BPS Condensates, Matrix Models and Emergent String

arXiv:hep-th/0607204v228Sep2006PreprinttypesetinJHEPstyle.-HYPERVERSIONBPSCondensates,MatrixModelsandEmergentStringTheorySamuelE.V´azquez1,†1DepartmentofPhysics,UCSB,SantaBarbara,CA93106†[e-mail]:AprescriptionisgivenforcomputinganomalousdimensionsofsingletraceoperatorsinSYMatstrongcouplingandlargeNusingareducedmodelofmatrixquantummechanics.Themethodinvolvestreatingsomepartsoftheoper-atorsas“BPScondensates”which,incertainlimit,haveadualdescriptionasnullgeodesicsontheS5.Inthegaugetheory,thecondensateissimilartoarepresentativeofthechiralringanditisdescribedbyabackgroundofcommutingmatrices.Exci-tationsaroundthesecondensatescorrespondtoexcitationsaroundthisbackgroundandtaketheformof“stringbits”whicharedualtothe“giantmagnons”ofHofmanandMaldacena.Infact,thematrixmodelapproachgivesaquantumdescriptionofthesestringconfigurationsandexplainswhytheinfinitemomentumlimitsuppressesthequantumeffects.Thismethodallows,notonlytoderivepartoftheclassicalsigmamodelHamiltonianofthedualstring(intheinfinitemomentumlimit),butalsoitsquantumcanonicalstructure.Therefore,itprovidesanalternativemethodoftestingtheAdS/CFTcorrespondencewithouttheneedofintegrability.Keywords:AdS/CFT,D-branes,MatrixQuantumMechanics,RotatingStrings.Contents1.Introduction12.TheGeneralIdeaofBPSCondensates53.1/2BPSCondensates:TheSU(2)Sector73.1Reviewof1/2BPSStatesDynamics83.2SU(2)StatesDynamics93.31/2BPSCondensates,StringBitsandGiantMagnons113.4ClassicalLimitandthedualStringTheory133.5MultipleGiantMagnonsandBoundStates164.1/4and1/8BPSCondensates:TheSU(3)Sector174.1HilbertSpaceandCanonicalStructure194.2LocalizationandtheClassicalLimit204.3SU(3)GiantMagnons?225.BackreactiontoBPSCondensates245.1HigherInteractionsandtheStrongCouplingLimit276.Discussion28A.Anti-NormalOrdering30B.SU(3)StringAction311.IntroductionUnderstandingthestrongcouplinglimitofnon-abeliangaugetheoriesisstillanoutstandingopenproblemintheoreticalphysics.MostofourunderstandingcomesfromConformalFieldTheories(CFTs).AccordingtotheAdS/CFTconjecture,atlargeNandlargebutfixed’tHooftcoupling,weshouldfindaneffectivegeo-metricaldescriptionofthesetheoriesintermsofperturbativestringtheoryonanasymptoticallyAdSbackground[1].Proving(ordisproving)thisconjectureisstillanimportantopenproblem.However,muchevidenceinitsfavorhavebeenfoundinrecentyears.Thebeststudiedexampleofthecorrespondenceisthedualitybetween1N=4SYMtheoryinfourdimensionsandstringtheoryonAdS5×S5.Thisisthemodelthatwewillstudyinthispaper.ThemostsignificantobstructiontoprovingthecorrespondenceisourinabilitytomakestrongcouplingcalculationsinSYMtheory.PuresupersymmetricstateshaveprotecteddimensionsandsoonecanmapthemdirectlytoSUGRAexcitationsonthedualstringtheory1.However,fornon-supersymmetricstatesoneneedstobemorecleverandfindawaytoextrapolateaweakcouplingcalculationtostrongcoupling.Asinanyperturbativeexpansion,thesecretistobecarefulinchoosingthe“background”stateoneisexpandingaround.Itwasrealizedafewyearsagothatonecandefinetheperturbationtheoryaroundthesupersymmetricsingletrace1/2BPSstatesoftheformTr(ZJ),(1.1)withZoneofthethreecomplexadjointscalarsofSYM.ForlargeJthesestatesaredualtopoint-likeclassicalstringsrotatingwithangularmomentumJalonganullgeodesicontheequatoroftheS5.Thefirst“small”non-BPSexcitationstobeunderstoodwheretheso-calledBMNstatesoftheform[3]XleiplTr(ZJ−lXZlX),(1.2)wherep∼1/J.Infact,theeffectivegeometryseenbythesmallstringsisaplanewaveandthusitallowstheexactcalculationofthespectruminthestringtheoryside.Itturnsoutthatforthesefastrotatingstringstheexpansionparameterintheirenergyisoftheform∼λ/J2.ItisthenpossibletoextrapolateaperturbativecalculationinSYMbyfirsttakingJlargeandthenλ→∞butwithλ/J2=fixed≪1.Onthestringtheoryside,however,thelimitistheopposite:λ≫1firsttogetperturbativestringtheory,andthenfocusontrajectorieswithJ≫1.Surprisingly,bothcalculationsagreeuptoorderλ3wherea1/Jdiscrepancyishasbeenfound[4].Itisbelievedthatthediscrepancyisjustanartifactoftheoppositeorderoflimitsandthefactthatkeepingλ/J2fixedentanglesboth1/√λand1/Jquantumcorrections.Onecanproceedalongtheselinesandconsiderstateswithmanytransverse“impurities”,e.g.Tr(ZXXZX···).(1.3)OnecancomputetheanomalousdimensionmatrixinthisbasisandithasbeenfoundthatonecanalsodefinelargechargelimitswithJX,JZ→∞thatallowtoextrapolatetheweakcouplingcalculationtostrongcoupling.Onthestringtheory1See[2]foramorecomprehensivereviewofthisandotheraspectsoftheAdS/CFTcorrespon-dence.2dualonefindthatthesestatescorrespondtolongsemiclassicalrotatingstrings2.Asbefore,oneconsidersalimitwhereλ/J2isheldfixed.Onecanevenmatchthesigmamodelactionsforthesestringsinthislimit[6,7,8],butoneagainfindsadiscrepancyatthreeloops[9].OnecanalsodefineanexpansionaroundBPSD-branestatescalledGiantGravi-tons.Thelimitsdescribedabovearealsousefulforthiscaseandonecanshowagreementwiththestringtheoryatoneloop[10,11].ItisapparentatthispointthatallofthecomparisonsdescribedaboverequirecertainamountofconfidenceinextrapolatingtheweakcouplingresultsinSYMtostrongcoupling.Onewouldliketobeabletodefineastrongcouplingexpansiondirectly.Thisisofcoursethe“holy