Strings from Quivers, Membranes from Moose

arXiv:hep-th/0204147v25May2002StringsfromQuivers,MembranesfromMooseSunilMukhia,b,†,MukundRangamanicandErikVerlindecaSchoolofNaturalSciences,InstituteforAdvancedStudy,Princeton,NJ08540,USAbTataInstituteofFundamentalResearch,Mumbai400005,IndiacDepartmentofPhysics,PrincetonUniversityPrinceton,NJ08544,USAFebruary1,2008hep-th/0204147PUPT–2033TIFR/TH/02-12AbstractWeconsiderN=2moose/quivergaugetheoriescorrespondingtoN1D3-branesataC2/ZN2singularityinthe“largemoose”limitwhereN1andN2arescaledtoinfinitytogether.Inthedualholographicdescription,thisscalinggivesrisetoamaximallysupersymmetricpp-wavebackgroundwithacompactlight-conedirection.WeidentifythegaugetheoryoperatorsthatdescribetheDiscreteLight-ConeQuantization(DLCQ)ofthestringinthisbackground.Foreachdiscretelight-conemomentumandwindingsectorthereisaseparategroundstateandFockspace.Thelargemoose/quiverdiagramprovidesausefulgraphicalrepresentationofthestringanditsexcitations.ThisrepresentationhasanaturalexplanationinaT-duallanguage.Thedualtheoryisanon-relativistictypeIIAstringwoundaroundtheT-dualdirection,andboundbyaquadraticNewtonianpotential.WeendwithsomecommentsonD-string/D-particlestates,apossiblelifttoM-theoryandtherelationtodeconstruction.†OnsabbaticalleavefromTataInstituteofFundamentalResearch,10/2001–9/2002.1.IntroductionItisbelievedthatgaugetheoriesencodethedynamicsofstrings.Attemptstophysicallyrealizethisidealedtothelarge-Nexpansionofgaugetheory[1].Inthiscontext,stringtheorieshaveindeedbeenseentoemergefromgaugetheories,mostnotablyinMatrixTheory[2,3]andintheAdS/CFTcorrespondence[4].Matrixtheorygives,inprinciple,aprecisedefinitionfornon-perturbativeM-theoryusinglarge-Nsupersymmetricquantummechanics.ToroidalcompactificationsofM-theoryarethendescribedbylarge-Ngaugetheoriesinvariousdimensions.Theconstructionofstringsintermsofgaugefieldscanbemaderatherexplicitinsomecases.ThebestexampleisMatrixStringTheory[5,6,7].Here,perturbativestringstatesandalsoD-branesintypeIIAstringtheorycanbeidentifiedintermsofoperatorsofatwo-dimensionalN=8supersymmetricgaugetheory.TheAdS/CFTcorrespondenceprovidedaveryexplicitmapbetweengaugetheoryoperatorsandstringtheorystatesinthesupergravitylimit[8,9].Althoughtheproposalcanbeimplicitlyextendedtothefullstringtheory,itprovesdifficulttoconstructstringsexplicitlyoutofgaugefieldsinthisframework.Recently,newinsightwasobtainedbyconsideringaspeciallimitoftheAdSbackgroundoftypeIIBstringtheory,inwhichitispossibletoconstructstringstatesfromgaugetheoryoperators[10].ThisisachievedbytakingaPenroselimit[11]ofthebackground,whichprovidesamaximallysupersymmetricsolutionoftypeIIBstringtheory[12],andrelatingthistoaspecialsubsetofoperatorsinN=4supersymmetricgaugetheory.Penroselimitsofspacetimesgenerallyleadtopp-wavebackgrounds,whichareexactbackgroundsforstringpropagation[13,14]andareexactlysolvableinthelight-conegauge[15,16].Inlight-conequantization,itisoftenusefultocompactifyanulldirection.ThisleadstotheDiscreteLight-ConeQuantization(DLCQ),whichprovidesaconvenientregulatorforstringtheory[17].Inthispicture,onehasinter-actingstringscarryingquantizedunitsofthelight-conemomentum,withtheminimalmomentumbeingcarriedbya“stringbit”.TheideaofDLCQismostnaturalinthecontextofMatrixTheory[18],whereithasbeenarguedthateachsectoroftheDLCQofM-theoryisexactlydescribedbyaU(N)matrixmodel.Althoughacom-pactnulldirectionmaylooksomewhatstrange,itisbestthoughtofasthelimit[19]ofaspacelikecompactification.OuraimistostudytypeIIBstringtheoryinaDLCQpp-wavebackground.ThePenroselimitsthathavebeenconsideredsofarintheAdS/CFTcontextalwaysleadtoapp-wavewithanon-compactnulldirection.WewillshowthatthereisanovelscalinglimitofaparticularAdSbackground,whichpreciselygivesrisetoaDLCQ1pp-wave.Theradiusofthenulldirectionisafinitecontrollableparameterofthisbackground.ThecorrespondinggaugetheoryisanN=2superconformal“moose”or“quiver”theory[20].Anumberoffascinatingaspectsofthegaugetheory/pp-wavecorrespondencewillunfoldasweexplorethismodel.AsinpreviousexamplesfollowingtheoriginalideasofRef.[10],wefindgaugetheoryoperatorswhichcanbeidentifiedtothestringgroundstate,zero-modeoscillatorsandexcitedoscillatorstates.However,thereareremarkabledifferences,andamuchricherstructureappearsduetotheDLCQnatureofthestringbackground.Wefindagaugetheoryoperatorthatdescribesastringgroundstateforeveryvalueofthe(positive)quantizedlight-conemomentumk.Wewillalsoconstructoperatorsthatdescribewindingstatesofthestringoverthenulldirection.ThegaugetheoryoperatorsautomaticallypossessallthepropertiesrequiredofDLCQstringstates.Ourconstructionturnsouttohaveatantalizingproperty:theoperatorsthatdescribeDLCQmomentumstateshavethestructureofa“string”ofoperatorsthatwindsaroundthemoose/quiverdiagram.Thisissuggestiveofastringwindingstate.Conversely,thegaugetheoryoperatorsthatdescribewindingoftheDLCQstringlookverymuchlikemomentumstates.ThissuggeststhatweshouldconsiderT-dualizationoftheDLCQpp-wavetofindthemostdirectcorrespondencewiththegaugetheory.WeperformtherequisiteT-dualityandfindatheoryofnon-relativisticstrings[21,22,23]boundinaharmonic-oscillatorpotential.Thestri