Topological anomalies from the path integral measure in supe(12)

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

Replacing

weobtaintheintegral

M2kµ→Mkµ(k+22(3.9)1

M2

g2¯kDθ2 d2k41

M2kpM2) 2 i¯)]gΦ(x,θ)k2(θθ i¯kDθ¯)]+k2(θθ

usingthataccordingto(2.28)

θ1→θ2k].By4limD(θ)δ(θ θ1)= 1(3.11)

whiletermswithoutDactingonδ(θ θ1)vanishforθ1→θ,andnotingthatonlythetermsintheintegraloforder1/M2orlargersurvivewhenMtendstoin nity,onecancon rmthatonlythetermstosecondorderintheexpansionsurvive.Infact,thesecondordertermscompletelycancelbecausetheterm

¯)D/M2k2(θθ

fromthecrosstermsofD/M2and1

2¯)k2DD/M¯)D/M2.¯(θθ= k2(θθ(3.12)2

(3.13)Wethusneedtoevaluateonlythe rstorderterms

d2k

4k2}[ gΦ(x,θ)D]δ(θ θ1)=i

ig

(2π)2exp[ k/4]=2

πΦ(x,θ)].(3.16)

Notethatthiscalculationremainsvalidforgeneral(non-derivative)interactionsdepend-ingonΦ;ifonehasapotentialV(Φ)insteadofgΦin(2.19),onemakesthesamereplacementin(3.16).

Inthespiritofthebackground eldmethod,onemayreplacethevariableΦ(x,θ)bythefullvariableφ(x,θ)totheaccuracyoftheone-loopapproximation.The nalresultfortheresultoftheJacobianofthepathintegralinMinkowskispaceisthusgivenby

lnJ=i d2xd2θω′(x,θ)[g