO(9) generators


Sij (j) yab pa pab
4


Sfijaba pb
2


,which can be identified with Eqn. (5.48), considering the replacement


Fa 2 2
'Qa


(5.111)









The coefficients

fijab = (ij)ab

naturally appear in the commutator between the generators of SO(9) and any

spinor operator T", as

[TU, T 3 (7j ra if ijab b


But there is more to it, the (7ij)ab can also be viewed as structure constants of

a Lie algebra. Manifestly antisymmetric under a -> b, they can appear in the

commutator of two spinors into the SO(9) generators

[Ta, Tb] (yij )abi fabiij
2

and one easily checks that they satisfy the Jacobi identities. Remarkably, the 52

operators Tij and T" generate the exceptional Lie algebra F4, showing explicitly

how an exceptional Lie algebra appears in the light-cone formulation of supergrav-

ity in eleven dimensions.

For Kostant solution T = O(O)f(x, y)

pa"T"q (F"a(O))(Tf(x, y)) = 0, (5.112)

Therefore

FPa(0) = Taf(x, y) 0. (5.113)

since [Lj, Faa] = 0, 0(0) and f(x, y) are both the highest weight states of the

SO(9) algebra, Si and Tij, and


S ((0) = Tijf(x, y) =0 (5.114)

To find the solution for the Kostant eqn., we have to choose the specific

representation, (5.101) and (5.79),(5.80), for the generators, and therefore the

solution is formed by the states of this representation.









Let us first show some useful relations based on the Dynkin diagram of 26 in

F4, shown at the end of the chapter, where the numbers 1, 2, 3, 4 near the arrow

represent the lowering operators T-(2-3), T-(3-4), T-(4), T-_ respectively. Using the

explicit formula, Eqn.(5.79), (5.80), (5.39) and (5.90), for those lowering operators,

the calculation shows,


T_,I(x + ix2)

T-(4)i(yo + .:1'.)

T-(3-4) i(7 + :, .)



T-(4) (Y2 .',,,)

T-q(-ys + .'i .)


1 ((7 )4b+ )4b (71i)12b + (72)12b) (yo .),

2 ((779)a0 7 i(1)a8 (89)aO (789a8) a = ( .,, .)j

- (()a7 7 757 )al5 5 67 ar7 + (767) al55

i(758)a7 (758)a15 + (768)a7 + ('768)a15) Ya 2 2 i/I

S ((79)a2 (779)al0 89)a2 (789g)al) Ya 5 .*

S((7i)45 i(7)4'13 i( )12,5 (7)12,13) Xi 3 + i4)


where we keep the coefficient of the states for the later antisymmetric construction

of the highest weights.

To verify the solutions for the Kostant equation, we need to identify the

generators,


T4 + iT12

T3 + iT1


v2T, ,


iT12

iT11


-V2[T-(4), T-


and also from eqn. (5.101), Kostant operator can be rewrite as,


FaTa = i [a0(T, + iT+8s) O(Ta iT+s)] .


(5.115)


The above explicit form shows that Kostant operators are just composed of

raising and lowering operators constructed by Ta. Since we only need to verify the

highest weight solutions, only the lowering operators are needed to be taken into

consideration. Explicit calculation shows that when acting the Kostant operator









on the highest weight solution, most of the terms will vanish due to 0, and 0", and

only few lowering operators (-(T3 T11) -L(T iT12)) need to be considered.

The solutions of kostant's equation form SO(9) triplets. For every repre-

sentation of F4, in dynkin label, [al, a2, a3, a4], there is a SO(9) triplet solution

associated [72],

(2+a2+a3+a4, a,, a2, a3) (a2, a,, 1+a2+a3, a4)(l1+a2+a3, a,, a2, 1+a3+a4) (5.116)

Now let us parameterize the F" by 08, and T0 by x, y, with Eqn.(5.101)

and (5.79). Kostant equation FPT" = 0 will have the solution in the form

(0, x,y) = (0)f(x, y).
The solution in the first level is when a =- a2 = 3 a= 4 = 0, (2000) (0010)

(1001) or (44) (84) (128). The highest weight solution is 0030407, 0007 and

000307 found before.

To find solutions in the higher level, notice two things,

1. The solutions are in the form of 0(0)f(x, y), where 0(0) 0003047, 0007 or

000307. 0(0) and f(x, y) are both the highest weights of the SO(9) subgroup

formed by generators, Lj = Sij + Tyi, eqn(5.2.4) and (5.79).

2. For the fundamental representation of F4(in dynkin label, al,..., a4 are all

zero except one ai 1), suppose the associated solution is 0(0)f(x, y), then

f(x, y) will be formed by the states of the F4 fundamental representation.
Further more, since Ti representations are homogeneous polynomial of x and

y, O(0)f(x, y)" is the solution for higher level(a, > 1).
Using the generalized form of the triplet solutions(5.116), the highest weight

solutions correspond to each fundamental representation of F4 are constructed as

follows:

1. al = a2 a= 3 = a4 1









F4 states are K = 1 t" ,v of 26 states. The highest weight solutions are

00030407(Xi + iX2), (5.117)

0007(yo + :,,.), (5.118)

000307(yo + .:1), (5.119)

where (xl + ix2) and (yo + :,/.) are the highest weight states of SO(9)
representations (1000) and (0001) respectively, and they are also the states
belong to the 26 of F4. Direct counting of the Dynkin label shows the above
solution is consistent with the general form (5.116). To verify the solution,
one need to use the properties of lowering simple root generators to traverse
through weight states,

FaT, (00030407(Xi + iX2))

p (9(T, + iT+s) a(T, iT+s)) (003047(X + 2)) = 0(5.120)


raT, (007(yo + :.))

(03(T iT11) + 04(T~ iT 1)) (007(yo + :,.))

S03 (_ -V2)[T-(4),T_ ,+O V2T_ ,) (007( +,.)) 0 .(5.121)

2. a2 = a3 = a4 = 0, a = 1
F4 states are represented by antisymmetric products of K = 2 copies of 26
states. From the general form of the solution, (5.116), we need to represent
the SO(9) highest weight state (0100) of 36 by the antisymmetric products
of the two copies of 26 of F4. This state is also the highest weight state
of 52 of F4. Since (26 x 26)a = 52 + 273, To form this state, use the
sixth highest weight states of 26, antisymmetrize the first and the sixth,
the second and the fourth, the third and the fourth, then choose proper








coefficients to combine them. This highest weight state of SO(9) is found is
to be ([xi + ix2,X3 + ix] + [Yo + .:I,- Y5- ., + [7 + :I- 12 -. ). It is
annihilated by all the simple roots raising operators of SO(9) and F4. The
highest weight solutions are

00030407([x1 + i2,X3 + ix4 + [0O + :'/- Y5 +I [i. + [ .2 n

0007([x1 + i 2, X3 + ix ] + [O + 5:'i., 5 *'i .] + [iY + :- 2 1),

000307([XI + ix2, 3 + i4] + [y0 + :'/. 5 i + [Y7 +: 2 :''n)

(5.122)

where ([xi + ix2, X3 + ix41 + [yo + :'- 5 : ]+ [Y7 + ,:'/ 12 )is
the highest weight of the SO(9) representation (0100), and here we denote

[a, b] = a[1]b[2] a[2]b[, antisymmetric product of 2 copies of a and b states.
The verification of this solution is similar to the previous case. For example,


aTa (o000([Xl + [i ix2, + [Xo I + :,,, + 5 .':,,, + [17 + .:,,-, 1 2
-i (03(T3 iT) + 04(T4 T2))

(0007([Xi + ix2,x3 + I4I + [YO + :',, :5 +' [Y7 + :'11-., Y2


(0007([ri + iz2,x3 + ix4] + [YO + : I Y + [Y7 + :'11-., 2
-i (-v2 3o007([i(7 + ;:,X3 + ix4] + [7 + :.'/i:.,-i(x- + iJ)])

+v/2040007([i(o + .'.),^ + i] + [4o + :',. 0i(3 + i 4)]))


(- 5.' 1)






(5.123)


3. al = a2 = a4 = 0, a2 = 1









F4 states, (273), are also represented by antisymmetric products of K = 2

copies of 26 states. The highest weight solutions are

00030407[x1 + i, Yo + :.], (5.124)

0007[yo + :'., Y7 + :',1], (5.125)

00307[X1 + i2, Yo + 1.], (5.126)

where [xi + ix2, Yo + .:/-] and [yo + .:,- Y7 + .:,/, .] are the highest weights of the

S0(9) representations (1001) C (16 x 9) and (0010) C (16 x 16)a respectively.

[xi + ix2, Yo + .:/-] is also the highest weight of 273 of F4.
4. a = a3 = a4 = 0, a2 1

F4 representation (1274) can be represented by Kronecker products of

S= 3 copies of 26 states. The highest weight state is simply the total

antisymmetrization of the highest three states in 26. The highest weight

solutions are

803407[I + X2 Yo + .:, Y7 + :, .], (5.127)

0007[Xl +ix2,yo + :'., y7 + .:,' ], (5.128)

000307[I + ix2,yo + :- Y7 + .], (5.129)

where [xi + ix2, Yo + .:I- Y7 + .:',1,.] is the highest weight of the S0(9)

representation 1010, and [a, b, c] is the antisymmetric products of 3 copies of

a, b and c states. It is also the highest weight state of 1274 of F4.















CHAPTER 6
SUMMARY

This dissertation includes two loosely connected parts. The main focus is

quantum aspects of NC field theories, including both perturbative and nonper-

turbative structures. In particular, perturbative behavior of NC supersymmetric

Wess-Zumino model is discussed in detail. It is shown that NCWZ model has

only wave function renormalization and UV finite as its commutative analog. It

is -i .--.- -i. ,l supersymmetric invariance of NCWZ model again leads to cancela-

tion which renders mass and vertex corrections UV finite. UV/IR mixing terms,

as a result of phase factors induced in the vertex, generally exist in all quantum

perturbation calculations. NC solitons, nonperturbative structure in NC field

theory, are interpreted as low energy manifestation of lower dimensional D-branes

in string theory. Through quantization of NC solitons, corrections to the energy

are calculated in detail. Energy of NC GMS solitons is found to be UV finite, and

also includes UV/IR mixing terms, which need not be surprising considering their

general existence in perturbation theory. UV/IR mixing terms in perturbative

theory are -ir-i.- -1- I as results of particles traveling in extra dimensions, which

in the context of string theory, are interpreted as low energy closed string modes

dual to high energy open string modes living on the brane. Existence of UV/IR

mixing terms in NC soliton energy r-tI--.- -1; these modes also interact with lower

dimensional D-branes. Properties of NC scalar solitons already make the UV/IR

terms an intriguing subject. Some NC solitons (GMS solitons) exist only at large

enough 0, and quantum corrections of which do not include UV/IR terms near

infinite 0 limit. There are many questions ready to be answered. How do UV/IR

terms affect the stability of NC solitons? Is there any interpretation from string









theory that such terms, which reveal the structure of string diagrams, cease to exist

at very large 0? What new insight can those UV/IR terms give with regard to the

interaction between D-brane and closed strings?

SuperPoincar4 symmetry is considered as basic space-time symmetry of fun-

damental theory. NC field theories, as low energy limit of string theory, explicitly

violate Lorentz symmetry. Thus it becomes important to understand the space-

time symmetries on which NC field theories are constructed. A representation of

deformed superPoincar4 algebra is obtained and commutation relations are cal-

culated in an intuitive way. Preservative of supersymmetry supports the attempt

to construct supersymmetric NC field theory directly from supersymmetric gen-

eralization of NC space. The presence of the B field on the boundary of D-brane

enables decoupling of low energy modes of string theory in certain limit, but also

yields noncommutativity explicitly broken Lorentz symmetry. If field theories are

fundamentally NC, there will be a very small upper bound of the NC parameter,

since in our space field theory seems to be Lorentz invariant in high precision. An

alternative explanation is to take into consideration the existence of a B field.

Indeed covariance of the theory is easily justified if the NC parameters are taken

to be 0^", where the indices transform accordingly under Lorentz rotation. Sym-

metry, nonlocality, causality and unitarity will continue to be important issues in

identifying NC field theories as realistic theories.

Solution of Kostant equation, as well as supersymmetry algebra representation

in 11 dimension, is considered as an attempt to construct zero slope limits of

string theory, if we believe they obey superPoincar6 symmetry and reduce to 11

dimension supergravity in low energy limit. The appearance of an infinite tower of

triplets is interesting and expected but construction of Lagrangian and interactions

for those multiplets still needs further work.
















APPENDIX
COMPUTER CODE

This program is designed to output matrix elements of 7i and 7ij (5.49).

//Define a complex number class and a sigma class,

//and a gamma class and a F_bar class.



#include

using namespace std;



#ifdef _MSC_VER

class F_bar;

stream& operator streamrem &, const F_bar &);

#endif



#ifdef _MSC_VER

class sComplex;

stream& operator streamrem &, const sComplex &);

#endif



//define a class sComplex which represents the complex numbers either

//imaginary or real.

class sComplex

{

public:

int value;