N-String Vertices in String Field Theory

arXiv:hep-th/9306029v14Jun1993FTUVIFICN-STRINGVERTICESINSTRINGFIELDTHEORYJ.BORDES1DepartamentdeF`ısicaTe´orica.UniversitatdeVal`encia.C./Dr.Moliner50.E-46100.Burjassot(Val`encia)Spain.,A.ABDURRAHMAN2andF.ANT´ONDepartmentofTheoreticalPhysics,UniversityofOxford1KebleRoad,Oxford,OX13NP,U.K.AbstractWegivethegeneralformofthevertexcorrespondingtotheinteractionofanarbitrarynumberofstrings.Thetechniqueemployedreliesonthe“comma”representationofStringFieldTheorywherestringfieldsandinteractionsarerepresentedasmatricesandoperationsbetweenthemsuchasmultiplicationandtrace.ThegeneralformulationpresentedhereshowsthattheinteractionvertexofNstrings,foranyarbitraryN,isgivenasafunctionofparticularcombinationsofmatricescorrespondingtothechangeofrepresentationbetweenthefullstringandthehalfstringdegreesoffreedom.January19931AlsoatIFIC,CentroMixtoUniversitatdeValencia-CSIC.Spain.2AlsoatMathematicalInstitute,24-29,St.Giles,Oxford,OX13LB,U.K.1INTRODUCTIONStringFieldTheoryhasprovidedaconsistentpictureforthetreatmentofOpenStrings[1].RecentlysomeadvanceshavebeenmadeintheformulationofaFieldTheoryforClosedStrings[2].Thisiswelcomedfromaphenomenologicalpointofviewsinceclosedstringsappeartogiveasuitablepictureofstringphysicsatlowenergy.Acompleteunderstandingoflowenergystringphysicsseemstorequirethetreatmentofstringsinthisframework.TheClosedStringFieldTheory(CSFT)proposedin[2]hastheparticularityofrequiringanon-polynomialactioninwhichateverysteponehastoincludeatermintheactioncorrespondingtotheinteractionofanarbitrarynumberofstringsoveraworldsheetgivenbytheso-calledrestrictedpolyhedra.Theedgesofthesepolyhedraplaytheroleofthemodularparametersandonerestrictstheregionofintegrationovertheseinaprescribedway.Inthetheory,theinteractiontermsareinterpretedastheoverlappingofclosedstringsinawaywhichresemblestheoriginaltheoryforopenstringsduetoWitten[1].Ontheotherhandthereisthesuggestionin[3]thattheoverlappingofclosedstringscanbeformulated,usingthepropertyofreparametrizationinvarianceofthestringamplitudes,astheoverlappingofstandardstringsegmentswhich,followingtheexampleoftheopenstrings,canbeconsideredashalf-strings.Inturn,oneinterpretsthestringfunctionalsasmatricesandtheinteractionbetweenstringsastheproductandtraceofthesematrices.Thisfeatureshowsthat,eveninthecaseofclosedstrings,thehalf-stringpictureisrelevantintheconstructionofthestringinteraction.However,theformulationfromaFockspaceapproachseemstobeaformidabletasksinceintheabsenceofacompactformulationoneshouldcalculateseparatelyeverytermintheaction.Hence,ourmainmotivationcomesfromthefactthat,following[3],theN-facedpolyhedradescribingtheinteractionofNstringsattheleveloftheaction,canbewrittenasareparametrizationofthevertexinacontactinteraction.Thus,asafirststep,onefindsusefultoworkoutthecontactinteractionvertexforanarbitrarynumberofstrings.ThepurposeofthisworkisthenthestudyoftheN–stringscontactinterac-tioninageneralform.Thiswillformthebaseofacompactformulationoftheclosedstringvertexinthenon-polynomialtheory.WeperformthecalculationforopenstringssincetheirformulationatthelevelofFockspaceisfirmlyset-tled.Ontheotherhand,theysharemanyofthepeculiaritiesoftheoperatorformulationforclosedstrings[4].Theresultsobtainedhereareverysimilar–fromthetechnicalpointofview–tothosefortheclosedstringsotheycanbeadapted,withminormodifications,tothelatter.Relatedtothispoint,wewillseethat,apartfromthefactthathalf-stringsplayaconceptualroleintheformulationofclosedstringinteractions,linkingtheCSFT[2]withtheapproachdescribedin[3],theyrevealasausefultechnicaltoolforthetreatmentofstringamplitudes,bothforopenandclosedstrings.InthetheoryofWitten[1]forstringfieldstheinteractionbetweenstringsisdefinedbya⋆(product)andR(integration)whicharedefinedthroughthejoiningofhalfstrings.Inparticular,theinteractionbetweentwostringstogive1athirdoneisdefinedfromthe⋆as:(ψ∗φ)[x]=ZDyψ[xL;y]φ[y;xR],(1.1)whereψrepresentsthestringfields.Theinteractiontakesplacethroughthejoiningofhalfstrings.Theoperationofintegration,whichallowsustoobtaininvariantquantities,isdefinedas:Zψ=ZDxψ[x;x].(1.2)Thereforethefundamentaldegreeoffreedomfromtheinteractionpointofviewisthehalfstring.Itisthenjustifiedtoadoptanapproachinwhichthehalfstringplaysexplicitlyanimportantrole.Wewillbaseourformalisminthe“comma”representationofstringfieldtheorydevelopedin[5].Inthe“comma”formulation,wesingleoutthemidpointandrepresentstringfieldsasmatrices.Thestringisdividedintoleftandrightpartswhichplaytheroleofrowandcolumnindicesofthosematrices.Interactiontakesplacesimplybymultiplying(product(1.1))andtakingthetrace(integration(1.2))overthematrices.TheadvantageofthisapproachisthatonecanhandleinacompactformtheNstringcontactinteractionforanyarbitraryN.Infact,thevertexforNstringsissimplygivenby:VN=Zdx(π2)Tr[A1.A2...AN],(1.3)whereAiarethematricesrepresentingthestringstates.WemustpointoutthattheformerequationdoesnotgivethetreelevelN-stringinteraction,butonlythecontactterm.Inordertogetthetreelevelamplitudeonecouldconsiderthereparametrizationapproachasdescribedin[3]togetthecorrectmodulispaceofparameters.Thishowever,isbeyondth