Topological anomalies from the path integral measure in supe(16)

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