Topological anomalies from the path integral measure in supe(3)

A fully quantum version of the Witten-Olive analysis of the central charge in the N=1 Wess-Zumino model in $d=2$ with a kink solution is presented by using path integrals in superspace. We regulate the Jacobians with heat kernels in superspace, and obtain

analysiswasmadeofordinaryandconformalmultipletsofcurrents,andordinaryandconformalmultipletsofanomalies;inparticular,aconformalcentralchargecurrentwasidenti edwhosedivergencecontained,inadditiontotermsduetoexplicitsymmetrybreaking,theanomalyinthecentralcharge.Thusthecentralchargecontainsananomalyandisitselftheanomalyofanothercurrent.

Inthisarticle,weintendtostudytheanomalystructureofthecurrentsofthesuper-symmetrickinkmodelinsuperspace.WeusethepathintegralformulationofanomalousWardidentities[12,13],andthepresentworkextendsasuperspaceanalysisofconformalanomaliesinQCDin3+1dimensions[14][15].Asuperspaceapproachtotheanomaliesinthecentralchargeandthekinkenergywas rstgivenin[8].Inthatarticleseveralregularizationschemeswereusedtoevaluatetheone-loopcorrections,inparticularahigherderivativeregularizationschemeinsuperspace.Weshallstartwithapathintegralapproach,andcomputetheJacobiansforgeneralizedsupersymmetrytransformationsus-ingasuperspaceheatkernel.Thiswillleadtoamultipletofanomalieswhichcontainsthetraceanomalyandthecentralchargeanomalyinadditiontothesupersymmetryanomalyandotherterms.Crucialinthisapproachisthatcarefulregularizationoftermsproportionaltothe eldequationoftheauxiliary eldyieldsnonvanishingcontributionstotheWardidentities.Intheliterature,somearticlesdealdirectlywiththecentralchargecurrentwhileinsomeotherarticlesthecentralchargeanomalyisobtainedfromasupersymmetrytransformationoftheconformalanomalyinthesupercurrent.WeshallderivethecentralchargeanomalybothdirectlybyevaluatingtheJacobian,andbyasupersymmetrytransformationoftheconformalanomalyinthesupercurrent,andshowthattheresultsagreewitheachother.

Webeginwithalocal(xandθdependent)supersymmetrytransformationofthescalarsuper eldφ(x,θ)ingthequadraticpartofthesuperspaceactionasregulator,weobtainaWardidentityinsuperspace(cor-respondingtoahierarchyofWardidentitiesinx-space)whichcontainstheone-loop µ(x),inadditiontoexplicitsym- µ(x),T µν(x),ζanomaliesincertainquantumcurrentsJ

metrybreakingterms.Theuseofasuper eldformulationofheatkernelsinstrictlyd=2Minkowskispace-timetoregularizetheJacobiansinthepathintegralformulationmani-festlypreservesordinaryrigidsupersymmetryatallstages.Asubtlepointistheproperidenti cationofthequantumcurrents.ThenaiveNoethercurrentsarenotconserved, (x)µwhichareconserved,so µ(x),T µν(x),ζbuttheWardidentitiescontainthecurrentsJ

thesecanbeusedtoconstructtime-independentcharges.Ifweweretosubstitutethe eldequationsintheseconservedquantumcurrents,wewouldobtaintheNoethercurrents,butthisisnotallowedatthequantumlevel.Infact,thecontributionsproportionalto eldequationsproduceanomalies,aswealreadymentioned.

Wealsoderivethesupersymmetryalgebraatthequantumlevel.Thisrequiresafullquantumoperatorapproachwhichincorporatesanomalies,ratherthanasemiclassi-calapproachbasedonDiracbrackets.Fortunatelythereexistssuchanapproach:theBjorken-Johnson-Low(BJL)methodwhichautomaticallyincorporatesthee ectsofsu-perconformalanomalies.TheBJLmethodhasbeenwidelyusedforcurrentalgebrasinthe1960’s[16].Itallowsonetorewriteresultsobtainedfrompathintegralsintooperatorrelations.Togofromthepathintegralresultstooperatorresultsonemustuse eld