Matrix Models and D-branes in Twistor String Theor

arXiv:hep-th/0511130v22Mar2006PreprinttypesetinJHEPstyle-HYPERVERSIONhep-th/0511130ITP–UH–21/05MatrixModelsandD-BranesinTwistorStringTheoryOlafLechtenfeldandChristianS¨amannInstitutf¨urTheoretischePhysikUniversit¨atHannoverAppelstraße2,D-30167Hannover,GermanyEmail:[e-mail],[e-mail]:Weconstructtwomatrixmodelsfromtwistorstringtheory:onebydimen-sionalreductionontoarationalcurveandanotheronebyintroducingnoncommutativecoordinatesonthefibresofthesupertwistorspaceP3|4→CP1.WecommentontheinterpretationofourmatrixmodelsintermsoftopologicalD-branesandrelatethemtoarecentlyproposedstringfieldtheory.Byextendingoneofthemodels,wecancarryoveralltheingredientsofthesuperADHMconstructiontoaD-braneconfigurationinthesupertwistorspaceP3|4.Eventually,wepresenttheanaloguepictureforthe(super)Nahmconstruction.Keywords:D-branes,SuperstringsandHeteroticStrings,IntegrableFieldTheories,MatrixModels.Contents1.Introduction22.HolomorphicChern-SimonstheoryonP3|432.1Thecomplextwistorcorrespondence32.2Therealtwistorcorrespondence42.3AntiholomorphicvectorfieldsonP3|4ε62.4FormsonP3|4ε72.5Holomorphicallyembeddedsubmanifoldsandtheirnormalbundles82.6HolomorphicChern-Simonstheory82.7Supersymmetricself-dualYang-Millstheory103.Matrixmodels113.1MatrixmodelofN=4SDYMtheory123.2MatrixmodelfromhCStheory123.3NoncommutativeN=4SDYMtheory153.4NoncommutativehCStheory163.5Stringfieldtheory184.IdentificationwithD-braneconfigurations194.1ReviewofordinaryD-braneswithinD-branes194.2SuperD-branes214.3TopologicalD-branesandthematrixmodels214.4InterpretationwithinN=2stringtheory214.5ADHMequationsandD-branes224.6SuperADHMconstructionandsuperD-branes244.7TheSDYMmatrixmodelandthesuperADHMconstruction254.8Extensionofthematrixmodel264.9D-branesinanontrivialB-fieldbackground275.DimensionalreductionsrelatedtotheNahmequations285.1TheD-braneinterpretationoftheNahmconstruction285.2ThesuperspacesQ3|4andˆQ3|4305.3Fieldtheoriesanddimensionalreductions305.4TheNahmconstructionfromtopologicalD-branes316.Conclusionsandoutlook32–1–1.IntroductionThebasicideaoftwistorstringtheory[1]1istheunionoftwistorgeometrywithCalabi-YaugeometryinthesupermanifoldCP3|4.ThisspaceissimultaneouslyasupertwistorspaceandaCalabi-YausupermanifoldandonecanuseitasatargetspaceforatopologicalB-model,whichcanbeshowntobeequivalenttoN=4supersymmetricallyextendedself-dualYang-Mills(SDYM)theory.ByincorporatingadditionalD-instantonsintothepicture,onecanobtainthefullN=4supersymmetricYang-Mills(SYM)theoryanduseitstwistorialandstringtheoreticaldescriptionforcalculatingamplitudesinthistheory.Foragoodoverviewoftheresultsinthisarea,seee.g.[3].AlreadyanumberofvariationsandreductionsoftheunderlyingsupertwistorspaceCP3|4anditsopensubsetP3|4=CP3|4\CP1|4havebeenconsidered[4]-[13].Inthispaper,wewanttodiscussdimensionalreductionsofthebosonicdimensionsofP3|4andconstructmatrixmodelsfromthetwistorstring.Forobtainingthese,wewillusetwomethods.StartingpointofbothisholomorphicChern-Simons(hCS)theoryonthenoncompactsupertwistorspaceP3|4,whichisarank2|4(complex)vectorbundleovertheRiemannsphereCP1andbecomesdiffeomorphictoR4|8×CP1afterimposingrealityconditionsonitssections.Here,R4|8isthemodulispaceof(real)holomorphicsectionsofP3|4.Inthefirstapproach,wewilldimensionallyreduceP3|4totherank0|4vectorbundleCP1|4overCP1.Viathetwistorcorrespondence,thisamountstoreducingthemodulispaceR4|8byitsbosoniccoordinatestoR0|8:CP1|4∼=R0|8×CP1.Onthefieldtheoryside,wewillobtainanactioncorrespondingtomatrixquantummechanicswithcomplextimeoverCP1|4.Forasimilarconstructionontheconifold,see[14].Thesecondmethodwillbetoimposeanoncommutativealgebraonthebosonicco-ordinatesofthemodulispaceR4|8,whichyieldsanoncommutativealgebraforthefibrecoordinatesofP3|4.Thisturnsthederivativesanditscoordinatesintooperatorsinaninfi-nitedimensionalFockspace,whichcanberepresentedbyinfinitedimensionalmatrices.Inthissense,hCStheorywillagainbereducedtomatrixquantummechanics,astheintegraloverthebosonicmodulibecomesatraceovertheFockspace.StartingfromhCStheorywithgaugegroupGL(n,C),thefirstmethodyieldsamatrixmodelwhosefieldcontenttakesvaluesintheLiealgebraofGL(n,C).Oneexpectsthismodeltobeequivalenttothesecondoneinanappropriatelimitn→∞.Furthermore,bothmodelscanbereducedbyintegratingovertheremainingbosoniccoordinateofCP1|4,whichleadstomatrixmodelsofN=4SDYMtheory.Havingdefinedthesematrixmodels,wewillelaborateontheirphysicalinterpretationanddiscusstheirrelationtothecubicstringfieldtheoryproposedin[15]aswellastheirrˆolesaseffectiveactionsforcertainD-braneconfigurations.TheD-braneinterpretationofthematrixmodelswillalwaysbetwofold:Ontheonehand,wehavephysicalD-branesintypeIIBsuperstringtheorywiththemodulispaceR4|8beingasubspaceoftheten-dimensionaltargetspace.Ontheotherhand,wehavetopologicalD-branesofB-typetopologicalstringtheoryinthesupertwistorspaceP3|4.ByextendingthematrixmodelonCP1|4,wewillbeabletocarryoveralltheingredientsof1See[2]foranalternativeformulation.–2–theD-braneinterpretationoftheADHMconstructionfromthemodulispaceR4|8tothesupertwistorspaceP3|4.Introducingfurtherdimens