Topological anomalies from the path integral measure in supe(3)

δ = ¯ψ(x),

δψ= µ (x)γµ +F(x) = (x) +F(x) ,

δF= ¯γµ µψ= ¯ ψ(x).

Thesuperchargewhichgenerates(2.4)

Q≡ ¯¯α (2.7)

+η¯γµθ µ¯ θα

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

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.

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

wherethesecondrelationisderivedbyperformingtheintegral

1(θ1 θ3)pθ2] d2θ2in

2¯D(θ1)δ(θ1 θ2)= exp[ iθ1pθ2],

D2(θ1)δ(θ1 θ2)= p2δ(θ1 θ2).

Usingtheseresults,weobtainthefollowingequations

Γ(x,θ)= 1

D2 1

12gΦD+(gΦ)2,

2gDΦ 1¯DD,(2.28)4

Γ2(x,θ)δ(θ θ1)=[

4p2 1

2gΦD+(gΦ)2]2δ(θ θ1).(2.29)

Weusetheheatkernelexp[(Γ/M)2]ing(2.29)we nd

(exp[1

[ 212gDΦ 1

M

M

2Γ(x,θ)}|θ,x 2

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

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

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

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