Topological anomalies from the path integral measure in supe(9)

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

orequivalently

L2(x,θ)=η(x,θ)Γ(x,θ)η(x,θ),

1Γ(x,θ)=

(2π)2eik(x y)exp[ iθ1¯kθ2]

1p2 i δ(x y),

µ.i

Thisequationyieldsthefollowingimportantrelations

1

3(2.25)1p2 i =δ(θ1 θ2),ThesymbolT denotes(covariant)timeorderinginthepathintegralapproach.Ithasthepropertythatitcommuterswithordinaryderivatives

1¯ 1JyieldsZ=h¯[ 2DD) ¯ 1δ(x dµexpi2DD)

y)δ(θ1 θ2).Weset¯h=1inmostplaces.