On non-abelian structures in open string field the

arXiv:hep-th/0105245v330May2001hep-th/0105245ITEP-TH-25/01YCTP-SS4-01Onnon-AbelianStructuresinFieldTheoryofOpenStringsAntonA.Gerasimov1andSamsonL.Shatashvili2*1InstituteforTheoreticalandExperimentalPhysics,Moscow,117259,Russia2DepartmentofPhysics,YaleUniversity,NewHaven,CT06520-8120Multi-branebackgroundsarestudiedintheframeworkofthebackgroundindependentopenstringfieldtheory.Asimpledescriptionofthenon-abeliandegreesoffreedomisgiven.Algebraofthedifferentialoperatorsactingonthespaceoffunctionsonthespace-timeprovidesanaturaltoolforthediscussionofthisphenomena.May19,2001*OnleaveofabsencefromSt.PetersburgBranchofSteklovMathematicalInstitute,Fontanka,St.Petersburg,Russia.1.IntroductionTheunderstandingoftherelationbetween(largeN)gaugetheoriesandstringtheories[1],[2],[3],[4](seealsodiscussionin[5]),isprobablyoneofthemostimportantproblemsinthemoderntheoreticalphysics.Duringthelastdecadethisconnectionshowedupinstringtheoryliteraturewithvariousfaces:multipleD-branesbackgrounds[6],MatrixTheoryproposal[7],AdS/CFTcorrespondence[8],[9],[10],variousconstructionsofthesolitonicobjectsintermsofmatrixdegreesoffreedom[11],[12],[13],[14].Allthisimpliesveryinterestinginterrelationbetweenthenon-abeliangaugedegreesoffreedom(ofopenstringsastheclosestrelativesofgaugetheoriesinthestringworld)andpurelygeometrical”gravitational”degreesoffreedom(ofclosedstrings).FromthemathematicalsideitseemstoprovidethemostintriguingexampleoftheunityofAlgebraandGeometry.ThesimplestexampleofsuchrelationisgivenbymultipleD-branebackground.Theimportantlessonlearnedfromtherecentstudiesoftheopenstringfieldtheoryisthenewpointofviewonsuchbackgrounds.AfewyearsagoonewouldthinkaboutD-branesassolitonsintheclosedstringtheory,butbynowitiswell-knownthatD-branescanbealsoconsideredassolitonsintheopenstringfieldtheory.ThiscanbeseenbothfromcubicCSstringfieldtheoryof[15]andbackgroundindependentopenstringfieldtheory[16],[17],[18],[19].InthelatteranyboundaryCFT(withworld-sheetbeingadisk)isacriticalpointforcorrespondingstringfieldtheoryaction;thisactionevaluatedonclassicalsolutioncoincideswiththeworld-sheetpartitionfunction.ThusanyD-branebackgroundshouldprovideaclassicalsolutionforthebackgroundindependentopenstringfieldtheorylagrangian1.AlthoughtheaboveargumentgivessimpleexplanationofthefactthatD-branesaresolitonsinopenstringfieldtheory,manyissuesremainunclarified.Forinstanceitisnotclearhowthenon-abeliangaugefieldsemergeinthecaseofthenearcoincidentD-branesinthisformalism.Theinterpretationoftheclosedstringsinthesetermsalsoremainsanopenquestion(seediscussioninlecturenotes[21]andcommentsbelow).1Infactthiswasalreadynoticedin[17]andreinterpretedinmoderntermsin[20]inordertoarguethatfieldtheoryofopenstringscontainsallpossiblestringybackgrounds,includingD-branesandclosedstrings,asclassicalsolutionsandthuscanserveasadefinitionoffull(consistent)off-shellstringtheory.1InthispaperwewilladdressthefirstquestionusingtheformalismofbackgroundindependentopenstringfieldtheorywhichturnedouttobeveryeffectiveinverifyingtheSen’sconjectures[22]asitwasdemonstratedin[23],[24],[25]2.Oneofthemainreasonsfortheappearanceofthenon-abelianalgebraicstructuresintheopenstringtheoryisanobviousgeometricalinterpretationoftheopenstringasasomekindofmatrix(openstringshavetwoends)actingintheappropriateHilbertspace[28],[29],[30],[31],[32].TheinfinitedimensionalityofthisHilbertspaceleadstosomeunusualphenomenaresponsibleforthepeculiarpropertiesofstringtheory.Letusstressthatthenon-commutativityisabasicpropertyofstringtheoryandisnotaspecialfeatureofthebackgroundswithnon-zerovacuumvalueoftheB-field.Theexistenceofthis”matrix”structureoftheopenstringtheoryallowsustoaddnon-abelianindexestotheopenstringwavefunctionwiththehelpofChan-Patonfactors.Basicallyonehasatensorproductofthestringy”matrix”algebraandsomeabstractmatrixalgebra.OneoftheimportantoutputsofthediscoveryoftheD-braneswastheunderstandingthatChan-PatonfactorsalsohaveageometricaloriginandnaturallyappearinmultipleD-branesbackgrounds[6].Wearguethatthenon-commutativestructureoftheopenstringtheoryonamanifoldMmaybecapturedinareasonableapproximationbythenon-commutativealgebraofdifferentialoperatorsonM.ThealgebraofdifferentialoperatorsonthemanifoldtogetherwithitsvariouscompletionsisthemodelofthematrixalgebraappearinginMatrixTheoryofstrings[7].Byconsideringthealgebraofdifferentialoperatorsinsteadofthe(abstract)infinitedimensionalmatricesoftheMatrixTheorywepartiallyloosebackgroundinde-pendencebutinsteadwegetaveryexplicitdescriptionofthenon-commutativedegreesoffreedom.Thealgebraofdifferentialoperatorsmaybeconsideredasanon-commutativedefor-mationofthealgebraoffunctionsonthecotangentbundleT∗Mdefinedbythecanonicalquantizationwiththesymplecticformω=Pidpi∧dxi.Thewidelydiscussednon-commutativestructureappearingwhenB-fieldisnon-zero[33],[34],[35],fromthispointofview,maybedescribedassomeB-dependentdeformationofthebasicsymplecticstruc-ture.Fulldescriptionofthenon-commutativestructureintheopenstringtheoryobviouslyincludesthedifferentialoperatorsonthespaceofthemapsofthe(holomorphic)diskstothespace-time.Someremarksonth