D-instantons and Matrix Models

arXiv:hep-th/9903050v516Jun1999hep-th/990305[e-mail]hematicsandTheoreticalPhysicsCambridgeUniversityCambridgeCB39EW,UKWediscusstheMatrixModelaspectofconfigurationssaturatingafixednumberoffermioniczeromodes.Thisnumberisindependentoftherankofthegaugegroupandtheinstantonnumber.Thiswillallowustodefinealarge-NclimitoftheembeddingofKD-instantonsintheMatrixModelandmakecontactwiththeleadingterm(themeasurefactor)ofthesupergravitycomputationsofD-instantoneffects.WeshowthattheconnectionbetweenthesetwoapproachesisdonethroughtheAbelianmodesoftheMatrixvariables.March19991.IntroductionOverthepastfouryearssometremendousprogressandinsightsaboutthenon-perturbativeandglobalbehaviourofsupersymmetricgaugetheory,superstringtheoryandsupergravityhaveappeared.Alltheseadvancesarefoundedonawebofconsistentcross-checkedconjecturesculminatingwiththeideaofM-theoryasthemotherofalltheories.Mostoftheimpressiveandexactnon-perturbativeresultswerederivedbyconsideringBPSsaturatedamplitudes.Duetothesaturationofthefermioniczero-modesthesetermsareprotectedbysomenonrenormalisationtheoremsandcanbecomputedbothintheperturbativeandnon-perturbativeregimes.Thisisthecaseoftheeight-fermiontermsinthethree-dimensionalsuperYang-Millstheory[1],theD-instantonsofthetypeIIbstringtheory[2],orthewrappedD1-branearoundtoriofdimensionssmallerthanfiveinthetypeItheory[3].Inthesecases,theD-instantonscontributionsbelongtoahalf-BPSmultipletofthetheory,andtheycomefromamplitudeswheresixteenoreightfermioniczero-modes,respectivelyforthetypeIIbandtypeItheory,havetobesoakedup.ThesurprisingaspectoftheseresultsisthateveninavacuumcontainingKD-instantonsitisonlynecessarytosaturatethesamefixednumberoffermioniczeromodes,independentlyofK.ThisisbecauseoftheexistenceofthresholdboundstatesofD0-branesandtheactionofT-dualitywhichexchangeKDp-branesontopofeachothersinglywrappedarounda(p+1)-toruswithoneD(p−1)-branewrappedK-timesaroundap-torus,andthefactthatthepresenceofwindingmodesdoesnotbreaksupersymmetry.1InthecontextofthecorrespondencebetweenthesupergravityresultsandtheCFTcomputation,thisindependenceoffermioniczero-modeswithrespecttotheinstantonnumberbecomesmuchmoreobscure.LetusconsiderforexamplethecaseofavacuumcontainingKD3-branes:itwasclaimedin[4]andimpressivelystrengthenbytheresultof[5]thatthelarge-Nclimitofthistheoryisincorrespondencewiththefour-dimensionalsuperYang-MillstheorywithgaugegroupSU(Nc)inthelarge-Nc’tHooftlimit(Ncg2YM=constant).ThisconfirmationusedasectorofthetheorywithafixednumberoffermioniczeromodesindependentlyofNc.ThepuzzleisthatinthesuperYang-Millscaseonewouldnormallythinkthatwhentherankofthegaugegroupincreasesthereareextrafermioniczero-modesandtheresultcannotmatchthesupergravityones.Infactitwasunderstoodby[5]thatanembeddingofaconfigurationofKinstantonsintheSU(Nc)grouphasa1ItshouldbenotedthatisnottrueforboundstatesofD-particleandanti-D-particle.1fixednumberoffermioniczero-modesindependentofthenumberofinstantonsandtherankofthegroup.Themainpurposeofthepresentarticle,istoexplainthatinthecontextofMatrixmodels,suchaconfigurationoffermioniczero-modescanberealized,andcanleadtoawayofdefiningalarge-NclimitoftheMatrixmodel.Insection2,wewillintroducethesupergravityaspectoftheD-instantoneffects,andinsection3wewillintroducetheMatrixmodelthatwewilluseinthefollowing.Section4containsadiscussionofthedynamicsofthesemodels,emphasingtheimportanceofconsideringagauge-invariantmodel.Insection5wecomputethepartitionfunctionoftheMatrixmodelwithvarioussymmetries,andmaptheseresultstothesupergravityresultsinsection6.Section7containsadiscussionofthisapproach.AppendixAcontainsanexplicitcomputationofthepartitionfunctionofthesupersymmetricMatrixmodelwithtworealsuperchargesandAppendixBsummarisesourconventionsfortheΓmatricesusedinthetext.2.TheSupergravitySideThetypeIIbchiralversionofthetendimensionalsupergravityispeculiarinseveralrespects.First,beingchiralwithmaximumnumberofsupersymmetries,withtwosetsofsixteenrealcomponentsuperchargesofthesamespace-timechirality,definedwithrespecttotheprojector(1±Γ11)/2,intendimensionsithasarichermodulispacethanitsnonchiralcounterpartthetypeIIasupergravity.Thesuperspaceformalismofthischiralsupergravitytheorywasworkedoutin[6]andwillbeusedhere.Thesuperspaceformalismusesasupermanifoldwithtenevencoordinates,xμ(μ=0,···,9),andsixteenoddcomplexcoordinates,2θα(α=1,···,16),andtheircomplexconjugates(θα)∗def=θ¯α,thewholesetpackagedinzMdef=(xμ,θα,θ¯α).Ateachpointinthesuperspacetherearesomelocalcoordinatesrelatedtotheone-formdzMbythevielbeindEAdef=dzMEAM,andasforusualRiemannmanifoldsthevielbeinareinvertible,2Thesecoordinateswillbeputincorrespondencewiththezeromodesofthematricialfermions(seesection6.2).2EMAEBM=δBA.ThetangentspaceisdescribedbythecoveringofthegroupSO(1,9)×U(1)B.TheU(1)Bfactorisalocalphasetransformationonthefermioniccoordinatesbyθα→expi2Γ11ξθα,θ¯α→exp−i2Γ11ξθ¯α.Itwasshownin[6]thatthisU(1)BfactorispreciselythefactorappearinginthecosetspaceparametrisationSU(1,1)/U(1)B≡Sl(2,R)/U(1)Bofthistheo