Matrix Representations of Holomorphic Curves on $T

arXiv:hep-th/9812184v120Dec1998hep-th/9812184MatrixRepresentationsofHolomorphicCurvesonT4LorenzoCornalbaCenterforTheoreticalPhysics1MassachusettsInstituteofTechnologyCambridge,MA02139,[e-mail]atrixrepresentationofcompactmembranesanalyticallyembed-dedincomplextori.Braneconfigurationsgiverise,viaBergmanquantization,toU(N)gaugefieldsonthedualtorus,withalmost-anti-self-dualfieldstrength.ThecorrespondingU(N)principalbundlesareshowntobenon-trivial,withvanishinginstantonnumberandfirstChernclasscorrespondingtothehomologyclassofthemembraneembeddedintheoriginaltorus.Inthecourseoftheinvestigation,weshowthattheproposedquantizationschemenaturallyprovidesanassociativestar-productoverthespaceoffunctionsonthesurface,forwhichwegiveanexplicitandcoordinate-invariantexpression.Thisproductcan,inturn,beusedthequantize,inthesenseofdeformationquantization,anysymplecticmanifoldofdimensiontwo.Dicember19981Visitingfrom:PrincetonUniversity,DepartmentofPhysics,Princeton,NJ08544,U.S.A.;[e-mail].1IntroductionMatrixtheory[2][7][9][8][3][4][5]isbelievedtodescribe,inthelimitoflargeN,thefunda-mentaldegreesoffreedomofM-theory.Infact,withinthesametheory,bothfundamentalparticlesandextendedobjectsaredescribedinaunifiedway.Itisindeedremarkablethatonecanstartwithatheoryofgluonsinacertaindimension(matrixtheoryisnothingbut9+1U(N)Super-Yang-Millstheorydimensionallyreducedto0+1dimensions)anddescribe,inadualway,aseeminglyunrelatedtheoryofgravityinadifferentspace-timedimension(10+1).Inlightofthisfact,itisimportanttobetterunderstandtherelationsthatexistbetweenthetwopointsofview,andtopreciselydescribehowtorepresentinmatrixlanguageobjectswhicharefamiliarfromtheM-theoryprospective,andwhicharedescribedatlowenergieswithin11-Dsuper-gravity.Itisofinterest,inparticular,toconsidermatrixtheoryconfigurationswhichrepresent,withinthegravitydescription,extendedmembranes.MorespecificallyonecanstudyBPSmembranestates,whichonlypartiallybreaksupersymmetry,andwhicharenotexpected,ongeneralgrounds,tobeeffectedbyquantumcorrections.Withintheframeworkofsupermem-branetheory[10][11],onecanshowthattheBPSconditionisequivalenttotherequirementthatthebranebeembeddedholomorphicallyinspace,anditisthereforenaturaltolookformatrixrepresentationsofholomorphiccurves.In[1]thequestionofrepresentationofholomorphiccurvesembeddedinnon-compactspacewasanalyzedindetail.Inparticulartheproblemwasrephrasedasaproblemingeometricquantization,withε∼L3P/RplayingtheroleofthePlanckconstant.Aspecificquantizationschemewasproposed,basedontheconceptofBergmanprojection,andthematricesrepresentingthecurveweretakentobeoperatorsactingontheinfinitedimensionalspaceofholomorphicfunctionslivingonthebrane.InordertopreservetheBPScharacteroftheconfiguration,oneneedstochooseaspecificinnerproductonthespaceoffunctions,whichisrelatedtoadeformationoftheK¨ahlerpotentialofthebrane.UsinganexplicitexpansionfortheBergmanprojection,thedeformationwasdeterminedasymptoticallyinε.Inthispaperweextendtheresultsof[1]totheinterestingcaseofholomorphiccurvesembeddedincomplextori.Thefirstmajordifferenceisthatthebranescannowbetakentobecompact.Thisrequiresanextensionofthequantizationschemeproposedin[1],inwhichwetaketheunderlyingHilbertspacetobethefinitedimensionalspaceofholomorphicsectionsofaspecificlinebundleoverthebrane.Thesecondmajordifferencewiththebasiccaseconsideredin[1]comesfromthefactthat,althoughthetargetspaceisstillflat,wecannotquantizedirectlythecoordinatefunctions,sincetheyaremultivaluedonthemembrane.WesolvethisproblemusinganextensionofT-duality[6]appropriatetothepresentcontext,andwerelatethebraneconfigurationsonthetorustoU(N)Yang-Millsconfigurationsonthedualtorus.TheresultingU(N)bundleisnon-trivial,eventhoughithasvanishinginstantonnumber.InfactthefirstChernclassofthebundlecorrespondstothehomologyclassofthemembraneembeddedintheoriginaltorus.MoreovertheBPScharacteroftheoriginalmembraneconfigurationisnowtranslatedinadualconditionfor1thecorrespondingU(N)gaugepotential.Moreprecisely,weshowthatthecorrespondingfieldstrengthFisalmost-anti-self-dual,inthesensethatF12+F34∼εF13−F24=F14+F23=0.Wealsoshowthattheconfigurationsdescribedarestablefortotopologicalreasons.Thebasisofmostofthediscussioninthispaperisthequantizationschemewhichisanalyzedindetailinthefirstpartofthepaper.Inparticularaveryimportanttoolwhichisanalyzedatlengthandusedrepeatedly,isaspecificnon-commutativeproduct(calledstar-productanddenotedwith⋆)betweenfunctionsonthebrane.Weshowthattheproduct⋆,whichwasintroducedin[1],isanassociativeoperation.Inparticular,ifwerecastourresultinthelanguageofdeformationquantization[14][15][16][19][17],weshowthattheformulaforthestar-productcanbeusedtoquantizeanysymplecticmanifoldofrealdimension2.Thestructureofthispaperisasfollows:insection2wereviewtheresultsof[1]andextendthemtoamoregeneralsetting,whichisneededinthesubsequentpartofthediscussion.Inparticularweintroducethegeneralconceptofquantizationusedthroughoutthepaper,andwedefineastarproduct⋆onthespaceoffunctions.Section3isentirelydevotedtoshowthattheproduct⋆isactuallyassociative