Holomorphic anomaly and matrix models

arXiv:hep-th/0702110v319Jul2007CERN-PH-TH/2007-031SPhT-T07/020HolomorphicanomalyandmatrixmodelsBertrandEynarda,MarcosMari˜nob1andNicolasOrantinaaServicedePhysiqueTh´eoriquedeSaclayF-91191Gif-sur-YvetteCedex,[e-mail],[e-mail],TheoryDivision,CERNCH-1211Geneva,[e-mail]nergiesofmatrixmodelscanbepromotedtomodularinvariant,non-holomorphicamplitudeswhichonlydependonthegeometryoftheclassicalspectralcurve.Weshowthatthesenon-holomorphicamplitudessatisfytheholomorphicanomalyequationsofBershadsky,Cecotti,OoguriandVafa.Wederiveaswellholomorphicanomalyequationsfortheopenstringsector.TheseresultsprovideevidenceatallgenerafortheDijkgraaf–VafaconjecturerelatingmatrixmodelstotypeBtopologicalstringsoncertainlocalCalabi–Yauthreefolds.1AlsoatDepartamentodeMatem´atica,IST,Lisboa,PortugalContents1Introductionandconclusions12Holomorphicanomalyandtopologicalstrings32.1Theholomorphicanomalyequations......................32.2ThelocalCalabi–YaucaseandtheDijkgraaf–Vafaconjecture.........63Reviewofmatrixmodels83.1Formalmatrixmodelsandalgebraicgeometry.................83.2Variationsofthematrixmodelfreeenergies..................103.2.1Variationswrtfillingfractions......................134Holomorphicanomalyequationsandmatrixmodels144.1Adirectproof...................................144.2Acombinatorialproof..............................161IntroductionandconclusionsTopologicalstringtheoryhasbeenafascinatinglaboratorytoexploreissuesinstringthe-orywithimportantconnectionstootherbranchesofphysicsandmathematics.Thebasicproblemintopologicalstringtheoryistocomputeandunderstandclosedandopenstringamplitudesondifferentgeometricbackgrounds.ParticularlyimportantamongtheseareCalabi–Yau(CY)manifolds.Differenttechniqueshavebeendevelopedforthecomputationoftheseamplitudes.FortypeBtopologicalstringsonCYmanifolds,apowerfulmethodtosolvetheclosedsectorofthemodelaretheholomorphicanomalyequationsof[6].Theseequationscontrolthe¯t-dependenceoftheclosedstringamplitudesF(g)(t,¯t),andwhencom-binedwithextraboundaryconditions,theyleadtoexplicitanswers,see[19,17]forrecentprogressinthisdirection.Ontheotherhand,insomespecialbackgroundsonecanuselargeNdualitiesandgeometrictransitionstocomputetheholomorphiclimitofF(g)(t,¯t),whichwillbedenotedbyF(g)(t).Inagroundbreakingpaper[11],DijkgraafandVafaconjecturedthatoncertainnoncompactCalabi–Yaumanifolds,wherethegeometryreducestoacomplexcurve,theF(g)(t)aregivenbythegenusgfreeenergiesofamatrixmodelinthe1/Nexpansion(see[24]forareview).ThecurveencodingtheCYgeometryisthenidentifiedastheclassicalspectralcurveofthematrixmodel.GiventheconnectionbetweentopologicalstringsandtypeIIsuperstrings,thishasmadepossibletocomputecertainprotectedquantitiesinsupersymmetricgaugetheoriesbyusingmatrixmodeltechnology.Theconnectionbetween1topologicalstringsonnoncompactCYmanifoldsandmatrixmodelshasbeenextendedrecentlytothemirrorsoftoricgeometries[25].TheconjectureofDijkgraafandVafawasverifiedattheplanarlevelin[11],andatgenusonein[22,12].Inthesimplecaseofthecubicmatrixmodel,evidencefortheconjecturewasgivenatgenustwoin[18],whereitwasshownthatthematrixmodelexpressionforF(g)(t)istheholomorphiclimitofaparticularsolutiontotheholomorphicanomalyequationsforthecorrespondinglocalcurve.InthispaperwewillgivefurtherevidencefortheDijkgraaf–Vafaconjectureandtherelatedresultsof[25],byusingrecentadvancesinthesolutionofmatrixmodelsatallordersinthe1/Nexpansionobtainedin[13,15,9]andmorerecentlyin[16].Oneoftheoutcomesoftheseadvancesisthat,asexplainedin[16],onecanextrapolatethematrixmodelprocedureanddefineaseriesofamplitudesF(g)(t)andcorrelationfunctionsWg(pk)foranyalgebraiccurveH(x,y)=0.Whenthiscurveistheclassicalspectralcurveofamatrixmodel,oneobtainsinthiswaythe1/Nexpansionofthefreeenergiesandcorrelationfunctions.Usingtheresultsof[16],wewillshowthatitispossibletoconstructnon-holomorphicfreeenergiesF(g)(t,t)andcorrelationfunctionswhichsatisfythefollowingconditions:•Intheholomorphiclimit¯t→∞theyreducetotheholomorphicamplitudesF(g)(t)associatedtothespectralcurve.•Theyareinvariantwithrespecttothesymplecticmodulargroupofthecurve.Thisisincontrasttotheirholomorphiclimit,whichdoesnothavegoodmodularproperties[16].•Theysatisfytheholomorphicanomalyequationsof[6].Thisgivesaproceduretoobtainthefullnon-holomorphiccouplingsF(g)(t,¯t)inthecon-textofmatrixmodels,byusingtherequirementofmodularinvarianceasin[1,10],andshowsthatthesecouplingsobeytheequationsof[6]thatcharacterizetopologicalstringam-plitudes.Thisimpliesinparticularthatthematrixmodelfreeenergyatgenusg(extendednon-holomorphicallyinthisway)mustbeequaltothetypeBfreeenergyofthenoncompactCalabi–Yaumanifoldsconsideredin[11,25],uptoaholomorphicmodularinvariantquan-tity.ProvingtheDijkgraaf–Vafaconjecturereducesnowtoprovingthattheseholomorphicinvariantfunctionsvanishatallgenera.Aninterestingspinoffofourworkisthatwecandeterminetheholomorphicanomalyequationsforopenstringamplitudeswithrespecttotheclosedmoduli¯t,atleastinthelocalcase.Adirectstringtheoryd