Topological anomalies from the path integral measure in supe(2)

Wealsoderivethesupersymmetryalgebraatthequantumlevel.Thisrequiresafullquantumoperatorapproachwhichincorporatesanomalies,ratherthanasemiclassi-calapproachbasedonDiracbrackets.Fortunatelythereexistssuchanapproach:theBjorken-Johnson-Low(BJL)methodwhichautomaticallyincorporatesthee ectsofsu-perconformalanomalies.TheBJLmethodhasbeenwidelyusedforcurrentalgebrasinthe1960’s[16].Itallowsonetorewriteresultsobtainedfrompathintegralsintooperatorrelations.Togofromthepathintegralresultstooperatorresultsonemustuse eld

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

equations,butthese eldequationssometimesyieldanomalies.TheBJLmethodtakessuchanomaliesintoaccount,andthisiscrucialinourcase.Forreaderswhoarenotfamiliarwiththistechniquewegiveadiscussionofthismethodintheappendix.WethuspresentafullyquantumversionoftheWitten-Oliveanalysis.ThedeformationofthesupersymmetryalgebrabyanomaliesissuchthattheBPSboundremainssaturatedduetouniformshiftsinenergyandcentralcharge.Inouralgebraicapproach,weinitiallyformulateallresultsintermsofatotalsuper eld,anddonotdecomposethissuper eldintoabackgroundandaquantumpart.Onlyattheenddoweneedtousesomeprop-ertiesofthekinkbackground,namelythefactthatthevacuumisannihilatedbyoneofthesupersymmetrycharges(thetime-independentcharge).

Wewouldliketosummarizeourresultsbrie y.ItisshownthattheNoethercurrentforordinary(nonconformal)supersymmetry

jµ(x)= [ +U( )]γµψ(x)

2withU( )=g( 2 v0)containsinthepresentformulationanapparentanomaly(1.3)

µjµ(x)=h¯g

2πγµψ(x)]α; µ(x)=0 µJ(1.6)

andtheassociatedtime-independentsupercharge

Q= α 0,α(x)dxJ(1.7)

whichgeneratesordinarysupersymmetry.Thisconservedcurrentcontainsthefollowinganomalouscomponent¯g µ(x))anomaly= h(γµJ

Divergenceswithrespecttoθinsuperspaceleadtoanomaliesinx-spacewhichareoftheformγµjµinsteadof µjµ.Similarly,theanomalyintheenergydensityisduetothetraceanomaly,whichitselfisduetoanotherθdivergenceinsuperspace.Fromasuperspacepointofview,θdivergencesandxdivergencesareequallyfundamental.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

isthenderivedbyusingtheThesupersymmetryalgebrafortheconservedchargeQ

BJLmethod.WerewriteWardidentitieswhicharederivedfrompathintegralsandwhichcontainthecovariantT timeordering,intermsofWardidentitiesattheoperatorlevelwhichcontaintheTtimeorderingsymbol.Weobtain2

α,Q β}= 2(γµ)αβP µ 2Z (γ5)αβi{Q

where

µ=P

=Z (1.9) 0µ(x),dxT

0(x)= dxζ =P 0,H

0(x).dxζ(1.10)

TheBJLmethod,unlikethesemi-classicalDiracbracket,incorporatesallthequantum µareconserved µandζe ects,inparticularsuperconformalanomalies.TheoperatorsTνquantities µ=0 µ=0, µT µζ(1.11)ν

butcontainsuperconformalanomalies

¯g¯(x)+h µ(x)=F(x)U(x) g ψψTµ

2πF(x),

h¯g

µ (x)γµ.(1.12) µ(x)γ5= µ (x)Uγµ+γµζ2π

Wederivetheseequationsfromthepathintegralformulation.Intheseequations,Tµµ(x)and(γµζµ)containonlythetermswhichexplicitlybreaksuperconformalsymmetry,asweshallshow.Thesearisefromthesuperpotential,are“soft”(theyhavelowerdimen-sionbecausetheyareproportionaltothedimensionfulg),andtherearenoanomalouscontributionstothesequantities.

Therelationsin(1.5)and(1.6)canbecombinedtogiveasimilarresultasin(1.12)

µ(x)=γµjµ(x) γµJh¯g

¯g µ(x)=ζµ(x)+hζ

24πF(x),Thesymbols(γ0)αβand(γ5)αβdenotethematrices(γ0)αγ(C 1)γβand(γ5)αγ(C 1)γβandareinourconventionsequalto iandiτ3,respectively,seeSection2.Theindicesαandβareequalto+or ,andforα=β=+one ndsthatthequantumanticommutatorhasthesameformastheclassicalrelation(1.1).

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

TheoperatorsTµν(x)andζµ(x)arespeci edby

Tµµ(x)anomaly=0,ζµ(x)anomaly=0(1.15)

µandcorrespondingtotheabsenceofananomalyinγµjµ,see(1.5),butalthoughbothζ

ζµareareconserved,Tµνisnotconserved

νTµν(x)=0,

similartothenon-conservationofjµ.

IntermsofTµνandζµ,thesupersymmetryalgebrareads

i{Q,Q}= 2(γ)Pµ+ α βµαβ(1.16) dxh¯g

1 (x)(γ5)αβ(1.17)π

whereweused 01=1and

Pµ=

Z= dxT0µ(x),

dxζ0(x)= H=P0,

dxζ0(x).(1.18)

Weseethatthesupersymmetryalgebraintermsoftime-independentchargeshasthesameformatthequantumlevelasattheclassicallevel,see(1.10).Thisagreeswith[8],whoseanalysisisbasedonthisobservation.In(1.16)wehaveusedchargesPandZ;Z explicitlyappearsontheright-handside,isfreefromanomaliesandtheanomalyofZ

µbutPstillcontainsasuperconformalanomalythoughTµisfreefromthetraceanomaly.

(Inotherwords,T00andT11haveequalanomalies,andtheanomalyinT00doublesthe intermscontributionin(1.14)proportionaltoF,ing(1.6)towriteQ

ofQ,alltheanomalytermsin(1.17)cancelseparatelyifonesplitso theanomalyfromPµ.Inthisway,alsointermsofQ,PµandZthequantumanticommutatorhasthesameformasclassically.However,QandParetime-dependent,sotheyarepysicallylessrelevant.)Thesetwoalternativewaysofwritingthealgebragiveriset …… 此处隐藏：5007字，全部文档内容请下载后查看。喜欢就下载吧 ……