2D Yang-Mills Theory and Topological Field Theory

arXiv:hep-th/9409044v19Sep1994.hep-th/9409044YCTP-P1[e-mail],YaleUniversity,NewHaven,CTContributiontotheProceedingsoftheInternationalCongressofMathematicians1994.WereviewrecentdevelopmentsinthephysicsandmathematicsofYang-Millstheoryintwodimensionalspacetimes.August2,1994;Revised,Sep.8,19941.IntroductionTwo-dimensionalYang-Millstheory(YM2)isoftendismissedasatrivialsystem.Infactitisveryrichmathematicallyandmightbethesourceofsomeimportantlessonsphysically.MathematicallyYM2hasservedasatoolforthestudyofthetopologyofthemodulispacesofflatconnectionsonsurfaces[2,26,39,40].Moreover,recentworkhasshownthatitcontainsmuchinformationaboutthetopologyofHurwitzspaces-modulispacesofcoveringsofsurfacesbysurfaces.Physically,YM2isimportantbecauseitisthefirstexampleofanonabeliangaugetheorywhichcanbereformulatedasastringtheory.Suchareformulationoffersoneofthefewwaysinwhichanalyticresultscouldbeobtainedforstronglycoupledgaugetheo-ries.Motivationsforastringreformulationincludeexperimental“approximateduality”ofstronginteractionamplitudes,weakcouplingexpansions[35],strongcouplingexpansions[38]andloopequations[30].Theevidenceissuggestivebutfarfromconclusive.In[20]D.GrossproposedthesearchforastringformulationofYang-MillstheoryusingtheexactresultsofYM2.Thisprogramhasenjoyedsomesuccess.AsuccessfuloutcomeforYM4wouldhaveprofoundconsequences,bothmathematicalandphysical.InordertodescribethestringinterpretationofYM2properlywewillbeledtoasubjectofbroadersignificance:theconstructionofcohomologicalfieldtheory(CohFT).Thisisreviewedinsection6.2.ExactSolutionofYM2LetΣTbeaclosed2-surfaceequippedwithEuclideanmetric.LetGbeacompactLiegroupwithLiealgebrag,P→ΣTaprincipalG-bundle,G(P)=Aut(P),A(P)=thespaceofconnectionsonP.TheactionforYM2istheG(P)-invariantfunctiononA(P)definedby:IYM=14e2RΣTTr(F∧∗F);F=dA+A2,∗=Hodgedual,e2=gaugecoupling.IYMisequivalenttoatheorywithaction:I(φ,A)=−12RΣTiTr(φF)+12e2μTrφ2;φ∈Ω0(M;g),μ=∗1,andTrisnormalizedasin[39]:18π2TrF2representsthefundamentalclassofH4(B˜G;ZZ),where˜GistheuniversalcoverofG.VariousdefinitionsofthequantumtheorywilldifferbyarenormalizationambiguityΔI=α1RR4π+α2e2Rμ.EquivalencetothetheoryI(φ,A)showsthatYM2isSDiff(ΣT)invariant(nogluons!)andthatamplitudesarefunctionsonlyofthetopologyofΣTande2a,wherea=Rμ.1TheHilbertspaceHGisthespaceofclassfunctionsL2(G)Ad(G)andhasanaturalbasisgivenbyunitaryirreps:HG=⊕RC·|Ri.TheHamiltonianisessentiallythequadraticCasimir:C2+α2.Theamplitudesarenicelysummarizedusingstandardideasfromtopologicalfieldtheory.LetSbethetensorcategoryoforientedsurfaceswitharea:Obj(S)=disjointorientedcircles,Mor(S)=orientedcobordisms,then:Theorem2.1:YM2amplitudesprovidearepresentationofthegeometriccategoryS.Thestateassociatedtothecapofareaais:eα1XRdimRe−e2a(C2(R)+α2)|Ri.ThemorphismassociatedtothetubeisXRe−e2a(C2(R)+α2)|RihR|,andthetrinionwithtwoingoingandoneoutgoingcircleis:e−α1XR(dimR)−1e−e2a(C2(R)+α2)|RihR|⊗hR|.Proof:Theheatkerneldefinesarenormalization-groupinvariantplaquetteaction♠Corollary:OnaclosedorientedsurfaceΣTofareaaandgenuspthepartitionfunctionisZ(e2a,p,G)=eα1(2−2p)XR(dimR)2−2pe−e2a(C2(R)+α2)(2.1)Theseconsiderationsgobackto[29].Aclearexpositionisin[39].3.YM2andthemodulispaceofflatbundlesAte2a=0theactionI(φ,A)definesatopologicalfieldtheory“ofSchwarztype”[8].In[39][40]WittenappliedYM2tothestudyofthetopologyofthespaceofflatG-connectionsonΣT:M≡M(F=0;ΣT,P)={A∈A(P):F(A)=0}/G(P).1Witten’sfirstresultisthat,forappropriatechoiceofα1,ZcomputesthesymplecticvolumeofM[39]:Z(0,p,G)=1#Z(G)ZMexpω=1#Z(G)vol(M)(3.1)1WetakeatopologicallytrivialPforsimplicity.Mthenhassingularities,buttheresultsextendtothecaseoftwistedP,whereMcanbesmooth[2].2whereZ(G)isthecenter,andωisthesymplecticformonMinheritedfromthe2-formonA:ω(δA1,δA2)=14π2RΣTr(δA1∧δA2).TheargumentusesacarefulapplicationofFaddeev-Popovgaugefixingandthetrivialityofanalytictorsiononorientedtwo-surfaces.Theresultextendstotheunorientablecase,andtheconstantα1canbeevaluatedbyadirectcomputationoftheReidemeistertorsion.Accordingto[39],(3.1)isthelargeklimitoftheVerlindeformula[36].LetSRR′(k)bethemodulartransformationmatrixforthecharactersofintegrablehigh-estweightmodulesR∈Pk+oftheaffineLiealgebrag(1)kunderτ→−1/τ[24].Ate2a=0wehave:Z=limk→∞eα1χ(ΣT)PR∈Pk+(S00(k)S0R(k))2p−2where0denotesthebasicrepresentation.Ontheotherhand,wemaychooseacomplexstructureJonΣTinducingaholomorphiclinebundleL→Mwithc1(L)=ω,andapplytheVerlindeformulatoget:limk→∞k−nPPk+(1S0R)2p−2=limk→∞k−ndimH0(ΣT;L⊗k)=limk→∞k−nhekc1(L)TdM,Mi=volM,wheren=12dimM.Using[24]onerecovers(3.1)witheα1=(2π)dimG/(p|P/L|volG)=(Yα02π(α,ρ))/p|P/L|,Pistheweightlattice,Lthelongrootlattice,andρtheWeylvector.ThefactthatthetrinionisdiagonalinthesumoverrepresentationsisthelargeklimitofVerlinde’sdiagonalizationoffusionrules.2Witten’ssecondresult[40]givestheasymptoticsof(2.1)fore2a→0(seta=1):Z(e2,p,G)e2→0∼1#Z(G)ZMeω+ǫΘ+O(e−c/e2)(3.2)e2=2π2ǫ,α2=(ρ,ρ),andcisaconstant.Θ∈H4(M;Q)is-roughly-thecharacteristicclassobtainedfromc2(Q)whereQ→ΣT×Mirr,istheuniversalflatG-bundle.3Θi