Topological anomalies from the path integral measure in supe(11)

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

=

= d2xd2θω′(x,θ) x,θ|exp{d2xd2θω′(x,θ)

y→x,θ1→θ12 D+gΦ(x,θ)]2}|θ,x

= ×lim

2exp{d2k

[212D+gΦ(x,θ)]2}δ(θ θ1)δ(x y)(3.3)dxdθ212MDgΦ(x,θ) 1

(2π)2

×limexp{θ1→θω′(x,θ)e ikx4p2 1

2

1gΦ(x,θ)D+g2Φ2]}eikxδ(θ θ1).1WerecallD=

2¯x,θ)D+1 (

(2π) ikxeexp{214p2 1

2gΦ(x,θ)D+g2Φ2]}eikxδ(θ θ1).(3.7)

Passingthefactoreikxthroughtheintegrandreplacesp→p+k,andtheoperatorDismodi edasfollows

1(γνθ)kν][D+i(γµθ)kµ]e ikxDeikx=

=11(kθ)D (kθ)(kθ)

¯kD+1=D iθ