Topological anomalies from the path integral measure in supe(17)

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.