Topological anomalies from the path integral measure in supe(4)

µ(x)= [ (x) F(x)]γµψ(x)J

h¯g=jµ

12ηµν[( ρ )( ρ ) FU]

4

h¯g¯ρ ρψ+2g ψψ¯],ηµν[ψγ4¯[γµ ν+γν µ]ψψ+ µν(x)= µ ν 1T

=Tµν(x)+ηµν

2¯g¯µγ5 ψ(x)+hψγ

φ,andthus

yieldsthetraceanomaly.Theparameterl(x)generateslocalLorentztransformations(orchiraltransformationssinceγ5istheLorentzgenerator),buttheLorentztransformationis,ofcourse,anomaly-freeforourvector-likemodel5anditsgeneratorvanishesidentically θα

¯µγ5ψ≡0.ψγ(4.11)

¯ψ,anditleadstotheWardTheparametertα(x)generatesthetransformationδF=t

identitycontainingthegamma-traceofthesupercurrent.

Themostinterestingcasearethetransformationswithcµ(x).Theparametercµ(x)generatesthetransformationsδF= µνcµ ν andδψ=cµγµγ5ψ,andthesetransfor-mationsyieldtheWardidentity

µ(x) cµζ1

2π µν ν (x).(4.12)

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

ThelastterminthisWardidentityisananomaly,andthisanomalyconstitutestheanomalouspartinthecentralchargecurrentitself.Thisisanunusualpointthatmayleadtoconfusion:theanomalyisproportionaltothecentralchargecurrentitself.Inthisrespectitresemblesneitherthetraceanomalynorthechiralanomaly:thetraceanomalycontainsacontractionofthecurrentwhilethechiralanomalycontainsadivergenceofthecurrent.Infact,ithasbeenshownin[10]thatthecentralchargecurrentistheanomalyinthedivergenceoftheconformalcentralchargecurrent(whichisexplicitlyx-dependent,justlikethedilationcurrent).Actually,theanomalycomesonlyfromtheδFvariationandnotfromtheδψvariation,seefootnote4.WeshalllaterexplicitlycomputetheanomalyfromtheδFvariationseparately,see(4.19).Inthatcaseone ndstherelation¯ν ψtermin(4.12)withouttheψγ

¯g µ(x)=ζµ(x)+hζ

2ηµν[ F2 FU+

4πF1(4.14)

Thisrelation,andothersin(4.9),wasobtainedbyextractingcontractedcurrentsfromtheWardidentity,andbygeneralizingthecontractedcurrentsandtheanomaliesinthecontractedcurrentstothecurrentsandtheanomaliesinthecurrentsthemselves.Thisdoesnotdeterminethecurrentcompletely.Wenowprovetheseuncontractedidentities.Webeginwith(4.14)andclaimthefollowingrelation

F FU+21

δF(x)

= h¯g 1¯(x)δψ

=21¯ w(x)θ2¯(x,θ)w(x)θQφ

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

integralintheformof(4.4)givestheidentity6(4.15).Thisanalysis xesthemagnitude

µoftheWeylanomaly.AttheendofSection5weshowthatTµdoesnotcontributetothe

traceanomaly,sothetraceanomalycomesonlyfromtheconservedtensor. µandjµ.WecanshowthatThelastrelationin(4.9)tobeprovenistheonewithJ

µ(x) jµ(x) = γµψ(x)(F(x)+U( )) J

δSγµψ(x) = γµψ(x)2π

byconsideringthevariation

δφ(x,θ)=δ(θ)¯ µ(x)γµ (4.17)

δF(x) = h¯g

ThecurrentTNµν(x)generatedbythevariationδφ(x,θ)=ξµ(x) µφ(x,θ)isgivenby Noether

δS= ( µξν)TNµν.Itreads

TNµν(x)= µ ν 1

¯[γµ ν+γν µ]ψ+1ψ¯µ µψ+2g ψψ¯],ηµν[ψγ

4¯[γµ ν γν µ]ψψ6=

=hg µν(x) ηµν¯T4 µν(x) ηµν[ F2 FU+1T4¯[γµ ν γν µ]ψψ2

whereweusedtheWeylanomalyin(4.15).Thelasttermismanifestlyantisymmetric,anditcan¯5γρ ρψ.Ityieldstheantisymmetricpart¯ρσγρ σψ= µνψγ¯[γµ ν γν µ]ψ= µνψ bewrittenasψ

ofthestresstensor,andisproportionaltothedivergenceoftheLorentzcurrent.Onemayshowthat¯g µTNµν(x)= h

µ 2π νF(x)),andthisimplies Tµν(x)=0.Inouranalysis,theconservedTµν,andTµνwhichis

µνandTµνaremanifestlynotconservedbutfreeofatraceanomaly,playabasicrole.NotethatbothT

symmetric.7Clearly,thenaiveequationofmotionF(x)+U( )=0cannotbeusedinthisderivation.

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

Ifonesetsvµ(x)= µv(x)in(4.20),onegeneratesthedivergenceofthecentralchargecurrentbuttheproceduregivesnoinformationforatopologicalcurrent.InanalogywithU(1)gaugetheory,weareconsideringthechangeofvariableAµ→Aµ+aµinsteadofAµ→Aµ+ µatogeneratethecurrent.

Itisimportanttorecognizethatalloperatorsappearingontheleft-handsidesoftherelationsin(4.7)havehighermassdimensionsthanthoseofthecorrespondingoperators µ(x)andζµ(x)are,respectively,dimension2andontheright-handsides.Forexample,ζ µ1operatorssincethecouplingconstantgcarriesaunitmassdimension.Similarly,γµJ µandjµandγµjµare,respectively,dimension3/2and1/2operators,thoughbothofJ µandTµare,respectively,dimension2and1aredimension3/2operators.Also,Tµµ µνandTµνaredimension2operators.Inthissensealltheoperators,thoughbothofT

compositeoperatorsontheright-handsidesof(4.7)aresoftoperators.Thissuggeststhatonlythe“hard”operatorsgenerateanomalies.Inthenextsectionweprovethisstatement.

5SupersymmetryalgebraofthequantumoperatorsIntheprevioussectionwegaveadirectderivationoftheanomaliesbasedonpathintegrals,butwealr …… 此处隐藏：4711字，全部文档内容请下载后查看。喜欢就下载吧 ……