Universal Aspects of Two-Dimensional Yang-Mills Th

arXiv:hep-th/9501099v121Jan1995MIT-CTP#2337BRX-TH-358BERC-94/101hep-th/9501099UniversalAspectsofTwo-DimensionalYang-MillsTheoryatLargeNMichaelCrescimanno†*‡CenterforTheoreticalPhysicsLaboratoryforNuclearScienceandDepartmentofPhysicsMassachusettsInstituteofTechnologyCambridge,Massachusetts02139U.S.A.HowardJ.Schnitzer†DepartmentofPhysicsBrandeisUniversityWaltham,MA02254U.S.A.ABSTRACTWeshowthatthelargeNpartitionfunctionsandWilsonloopobservablesoftwo-dimensionalYang-Millstheoriesadmitauniversalfunctionalformirrespectiveofthegaugegroup.WedemonstratethatU(N)QCD2undergoesalargeN,third-orderphasetransitionontheprojectiveplaneatanarea-couplingproductofπ2/2.Weusethisasalemmatopro-videadirecttranscriptionofthepartitionfunctionsandphaseportraitsofYang-MillstheoryfromtheU(N)onRP2atlargeNtotheotherclassicalLiegroupsonS2.WecomputetheexpectationvalueoftheWilsonloopsinthefundamentalrepresentationforSO(N)andSp(N)onthetwosphere.Finallywecomparethestrongandweakcouplinglimitoftheseexpressionswiththosefoundelsewhereintheliterature.*ThisworkissupportedinpartbyfundsprovidedbytheU.S.DepartmentofEnergy(D.O.E.)undercooperativeagreement#DE-AC02-76ER03069andbytheDivisionofAppliedMathematicsoftheU.S.DepartmentofEnergyundercontract#DE-FG02-88ER25065.‡AfterSept.’[e-mail],CPO325,PhysicsDepartment,BereaCol-lege,Berea,KY,40404.†ThisworksupportedinpartbyfundsprovidedbytheU.S.DepartmentofEnergy(D.O.E.)undergrant#DE-FG02-92ER40706.0SECTION1.IntroductionRecentworkbymanyauthorsstronglysuggeststhatpureYang-Millsgaugetheoryintwo-dimensionsisequivalenttoastringtheory1−12.TheinterestinunderstandingsuchasimpletheoryinsuchgreatdetailistoshedsomelightonthestrongcouplinglimitofpureQCDinfourdimensions.Ofcourse,thestringpictureoftwo-dimensionalYang-Millsisinterestinginit’sownrightasatestinggroundfornon-perturbativeanalysesofaquantumfieldtheory.Studyingtwo-dimensionalpureYang-MillsfieldtheoryinthelargeNlimitisthestartingpointformakingthecorrespondencewithstringtheory.Forexample,asshowninRefs.[1,3,4],agaugetheorybasedonSU(N)atlargeNsplitsintotwocopiesofa“chiral”theorythatencapsulatesthegeometryofthestringmaps.ThechiraltheoryassociatedtoYang-Millstheoryonatwo-manifoldMisasumovermapsfromatwo-dimensionalworldsheets(ofarbitrarygenus)tothemanifoldM.Thisleadstoaexpansionin1/Nforthepartitionfunctionandobservablesthatconvergesforallarea-couplingproductfortargetspaceMofgenusoneandgreater.Thestringexpansionbreaksdownforsmallcouplingwhenthetargetspaceisthetwo-sphere,i.e.forYang-MillsonS2.Simplyput,forthespherethereareaninfinitenumberofstringdiagramsthatcontributeto,forexample,theleadingtermoffreeenergy.Thus,thisterm,asformulatedintermsofstringperturbationtheory,mayfailtoconvergeatacertainvalueforthecoupling.ThattheremustindeedbesuchadifficultywiththestringexpansionwasfirstshownbyDouglasandKazakov13(seealsoRef.[14].)Inparticular,theyshowedthatthefreeenergyofQCD2onthesphereundergoesathirdorderphasetransitionatavalueofthecouplingA=Ac=π2.ExtensionsoftheirworkappearinRefs.[15-18]inwhichthisrelevanceofthisphasetransitionwasbetterunderstoodfromthepointofviewoflarge1Ntechniqueanditsapplicationtothe“chiral”sectorofthestringypictureofQCD2wascarriedout.Finally,Boulatov19,andDaul,etal.20havecomputedtheexpectationvalueoftheWilsonloop(inthefundamentalrepresentation)onthesphereusingthelargeNsaddlepointfoundintheworkofRefs.[13,14].InthisnotewepresentthelargeNanalysisforQCD2ontherealprojectiveplaneandcomputethepartitionfunctiononthesphere(andontheprojectiveplane)fortheclassicalLiegroupsSO(N)andSp(N)(byconventionSp(N)hasrankN/2).WealsomodifythetechniquesofRefs.[19,20]tostudytheexpectationvalueoftheWilsonlineinthefundamentalrepresentationforthesegroups.WefindthatatlargeNitispossibletoadapt“technology”developedinRefs.[13-17]totheseothercases.Weweresurprisedbythesimplicityoftheapplicationofthesetechniquesandfindanunderlyinguniversalitythathasimplicationsfortheobservablesandforthestringinterpretationofthetheory22.Insection2wedescribethelargeNanalysisofU(N)QCD2ontherealprojectiveplane.TheseresultsserveasalemmawhichallowsonetocomputethelargeNphasesofthegaugetheoriesSp(N)andSO(N)Theseconclusionsarepresentedinsection3.Insection4wecomputetheWilsonloopinthefundamentalrepresentationonthetwosphereforthesegaugegroups.TherewemakecontactwiththeknownresultsinstrongandweakcouplingandfurtherdescribetheuniversalanalyticformoftheWilsonloopandpartitionfunction.SECTION2:ThePhasesofU(N)QCD2ontheProjectivePlaneAswasdescribedinRefs.[13,14],thepartitionsumforU(N)QCD2onthesphereis,forlargeN,dominatedbyasinglerepresentation.IngeneralthepartitionsumforQCD2isZM=Z[DA]exp(−14g2ZMTr(F∧∗F)).(2.1)2andspecializingtothecasewhereM=S2,(2.1)hasbeenshowntobe*ZS2=XRdim(R)2e−AC2(R)/2N(2.2)whereC2(R)isthequadraticCasimirfortherepresentationRanddim(R)itsdimension.InthelargeNlimititisalwayspossibletowrite(2.2)intermsofa(quantummechanical)pathintegralinacontinuousvariablehthatislinearlyrelatedtotherowlengthsoftheYoungdiagramsforrepresentationsZS2=Z[Dh]e−N2Seff(h).(2.3)ForafixedgaugegroupGatlargeNthepartitionsumtakestheformof