On Geometry and Matrix Models

arXiv:hep-th/0207106v111Jul2002hep-th/0207106HUTP-02/A030ITFA-2002-24OnGeometryandMatrixModelsRobbertDijkgraafInstituteforTheoreticalPhysics&Korteweg-deVriesInstituteforMathematicsUniversityofAmsterdam1018TVAmsterdam,TheNetherlandsandCumrunVafaJeffersonPhysicalLaboratoryHarvardUniversityCambridge,MA02138,USAAbstractWepointouttwoextensionsoftherelationbetweenmatrixmodels,topologicalstringsandN=1supersymmetricgaugetheories.First,wenotethatbyconsideringdoublescalinglimitsofunitarymatrixmodelsonecanobtainlargeNdualsofthelocalCalabi-YaugeometriesthatengineerN=2gaugetheories.Inparticular,adoublescalinglimitoftheGross-Wittenone-plaquettelatticemodelgivestheSU(2)Seiberg-Wittensolution,includingitsinducedgravitationalcorrections.Secondly,wepointoutthattheeffectivesuperpotentialtermsforN=1ADEquivergaugetheoriesissimilarlycomputedbylargeNmulti-matrixmodels,thathavebeenconsideredinthecontextofADEminimalmodelsonrandomsurfaces.TheassociatedspectralcurvesaremultiplebranchedcoversobtainedasVirasoroandW-constraintsofthepartitionfunction.July,20021.IntroductionIn[1]wehaveshownhowtheeffectivesuperpotentialinN=1supersymmetricgaugetheories,thatareobtainedbybreakinganN=2superYang-Millstheorybyaddingatree-levelsuperpotentialW(Φ)fortheadjointscalarΦ,canbecomputedbyalargeNhermitianmatrixmodels.Moreprecisely,theeffectivesuperpotentialoftheN=1theoryconsideredasafunctionofthegluinocondensatesSiis—apartfromtheuniversalSilog(Si/Λ)termscomingfromthepureN=1Yang-Millstheory—givenexactlybyaperturbativeseriesthatiscomputedbytheplanardiagramsofthematrixmodelwithpotentialW(Φ).Furthermore,thecontributionsofthisgaugetheorytotheinducedsupergravitycorrectionsR2F2g−2(withRtheRiemanncurvatureandFthegraviphotonfieldstrength)aresimilarlycomputedexactlybythegenusg0matrixdiagrams.Thisgaugetheory/matrixmodelcorrespondencewasaconsequenceofthelargeNdualitiesof[2,3,4]thatrelatethecomputationofholomorphicF-termsintheworld-volumetheoriesofD-branestopartitionfunctionsoftopologicalstringsinlocalCalabi-Yaugeometries—arelationthatwasfurtherexploredin[5,6,7,8,9,10,11].Inthesimplestcasetheselocalnon-compactCalabi-Yaumanifoldstaketheformvv′+y2−W′(x)2+f(x)=0.OnefindsthatintheB-modeltopologicalstringthetree-levelfreeenergycanbecomputedintermsoftheperiodsofthemeromorphicdifferentialydxontheassociatedRiemannsurfacey2−W′(x)2+f(x)=0.Aswearguedin[1]thiscurveandtheassociatedspecialgeometryarisesnaturallyfromthelargeNdynamicsofthematrixintegralwithactionW(Φ).Butweshouldstressagainthattherelationwithmatrixmodelsgoesbeyondtheplanarlimit.ThehighergenusstringpartitionfunctionsFgandtherelatedgravitationalcouplingsofthegaugetheoriesareexactlycomputedinthe1/Nexpansionofthematrixmodels.Letusbrieflysummarizetheseconnections,formoredetailssee[1].WestartwiththehermitianmatrixintegralZdΦ·e−S(Φ)withactionS(Φ)=1gsTrW(Φ),1andW(x)isapolynomialofdegreen+1.Thematrixintegralcanbereducedtoanintegralovertheeigenvaluesx1,...,xNofΦinthepotentialW(x).Intheclassicallimitgs→0,whereoneignorestheinteractionsamongtheeigenvalues,theequationofmotionisgivenbyy(x)=gs∂S∂x=W′(x)=0.(1.1)Theassociatedclassicalspectralcurveisy2−W′(x)2=0,(1.2)wherex,ycanbeconsideredascomplexvariables.WritingW′(x)=Qi(x−ai)weseethatthissingulargenuszeroplanarcurvehasndoublepointsatthecriticalpointsx=ai.Sometimesitcanbehelpfultothinkofthe(x,y)-planeasaphasespace,withythemomentumconjugatetox,asgivenbytheHamilton-Jacobiequation(1.1).Then(1.2)hasaninterpretationasthezero-energylevelsetofthe(bosonicpartofthe)supersymmetricquantummechanicsHamiltonianassociatedtothesuperpotentialW(x),andS(x)canbethoughtofasthesemi-classicalWKBactionoftheassociatedquantummechanicalgroundstateΨ(x)∼e−S(x).Classically,theNeigenvalueswillclusteringroupsofNiinthecriticalpointsaiwheretheywillformsomemeta-stablestate.Therelativenumberofeigenvaluesorfillingfractionofthecriticalpointaiwewilldenoteasνi=Ni/N.Ifgsisnotzero,wehavetotakeintoaccounttheCoulombinteractionthatresultsfromintegratingouttheangular,off-diagonalcomponentsofthematrixΦ.TheequationofmotionofasingleeigenvaluexinthepresenceoftheDysongasofeigenvaluesx1,...,xNisnowmodifiedtoy=W′(x)−2gsNXI=11x−xI.WewillnowtakethelargeN’tHooftlimitkeepingbothμ=gsNandthefillingfractionsνifixed.Inthiscaseeachcriticalpointhasitsown’tHooftcouplingμi=gsNi=μνi.ThecollectivedynamicsoftheseeigenvaluesinthelargeNlimitcanbesummarizedgeometricallyasfollows.Eachofthendoublepointsx=aigetsresolvedintotwo2branchpointsa+i,a−i.TheresultingbranchcutsAi=[a−i,a+i]arefilledbyacontinuousdensityofeigenvaluesthatbehaveasfermionsandspreadoutduetothePauliexclusionprinciple.ThisprocessofsplittingupofdoublepointsisveryanalogoustotransitionfromtheclassicaltothequantummodulispaceintheSeiberg-WittensolutionofN=2supersymmetricgaugetheories[12]—arelationthatwasexplainedin[11].Theresolutionofdoublepointsiscapturedbydeformingtheclassicalspectralcurve(1.2)intothequantumcurvey2−W′(x)2+μf(x)=0,(1.3)wherethequantumdeformationf(x)isapolynomialofdegreen−1.ThefillingfractionνiisrelatedtothesizeofthebranchcutAi.Roughly,thehighertheproportionofeigenvalueatthecriticalpointai,the