The Dual Variables Of Yang-Mills Theory And Local

arXiv:hep-th/9507036v327Aug1995TAUP-2274-95hep-th/9507036July6,1995The“Dual”VariablesOfYang-MillsTheoryAndLocalGaugeInvariantVariables.O.GanorandJ.SonnenscheinSchoolofPhysicsandAstronomyBeverlyandRaymondSacklerFacultyofExactSciencesTel-AvivUniversityRamatAvivTel-Aviv,69987,IsraelAbstractAfteraddingauxiliaryfieldsandintegratingouttheoriginalvari-ables,theYang-Millsactioncanbeexpressedintermsoflocalgaugeinvariantvariables.ThismethodreproducestheknownsolutionofthetwodimensionalSU(N)theory.Inmorethantwodimensionstheactionsplitsintoatopologicalpartandapartproportionaltoαs.WedemonstratetheprocedureforSU(2)inthreedimensionswherewereproduceagravity-liketheory.Wediscussthefourdimensionalcaseaswell.Weuseacubicexpressioninthefieldsasaspace-timemetrictoobtainacovariantLagrangian.Wealsoshowhowthefour-dimensionalSU(2)theorycanbeexpressedintermsofalocalactionwithsixdegreesoffreedomonly.Contents1Introduction12Reformulationofthetheory33Rederivingthe2DpartitionfunctionforSU(N)onatorus73.1TheSU(2)partitionfunctiononatorus............73.1.1Theproblemnearg=0.................93.2GeneralizationtoSU(N).....................114ThreedimensionalSU(2)144.1Theaction.............................144.2GoingroundΔ=0........................154.3Thefullfunctionalintegral....................175FourdimensionalSU(2)185.1Algebraicidentities........................185.2Gaugeinvariantvariables.....................195.3Reductiontoconformalmetrics.................206Discussion211IntroductionOneofthefascinatingpropertiesofcertainquantumfieldtheoriesistheexistenceofastrong-weakdualityofthecouplingconstant[1].Recently,variousnoveldualitieswerediscoveredbothinthecontextofsupersymmetricgaugetheories[5,6]andtopologicalfieldtheories[2]aswellasinstringtheories[7].Incertaintheories,likethecompactifiedbosonin2Dortheabeliangaugetheoryin4D,thedualitytransformationswereperformedbyaddingauxiliaryfieldsandintegratingouttheoriginalvariables[3,4,5].Inthispaperweexplorethisprocedureinnon-Abeliannon-supersymmetricYang-Millstheories.Itiswellknownthatthelatterdonotpossessastrong-weakdualityinvariance.(Infactsuchadualityisusuallymeaninglesswhenthecouplingconstantisrunning.)However,theexcitingexactresultsde-1rivedrecently[5]taughtusthatthestudyoftheresultingactionintermsoftheauxiliaryfieldsmaybeveryfruitful.Aftertheintegrationovertheoriginalvariables,theresultingactioncanbeexpressedintermsoflocalgaugeinvariantvariables.HavingagaugeinvariantdescriptioncouldbeimportantforthelargeNlimitofSU(N)Yang-Millstheory.Thecorrelationfunctionsofpropergaugeinvariantvari-ablesvanishas1/N2andthusthosevariablescanbeconsideredclassical(Obviously,noteverygaugeinvariantvariablehasthisproperty).Thisistheessenceoftheoriginalmaster-fieldidea[8].IntheapproachofMigdalandMakeenko[9,10]thegaugeinvariantvariablesarethenon-localWil-sonloops.Morerecently,GrossandGopakumar[11]suggestedadifferentapproachwherethemasterfieldislocalbutnotgaugeinvariant.Thismas-terfieldcorrespondedtothegaugefieldbutasanon-commutativerandomvariable.InthepresentpaperweobtainactionswithlocalgaugeinvariantvariablesfortheSU(2)theory.Intwo-dimensionsweobtain,bytheaboveprocedure,theSU(N)parti-tionfunctiononatorus.Thiscorrespondstothewell-knownresult[17][18]whichisgivenasasumoverrepresentationsofSU(N).Inourcase,eachrepresentationcorrespondstoadifferentconfigurationoftheauxiliaryfield(whosevaluebecomesquantized).Inmorethantwodimensionstheactionthatweobtainsplitsintoatopologicalfieldtheoryplusatermproportionaltothecouplingconstantαs.Thetopologicalfieldtheorydescribestheflatgaugeconfigurationsatαs=0.WedemonstratethisbyreproducingLunev’sresult[24]forthethreedimensionalSU(2)theory.Itisexpressedasatheorysimilarto3Dgravity(thetopologicalpart)plusanon-covariantcouplingproportionaltoαs.Thetopologicalfieldtheoryofflatgaugeconnectionswasintroducedin[19]inrelationto2Dtopologicalgravity.In[20]theoriesofflatgaugecon-nectionsofdifferentgroupsandforD2wherewrittendown.The2Dcaseswerethendiscussedin[21][22].In[22]theexactinstantonexpansionofthe2Dpartitionfunctionwasobtainedbyexpressingtheactionasatopologicalperturbation(proportionaltoαs)totheflatgaugeconnectiontheory.Thetopologicaltheoryofflatgaugeconnectionsin4Dwasrecentlydiscussedin[23]inrelationtoatwistingofthesuper-symmetricN=4Yang-Millstheory.Continuingto4Dingaugeinvariantvariables,weobtainanexpression2fortheSU(2)actionintermsofa(non-positive)metricgμνandachiralspin-2field.WethenshowthatwecanrestricttoconformalmetricsandthusobtainadescriptionofSU(2)in4Dintermsofalocalactionofsixgaugeinvariantfieldsonly.Historically,ourprocedurewaspioneeredbyHalpern[12,13,14].Inpar-ticular,ourstartingpointisHalpern’s1977fieldstrengthformulation[12]ofYang-Mills.Halpernalsousedthisformulationtoobtainagaugeinvariantformulation[13]oftheself-dualsectorofYang-Mills,whichisaprototypeofourgaugeinvariantformulationofthefulltheory.Thefieldstrengthformu-lationwasalsoemployedbyFradkinandTseytlin[16]In[24]thegravitationaldescriptionof3DSU(2)wasfound.In[25]agravity-liketheoryof3DSU(2)wasfound,butitseemsdifferentfromourdescription.Inarecentwork[26]morerelati