hep-th990 On the Algebraic K-theory of The Massive

arXiv:hep-th/9905034v15May1999CBPF-NF-022/99LNCC-RP-013/99hep-th/990????May1999OntheAlgebraicK-theoryofTheMassiveD8andM9BranesIonV.VanceaDepartmentofTheoreticalPhysics,StateUniversityofRiodeJaneiroRuaSa˜oFranciscoXavier,524-Maracan˜a,[e-mail]tionbetweentheD8-braneswrappedonanorientablecompactmanifoldWinamassiveTypeIIAsupergravitybackgroundandtheM9-braneswrappedonacompactmanifoldZinamassived=11supergravitybackgroundfromtheK-theoreticpointofview.ByspeculationgontheuseofthedimensionalreductiontorelatethetwotheoriesindifferentdimensionsandbyinterpretingtheD8-branechargesaselementsofK0(C(W))andthe(inequivalentclassesof)spacesofgaugefieldsontheM9-branesastheelementsofK0(C(Z)ׯk∗G)aconnectionbetweenchargesandgaugefieldsisarguedtoexists.ThisconnectionisrealizedasamapbetweenthecorrespondingalgebraicK-theorygroups.1.IntroductionAstheresultoftheanalysisofthenon-BPSbranestates[1]thepictureofthechargesoftheDp-braneswrappedonan(orientable)compactmanifoldWaselementsofthetopologicalK-theoryofWemerged[2,3].ThechargesofallpossibleDp-braneconfigura-tionsactuallytakevalueintheabeliangroupsK0(Y),K−1(Y)andKO0(Y)forTypeIIB,TypeIIAandTypeIbranes,respectively,whereYisthed=10spacetimeandW⊂Y.ByimposingthetadpoleanomalycancellationinTypeIIBandTypeItheories,thegroupsreducetogK0(Y)andgKO0(Y),respectively[3].InTypeIIAtheorythenon-existenceofanyRRboundarystateguaranteesthatthereisnoRRspacetimetadpoleanomaly[4].Severalimportantquestionshavealreadybeenaddressedinliteratureintheframeofthistheory.ThelistincludesthepossibilityofusingtheGrothendieckgroupsandthederivedcategoriesinthestudyofbranecharges[9],thecomputationofbranechargesinvariousbackgrounds[5,7],theanalysisofT-dualityofnon-BPSstates[6,8,12]andtheclassificationofdescentanddualityrelationsamongbranes[10].Anotherimportantproblempointedoutin[3]istheunderstandingoftheK-theoryclassificationofbranechargesfromelevendimensionalpointofview.ThemotivationforsuspectingaconnectionbetweenTypeIIAbranechargesandsomed=11objectscomesfromtheremarkthat,ononehand,theM-theorycompactifiedonS1istheTypeIIAtheoryand,ontheotherhand,theK-theorygroupofTypeIIAbranechargesisgK−1(Y)=gK0(Y×S1)1.ThepresenceofS1inboththeoriessuggeststhatthecircleshouldbeactuallythesame.However,themajorobstructioninrealizingthisideainaconcretemanneristhefactthatthereareno10-branesinM-theory.ThispreventsusfromgivingasensiblephysicalinterpretationofK-theoryofTypeIIAbranesinelevendimensions[3,4].OnewaytocircumventthisdifficultyistouseaK-theorythatsatisfiesthefollowingconditions:i)itallowsaphysicalinterpretationofitselementsind=11andii)itrepresentstheTypeIIAD-branechargesind=10.AtheorythatsatisfiesthetwoconditionsaboveisthealgebraicK-theory[14,15].ThegroupK0(C(X))classifiesthefinitelygeneratedprojectiveC(X)-moduleswhicharejustthespacesofsectionsofvectorbundleswithbasemanifoldX.Sincethesesections,attheirturn,canbeinterpretedasgaugefieldsonX,K0(C(X))satisfiesi)above.Theconditionii)isautomaticallysatisfiedsincebyconstructionK0(C(W))=K0(W)[15].TheothercrucialingredientneccesarytodescribetheD-branechargesfromtheelevendimensionalperspectiveisamapbetweenthebranesandthecorrespondingobjectsin1IngeneralonecanreplacethegK0(Y)groupwithK0(Y)groupbecausethecorrespondingK-theoriesareactuallyKc-theories,i.e.thevectorbundlessatisfy,fromphysicalrequirements,someappropriatecompactsupportconditions[3].1d=11.ThismapcanalternativelybethoughtasamapbetweenTypeIIAtheoryandthed=11theoryinwhichtheobjectsaredefined.WehaveseenabovethatifthealgebraicK-theoryistobeusedtheobjectsarethespacesofgaugefields.Then,asthepreviousdiscussionandtheoriginalformulationoftheproblemsuggests,thesoughtfortheoryind=11shouldbeM-theoryorarelatedone.Anotherwaytothinkofthisistonoticethatthemapbetweend=10andd=11theoriesshouldappearfromanaturalconnectionbetweenthese.ForM-theoryandTypeIIAtheorythereissuchofconnectiongivenbythedimensionalreduction.InthefollowingweshallillustratetheseideasonasystemformedfromD8andM9braneswrappedoncompactmanifoldswhichareembeddedinmassiveTypeIIAandmassived=11supergravitybackgrounds,respectively.Themassived=11supergravityistheoneproposedin[17].Themotivationforchoosingthissystemreliesonthefollowingknownfacts.TheD8-braneisthehigheststableTypeIIAD-braneanditschargestakevalueinK0(W).Evenif8-brane-antibraneconfigurationsdonotcontainalllowerdimensionalbraneconfigurations,aswaspointedoutin[4],theycontainlowerdimensionalbranecharges.ThismakesK0(W)anontrivialinterestingobject.ItisalsoknownthataD8-branecanbeobtainedfromaM9-branebydoubledimensionalreduction2.AM9-branemovesfreelyinamassived=11supergravitybackgroundwithaKillingisometry[16,17,18].Themassived=11supergravityisconnectedtothemassiveTypeIIAsupergravitybydimensionalreduction.Moreover,itssolitonicsolutionsincludeallM-branesfromwhichallTypeIIAbranescanbeobtainedbydirectordoubledimensionalreductions.3Weanalysethepossibilityofusingthedimensionalreductiontoconnecttheobjectsofinterestinthetwotheories.InordertoconstructamapbetweenthespacesofgaugefieldsontheM9-braneandtheD8-branechargeshavetoassociate