Topological anomalies from the path integral measure in supe(13)

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

Forexample,fortheclassoftransformations

¯Dφ+c(D¯ )φ= ¯Dφ+1δφ(x,θ)=

withaconstantc,oneobtainsfortheJacobian

ig

)2g1¯D( Φ(x,θ))+(c 2¯ )φ)(D(3.18)

2

2

where

L=1¯β)(D¯αφ)Qβφ+ ¯αQαL](Dα 2gφ3 gv0φ.(4.2)(4.3)3

Anytransformationofφ,whetheritisasymmetryoftheactionornot,leadstoacorrespondingWardidentity,butusingalocalsupersymmetrytransformationhastheadvantagethatoneobtainsahierarchyofWardidentititesinx-spacewhichcontaintheWardidentitiesforordinaryandconformalsupersymmetry.Theseare,ofcourse,theWardidentitiesweareinterestedin,andweexpectinthismultipletofWardidentitiesalsoto ndaWardidentityforthecentralchargecurrent.

Forconstantsuper elds α,theactionisinvariant,butforlocal α,thevariationofSisporportionaltotheNoethercurrent.OnethusobtainsthefollowingWardidentityforcorrelationfunctions

i

g2¯β)(D¯αφ)Qβφ+ ¯αQαL]φ(x1,θ1)...φ(xn,θn) (Dα

= i

2¯g¯αφ)Qβφ]+QβL= hDα[(D