Introduction to M(atrix) theory and noncommutative

arXiv:hep-th/0012145v329Jul2001IntroductiontoM(atrix)theoryandnoncommutativegeometryAnatolyKonechny1∗andAlbertSchwarz2†1DepartmentofPhysics,UniversityofCaliforniaBerkeleyandTheoreticalPhysicsGroup,MailStop50A-5101LBNL,Berkeley,[e-mail]2DepartmentofMathematics,UniversityofCaliforniaDavisDavis,[e-mail]1,2008AbstractNoncommutativegeometryisbasedonanideathatanassociativealgebracanberegardedas”analgebraoffunctionsonanoncommutativespace”.ThemajorcontributiontononcommutativegeometrywasmadebyA.Connes,who,inparticular,analyzedYang-Millstheoriesonnoncommutativespaces,usingimportantnotionsthatwereintroducedinhispapers(connection,Cherncharacter,etc).ItwasfoundrecentlythatYang-Millstheoriesonnoncommutativespacesappearnaturallyinstring/M-theory;thenotionsandresultsofnoncommutativegeometrywereappliedverysuccessfullytotheproblemsofphysics.Inthispaperwegiveamostlyself-containedreviewofsomeaspectsofM(atrix)theory,ofConnes’noncommutativegeometryandofapplicationsofnoncommutativegeometrytoM(atrix)theory.ThetopicsincludeintroductiontoBFSSandIKKTmatrixmodels,compactificationsonnoncommutativetori,areviewofbasicnotionsofnoncommutativegeometrywithadetaileddiscussionofnoncommutativetori,MoritaequivalenceandSO(d,d|Z)-duality,anelementarydiscussionofinstantonsandnoncommutativeorbifolds.Thereviewisprimarilyintendedforphysicistswhowouldliketolearnsomebasictechniquesofnoncommutativegeometryandhowtheycanbeappliedinstringtheoryandtomathematicianswhowouldliketolearnaboutsomenewproblemsarisingintheoreticalphysics.Contents1Introduction32Yang-Millstheoryreducedtoapoint43Matrixmodels83.1IKKTmatrixmodel...............................8∗ResearchsupportedbytheDirector,OfficeofEnergyResearch,OfficeofHighEnergyandNuclearPhysics,DivisionofHighEnergyPhysicsoftheU.S.DepartmentofEnergyunderContractDE-AC03-76SF00098andinpartbytheNationalScienceFoundationgrantPHY-95-14797.†ResearchsupportedinpartbyNSFgrantDMS-980100913.2BFSSmatrixquantummechanics........................83.3BoundstatesandscatteringinBFSSmodel..................104Compactifications114.1Compactificationonacircle.RelationbetweenIKKTandBFSSmodels...114.2CompactificationonaregularT2........................134.3CompactificationonanoncommutativeT2...................144.4CompactificationsonTdandTdθ.........................154.5Noncommutativegeometryfromaconstantcurvaturebackground......155Noncommutativegeometry175.1Algebrasoffunctionsandvectorbundles....................175.2NoncommutativeRdspaces...........................185.3Endomorphismsandconnections........................195.4Involutivealgebras................................225.5Noncommutativetori...............................225.6Projectivemodulesovernoncommutativetori.................245.7Connectionsonnoncommutativetori......................265.8K-theory,Cherncharacter............................285.9Moduleswithnondegenerateconstantcurvatureconnection.........325.10Heisenbergmodulesasdeformationsofvectorbundles.............366NoncommutativeYang-MillsandsuperYang-Millstheories376.1YMandSYMonfreemodules..........................376.2SYMonarbitraryprojectivemodules......................396.3BPSstatesonT2θ.................................406.4Supersymmetryalgebra..............................456.5Topologicaltermsfromgeometricquantization.................476.6Spectrumoftranslationoperators........................486.7EnergiesofBPSstatesind=2,3,4.......................497Moritaequivalence517.1Moritaequivalenceofassociativealgebras....................517.2GaugeMoritaequivalence............................537.3InvarianceofBPSspectrum...........................578Noncommutativeinstantons588.1InstantonsonT4θ.Definitionandasimpleexample...............588.2Instantonaction..................................609Noncommutativeorbifolds629.1Noncommutativetoroidalorbifolds.......................629.2K-theoryoforbifolds..............................649.3K-theoryofnoncommutativeZ2orbifolds....................6610Literature6721IntroductionFirstofallwewouldliketogiveanexpositionofsomebasicfactsaboutM(atrix)theorythatiscompletelyindependentofanystringtheorytextbooks.WewillconsiderM(atrix)modelasastartingpointandwewillshowthatstringtheorycanbeobtainedfromit.MorepreciselyM(atrix)theoryshouldbeconsideredasanonperturbativeformulationofstringtheory.OursecondgoalistogiveanexpositionofConnes’differentialnoncommutativegeometryandtoshowthatitarisesverynaturallyintheframeworkofM(atrix)theory.Wewillshowthatnoncommutativegeometrycanbeusedintheconsiderationofdualities,inanalysisofBPSstates,etc.Weaddressthepresentreviewtomathematicianswhowouldliketolearnaboutsomemathematicalproblemsarisingattheforefrontofmoderntheoreticalphysicsandtophysi-cistswhowouldliketostudysomebasicnotionsofnoncommutativegeometryandseehowtheycanbeappliedtophysics.WedonotassumethatamathematicianreadingthisreviewhasanypreliminaryknowledgeofstringtheoryorM-theory.Howeversomeacquaintancewiththebasicnotionsofsupersymmetryisdesired(forexampleseeIASschoollectures[12]