Topological anomalies from the path integral measure in supe(8)

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

anti-commutewitheachother.Wehave

¯µψDαφ(x,θ)=ψα+θαF+(γµθ)α µ +(γµθ)αθ

and

¯+2ψθF¯+2(ψγ¯µθ) µ ¯(x,θ)Dφ(x,θ)=ψψDφ

¯[FF µ µ ψγ¯µ µψ]+θθ

¯µθ=0.whereweusedθγ

Wenextnotethat

¯) 2+1φ3(x,θ)= 3+3(θψ

2(2.9)(2.10)¯)(ψψ¯).Wethuschoosetheaction(θθ

dxdθL(x,θ)=

2dxd2θ[1

32gφ3(x,θ) gv0φ(x,θ)]

=dx{1

2 2Thedeltafunctionisde nedbydθ1δ(θ1 θ2)=1andgivenby

1δ(θ1 θ2)=(θ1 θ2)(θ1 θ2).

¯ThepotentialVinL=T VisgivenbyV= FU+gψψ where

2U( )≡g( 2 v0).¯)=1(θθ(2.13)(2.14)(2.15)

Weuseacouplingconstantgwhichisrelatedtothecouplingconstantλusedinotherarticlesby µ0m0,v0==g=2λ

2µ

istherenormalizedmesonmass.

Toapplythebackground eldmethod,wedecomposethe eldvariableasfollows

φ(x,θ)=Φ(x,θ)+η(x,θ)(2.17)

whereΦ(x,θ)isthebackground eldandη(x,θ)isthequantum uctuation.Wethenconsiderthepartsofthe(super eld)Lagrangianwhicharequadraticinη

L2(x,θ)=1