Topological anomalies from the path integral measure in supe(15)

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

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)