Closed String Field Theory Quantum Action and the

arXiv:hep-th/9206084v123Jun1992IASSNS-HEP-92/41MIT-CTP-2102hep-th/9206084June1992CLOSEDSTRINGFIELDTHEORY:QUANTUMACTIONANDTHEB-VMASTEREQUATIONBartonZwiebach⋆SchoolofNaturalSciencesInstituteforAdvancedStudyOldenLanePrinceton,NJ08540ABSTRACTThecompletequantumtheoryofcovariantclosedstringsisconstructedindetail.ThenonpolynomialactionisdefinedbyelementaryverticessatisfyingrecursionrelationsthatgiverisetoJacobi-likeidentitiesforaninfinitechainofstringfieldproducts.ThegenuszerostringfieldalgebraisthehomotopyLiealgebraL∞encodingthegaugesymmetryoftheclassicaltheory.ThehighergenusalgebraicstructureimpliestheBatalin-Vilkovisky(BV)masterequationandthusconsistentBRSTquantizationofthequantumaction.FromtheL∞algebra,andtheBVequationontheoff-shellstatespacewederivetheL∞algebra,andtheBVequationonphysicalstatesthatwererecentlyconstructedind=2stringtheory.Thestringdiagramsaresurfaceswithminimalareametrics,foliatedbyclosedgeodesicsoflength2π.Thesemetricsgeneralizequadraticdifferentialsinthatfoliationbandscancross.Thestringverticesaresuccinctlycharacterized;theyincludethesurfaceswhosefoliationbandsareallofheightsmallerthan2π.⋆Permanentaddress:CenterforTheoreticalPhysics,MIT,Cambridge,Mass.02139.SupportedinpartbyD.O.E.contractDE-AC02-76ER03069andNSFgrantPHY91-06210.1.IntroductionandSummaryofResultsOnecanhardlyoverstatetheneedforacompleteformulationofstringtheory.Thisnecessityarisesbothataconceptuallevelandatacomputationallevel.Attheconceptuallevelwedonotyetknowtheunderlyingprinciplesofstringtheory.Suchunderstandingappearstobeaprerequisiteforaddressingdeepissuesofquantumgravityinthecontextofstringtheory.Atthecomputationallevelwedonothaveyettheabilitytocalculatenonperturbativeeffects.Sucheffectsseemtobeneededinordertoobtainrealisticstringmodels.Thesubjectofthispaperisthecovarianttheoryofclosedstringfields[1–8]Thistheory,Ibelieve,representsconcreteprogressintheconstructionofacompleteformulationofstringtheory.ThecurrentformulationofclosedstringfieldtheoryisbasedintheBRSTapproach,whichoriginatedintheworkofSiegel[9].ThetheoryappearstobethenaturalclosedstringanalogoftheinteractingopenstringfieldtheoryconstructedbyWitten[10].Aswewilldiscuss,itsucceedsingeneratingtheperturbativedefinitionofthetheorystartingfromanactionbasedonagaugeprinciple.Italsomakesconceptuallymanifestthefactorizationandunitarityofthestringamplitudes.Theclosedstringfieldtheoryisapparentlythefirstfieldtheoryforwhichthemostsophisticatedmachineryforquantization,theBatalin-Vilkovisky(BV)field-antifieldformalism[11–13],isnecessaryandusefulinitsfullform.Atpresent,however,theclosedstringfieldtheoryisnotyetacompleteformulationofstringtheory.Itsmostglaringshortcomingisthatitsformulationrequiresmakingachoiceofaconformalbackground.Inacompleteformulation,thisbackgroundshouldariseasaclassicalsolution.Therefore,thequestionwhetherstringfieldtheoryisthe‘right’approachtostringtheory,isatpresentthequestionwhetherstringfieldtheorycanbeformulatedwithouthavingtochooseaconformalbackground.Wemaybeencouragedbythefactthatdespiteinitialdifficultiestheconstructionofcovariantclosedstringfieldtheorywaspossible.Theresultingtheory,aswillbeelaboratedhere,hasaninterestingandnovelalgebraicstructurearisingfromsubtlepropertiesofmodulispacesofRiemannsurfaces.Moreover,asrecentworkofSenindicates,thestringfieldtheory,whilenotmanifestlyindependentoftheconformalbackgroundchosentoformulateit,doesincorporatesuchbackgroundindependencetosomedegree[14].Thereisthereforelittleevidencethatwefacedifficultiesthatcannotaltogetherbesolvedintheframeworkofstringfieldtheorybyextensionsormodificationsofthecurrentformulations.Therealobstructiontotheconstructionofclosedstringfieldtheorywas,allalong,theabsenceofadecompositionofthemodulispacesofRiemannsurfacescompatiblewithcovariantFeynmanrules.Adecompositionsuitablefortheclassicaltheory(genuszerosurfaces)wasfoundinRefs.[1,2]byincorporatingandgeneralizingtheclosedstringextensionoftheopenstringvertex[10]andthetetrahedronofRef.[4].Themathematicallypreciseproofthatthiswasacorrectdecompositionrevealedthatthestringdiagramsweregivenbyminimalareametrics[8],andthissuggestedanaturalwaytogeneralizetheconstructionforhighergenussurfaces.ThedecompositionofmodulispaceforthequantumtheorywasaddressedinRef.[5]usingminimalareametricsandtheideasof[7],andthealgebraicstructureoftheresultingtheorywassketchedinRefs.[5,6].Theassumptionsusedintheseworksregardingplumbing2propertiesofminimalareametricshavenowbeenestablishedtoalargedegreeinrecentworksbyRanganathan,Wolf,andtheauthor[15,16].Eventhoughthemaindifficultywasgeometrical,workingoutthealgebraicimplicationsofthegeometricalconstructionsisbothinstructiveandnecessary.Thisamountstogivingtheexplicitconstructionofthestringfieldvertices,withduecareoftechnicalissueslikeantighostinsertions,andprovingthesetofidentitiestheysatisfy.Thesepropertiesmustalsobeshowntoguaranteetheconsistencyofthequantumtheoryconstructedwiththestringfieldvertices.Thisisoneofthemainpurposesofthepresentpaper,itamountstogivingthedetailedderivationandconstructionofthequantumclosedstringfieldtheorysketchedinRefs.[5,6].Essen