Dipoles, Twists and Noncommutative Gauge Theory

arXiv:hep-th/0008030v218Aug2000PreprinttypesetinJHEPstyle.-PAPERVERSIONhep-th/0008030;PUPT-1946Dipoles,TwistsandNoncommutativeGaugeTheoryAaronBergmanandOriJ.GanorDepartmentofPhysics,JadwinHallPrincetonUniversityNJ08544,USAAbstract:T-dualityofgaugetheoriesonanoncommutativeTdcanbeextendedtoincludefieldswithtwistedboundaryconditions.TheresultingT-dualtheoriescontainnovelnonlocalfields.Thesefieldsrepresentdipolesofconstantmagnitude.Severaluniquepropertiesoffieldtheoriesonnoncommutativespaceshavesimplercounterpartsinthedipole-theories.Keywords:NoncommutativeGeometry,T-duality.Contents1.Introduction12.ReviewofT-dualityinNoncommutativeGaugeTheories23.TheT-dualofaTwist64.PropertiesofDipoleTheories94.1Themodified⋆-product94.2Seiberg-Wittenmap104.3Rationaldipoles104.4S-duality114.5Generalizationtothe(2,0)theory135.Discussion141.IntroductionGaugetheoriesonanoncommutativeTdpossessaT-dualitythatactsonthemetricGijandanti-symmetric2-formΘij[1]-[15].InthispaperwewillexploretheactionofT-dualityonnoncommutativefieldtheorieswithtwistedboundaryconditions.SupposewetakeascalarfieldΦ(x)withboundaryconditionsΦ(x1+2πn1,...,xd+2πnd)=ei(n1α1+···+ndαd)Φ(x1,...,xd),where(x1,...,xd)arecoordinatesonTdwithperiod2πand(α1,...,αd)are“twists.”Thequestionis:whathappensafterT-duality?WewillshowthattheT-dualofsuchatheorycontainsnonlocalfieldsthatbehaveasconstantdipoles,evenwhenthenoncommutativityisturnedoff.Thedipole-vectorandthetwists,αi,togetherforma2d-dimensionalrepresentationoftheSO(d,d,Z)T-dualitygroup.Thepaperisorganizedasfollows.Insection(2)wereviewtheproofofT-dualityinnoncommutativegaugetheories.Insection(3)weextendT-dualitytoactonthetwists.Wedefinethedipoletheoriesandshowthatthetwists,αi,andthedipole-vectorstransformintoeachotherunderT-duality.Insection(4)weexplorethepropertiesof1thedipoletheories.Thesehaveseveralfeaturesthatarereminiscentofnoncommutativefieldtheories,althoughthesefeaturesseemtohaveamuchsimplerversioninthedipoletheories.Forexample,wecandefineamodifiedproductoffields,andwedescribetheanalogofthe“Seiberg-Wittenmap”[15]tolocalvariables.WealsoshowthatwhencompactifiedonS1,thedipoletheoriesreducetoordinaryquivertheories[16]whenthedipole-vectorisarationalfractionofthecircumferenceofS1.Beforeweproceed,letusnotethatdipoles,inthecontextofnoncommutativity,arealsodiscussedin[17]andinanupcomingpaper[18].2.ReviewofT-dualityinNoncommutativeGaugeTheoriesUnlikethecommutativetheory,noncommutativeYang-MillstheoryexhibitstheT-dualityofstringtheory.T-dualityofnon-commutativetoriwasfirstinvestigatedin[1]inthecontextofcompactificationsofMatrixtheory.Inthecontextofnoncommutativegeometry,T-dualityisimplementedasaMoritaequivalencebetweentheC*-algebrasthatarethenoncommutativetori[3,4].Inthislanguage,vectorbundlescorrespondtoprojectivemodulesandtheMoritaequivalenceisgivenbyabimodulewhichal-lowsustomapmodulesoveronetorustomodulesovertheother[13].T-dualityofnoncommutativegaugetheoriesisalsoreviewedin[5]-[15].Theseconsiderationsaresomewhatabstract,however,sowewillnowgiveanex-plicitrelationbetweenadjointfieldsonnon-commutativetori.Thiswillleadtoaconstructionforthetransformationofthecovariantderivativeandthegaugeconnec-tion.ThetheoriesweworkwithareU(n)gaugetheorieswithmunitsofelectricflux.WewillshowthatanysuchtheoryisdualtoaU(gcd(n,m))theorywithnofluxandadifferentnoncommutativityparameter.ThestatementofT-dualityisthatanypairofT-dualtheoriescorrespondtothesamezerofluxtheory.OurpresentationoftheT-dualityofthefieldsisaslightgeneralizationoftheconstructionin[6].WewillworkonT2withnoncommutativityparameterθ.Thismeansthatweworkwiththealgebraoffunctionsonthetorussubjecttothefollowingrelation:[x,y]=2πiR2θ(2.1)where(2πR)2istheareaofthetorus.WecollectsomeusefulfactsthatfollowfromthisrelationandtheBaker-Campbell-Hausdorffformula:eABe−A=eAd(A)B,(2.2)2logeAeB=A+B+12[A,B]+...,(2.3)[x,f(x,y)]=2πiR2θ∂yf(x,y),[y,f(x,y)]=−2πiR2θ∂xf(x,y),(2.4)Fromequation(2.2),weobtaintheusefulrelationeaxf(x,y)e−ax=e2πiaR2θ∂yf(x,y)=f(x,y+2πiR2aθ)(2.5)Hereandupuntiltheendofsection(3),aproductindicatesthenoncommutative⋆-product.Abundleoverthetoruswithnonzerofluxisgivenbyapairoftransitionfunctionssuchthatanadjointsectiontransformsas:Ψ(x+2πR,y)=Ω1(x,y)Ψ(x,y)Ω1(x,y)−1Ψ(x,y+2πR)=Ω2(x,y)Ψ(x,y)Ω2(x,y)−1(2.6)Aconsistentchoiceoftransitionfunctionsis:Ω1=eimy/nRUΩ2=V(2.7)whereUandVarematricessatisfyingUV=e2πim/nVUUn=Vn=1(2.8)Letgcd(m,n)=ν,˜m=m/ν,and˜n=n/ν.Wedefinethefollowing˜nטnmatrices:ukl=e2πikm/nδklvkl=δk+1,lk,l∈Z/˜nZ(2.9)OurchoiceforUandVwillbethen×nmatricesthathaveνcopiesofuandvalongthediagonal.Inthecaseofν=1,thesearethematricesof[6].WecannowputthefieldΨintoastandardform.Following[6],wenoteΨ(x+2πR˜n,y)=Ω˜n1Ψ(x,y)Ω−˜n1=Ψ(x+2πRθ˜m,y)(2.10)Therefore,wehavethefollowingperiodicityconditions:Ψ(x+2πR(˜n−˜mθ),y)=Ψ(x,y),Ψ(x,y+2πR˜n)=Ψ(x,y)(2.11)andwecandoaFourierexpansion:Ψ(x,y)=Xs,t∈Zeisx/(˜n−˜mθ)e−ity/˜nΨs,t(2.12)3Ψs,tisan×nmatrixwhichwetreatasamatrixofν×νblocks.Thus,wehavethe˜nטnmatrixΨf,gs,twithf,g∈Zν.Weexpandthismatrixintermsoftheuandvmatrices:Ψf,gs,t=Xi,j∈Z/˜nZcf,gs,t,i,jviuj(2.13)In[6]itisshownthat,onceweimposetheboundary