Composite Operators and Topological Contributions in Gauge Theory

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

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aCompositeOperatorsandTopologicalContributionsinGaugeTheoryJungjaiLee1DepartmentofPhysics,DaeJinUniversity,PoCheon,GyeongGi487-711,KoreaYeongDeokHan2DepartmentofPhysics,WoosukUniversity,Hujeong,Samrye,Wanju,Cheonbuk565-701,KoreaAbstractInD-dimensionalgaugetheorywithakinetictermbasedonthep-formtensorgauge eld,weintroduceagaugeinvariantoperatorassociatedwiththecompositeformed

fromaelectric(p 1)-braneandamagnetic(q 1)-braneinD=p+q+1spacetimedimensions.Byevaluatingthepartitionfunctionforthisoperator,weshowthattheexpectationvalueofthisoperatorgivesrisetothetopologicalcontributionsidenticaltothoseingaugetheorywithatopologicalChern-SimonsBFterm.

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

Theexoticstatistics,anyonsandfermion-bosontransmutationforthecompositestateofapointchargeandamagneticvortexhasbeendiscussedbyWilczek[1].In(2+1)dimensionalspacetime,theanyonshaveawell-knownphysicalrealizationwheremagnetic uxtubesareattachedtochargedparticlesandtheAharonov-Bohmphaseresultingfromadiabatictransportofthecompositesgivethemfractionalexchangestatistics.ItwasgeneralizedtothefactthatthecompositeofaclosedNambuchargedstringandapointvortexpresentedtheunusalstatisticsin(3+1)dimensions[2].

Also,Polyakovshowedthefermion-bosontransmutationin(2+1)dimensionsbyinvestigatingthesmallmomentumbehaviorofascalar eldinteractingwithtopo-logicalChern-simonsterm[3].ThisChern-Simonsmechanismofstatisticaltransmu-tationwasshowntoholdforstring-likeobjectinteractingwithtopologicalBFtermin(3+1)dimensions[4][5].Thetopologicalquantum eldtheorywhichgivestheappropriategeneralizationtohigherdimensionshasbeenstudied[6].Chern-Simonstheorygivestherepresentationsoflinkingnumbersofcurvesin3dimensions,BFtheoriesprovidethepathintegralrepresentationsofthelinkingandintersectionnumbersofgenericsurfacesinDdimensions[5][6].

Inthesestatisticalphenomenalikefermion-bosontransmutation,eventhoughthetopologicalterm(Chern-SimonstermorBFterminhigherdimensions)playsanessentialrole,suchtopologicalcontributionsmayarisefromthegaugetheorywiththekinetictermaselectromagnetism[7].InWilczek’swork[1],iftwocompositesofachargedparticleandapointvortexareinterchanged,anadditionalphasefactorappearsinallgaugeinvariantobservables,thisisbecauseeachcompositeshouldbecovariantlytransportedinthegaugepotentialoftheother.Sothecompositeshavetheunusualstatistics.

Recentlytherehasincreasinglybeenthetheoreticalevidencefortheextended

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

objects(string,membrane,...,p-braneetc.)asthefundamentalconstituentsoftheuniverse[8].Inthispaper,inordertoseeifthetopologicalcontributionsarisefromthehigherdimensionalgaugetheorieswiththeminimalMaxwellkineticterm,wewillapplyPolyakov’spathintegralarguments[3]tothecompositeofelectric(p 1)-braneandmagnetic(q 1)-braneinD(=p+q+1)spacetimedimensions.Theapplicationto(2+1)or(3+1)dimensionalgaugetheoryshowedthattheexpectationvalueofthecompositeoperatorsinvolvessometopologicalcontributionsassociatedwithlinkinvariants[7][9].

Westartbyintroducingangaugeinvariantoperatorassociatedwiththecom-positeofanelectric(p 1)-braneandamagnetic(q 1)-braneinDspace-timedimensions

X(Wq;Wp)=exp( iWq (Fp+1))exp(iWpAp).(1)

Thisoperatordependonthep-dimensionalworldvolumeWpandq-dimensionalworldvolumeWqinDspacetimedimensions.HereApisap-formgauge eldwhichcouplestoelectric(p 1)-brane,andFp+1isthecorresponding(p+1)-form eldstrength.TheoperatorX(Wq;Wp)shouldbecalledtheextended’tHooft-Wilsonoperator.Infact,the rstfactoristhecovariantversionofthe’tHooftoperatororiginatedfromtheclosedhyper-pathof(q 1)-brane,thusitshouldbesatis edthatD p 1=q,because (Fp+1)is(D p 1)-form eldinDdimensions.Ouraimistocalculatethetransitionamplitudeforthepairofa(p 1)-braneanda(q 1)-brane,andtoinvestigatetheformofthepartitionfunction,andthentoseehowthetopologicalcontributionsarise.Whenthe(p 1)-braneand(q 1)-braneofthecompositeevolvefromtheinitialcon gurationPitothe nalcon gurationPf,wehavethetransitionamplitude

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

G(Pi,Pf)=

Wp(Pi,Pf)Wq(Pi,Pf) e Sp Sq<XE(Wp,Wq)>,(2)

wherethesumisoverallp-dimensionalworldvolumeWpandq-dimensionalworldvolumeWqinterpolatingbetweentheinitialandthe nalcon gurationsPiandPfofthe(p 1,q 1)-branecomposite.ThepurelygeometricalactionSpfora(p 1)-branewithaclassicaltrajectory,parametrizedby

xµ=xµ(ξ0,...ξp 1),(3)

isthep-dimensionworld-volumeinducedonthetrajectorybytheexternalDdi-mensionalmetrictimes(p 1)-branetensionTp

Sp=Tp Wpd(Vol),(4)

wheretheRiemannianvolumeelementisgivenintermsofthedeterminantoftheworld-volumemetrich=det(hij),by

d(Vol)=dξp 1∧...∧dξ0

×exp( i

2(p+1)!Wq F)exp(i 2dDxFp+1)WpAp),(8)

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

whereSqisthegeometricalactionfora(q 1)-brane.

PerformingtheintegrationoverAµ1µ2...µpleadsto

<XE(Wp,Wq)>=exp Wpdxµ1µ2...µp Wpdyµ1µ2...µp1

1

xµ1(

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

isavector eldnormaltothehyper-surfaceWp.Thisregularizationcontributesto

RtherenormalizationofthestringTp→Tp.

Indiscussion,theclassicalactionofthisgaugetheoryisnottopological,howdoesitgiverisetothetopologicalcontributionsidenticaltothoseoftopologicalgaugetheorywithaChern-Simonsterm(orBFterminhigherdimensions)?Thoughtheactionisdynamical,theoperatorbeingconsideredisthecompositeformedfromaelectric(p 1)-braneandamagnetic(q 1)-branetowhichap-formgauge eldcouplesinDdimensions.So,whentheyevolveandformaclosedhyper-path,theymaybemutuallylinked.Thatis,iftwocompositesareinterchanged,eachcompositeshouldbecovariantlytransportedinthep-formgaugepotentialoftheother.Thusthetopologicalcontributionsareoriginatedfromtheelectro-magneticdualstructureofthecompositesource.Inmoremathematicaldetails,thepropagatorinthetopologicalChern-Simonstheoryisgivenastheinverseoperatorofthedi erentialoperator” ”,thatisthepropagatorcontainstheantisymmetric tensor,butthegaugeinvariantsourcetermdoesnot.Ontheotherhand,inthepresentgaugetheorywithakineticterm,thepropagatorwhichistheinverseofLaplacianoperatordoesnotcontain tensor,whileoneofthecurrentsourcesJandKtowhichthegauge eldcouplesdoesit.Ifwewritetheineractiontermsindetails,

SI= i

=

where

Jµ2...µp+1(x)= i

y

Kµ1...µp(D)µp+2...µDδ(x y)dy,(11)µ1 Wq( F)D p 1+i WpAp(10)dDx(J+K)µ1µ2...µpAµ1µ2...µp,(x)=i Wpδ(x y)dyµ1...µp.(12)

In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

ThereforetheexpectationvalueofcompositeoperatorcangivethelinkingnumberidenticaltothatintopologicalBFtheory.

Inthistheory,sincethelinkingnumberisnotself-linkingone,inanyspacetimedimensionsthesetopologicalphasescanbeoccurred.InhigherdimensionalChern-Simonstheorythepossibletopologicalcontributionsstemingfromtheselflinkingofthesinglegaugeinvariantoperator,however,arelimitedonlytothespacetimedimensionsD=3,7,11,15,sinceChern-Simonstermcanbeconstructedonlyinodddimensionsandtheself-linkingoftheobjectwithevendimensionalstructurevanish[10][11].

Inconclusions,wehaveshownthetopologicalcontributionsfromtheexpecta-tionvaluesofthecompositeoperatorX(Wq,Wp)inthegaugetheorywithaminimalkinetictermareidenticaltothoseobtainedforWilsonhypersurfacesintopologicalquantum eldtheorywithaBFChern-SimonsterminDdimensions.Thistopolog-icalphasefactorsoriginatefromthedistinctwaysinwhichthe(p 1)-braneworldsurfacemaybraidaroundthe(q 1)-braneworldsurfacewhentheyevolveinDdimensionalspace-time.Thisresultcanbegeneralizedtothenon-intersectingmultip-branehypersurfacesandthenon-Abelianextensiondeservesfutherinvestigations.Sincestatisticshasbarelybeentouchedforobjectssuchasstring,membrane,p-brane,itisverytemptingtoanalyzethegeneralizedstatisticsindi erentcontexts.Acknowledgements

ThisworkwassupportedbytheResearchFundofWoosukUniversity.

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In $D$-dimensional gauge theory with a kinetic term based on the p-form tensor gauge field, we introduce a gauge invariant operator associated with the composite formed from a electric $(p-1)$-brane and a magnetic $(q-1)$-brane in $D=p+q+1$ spacetime dimen

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