Dilaton Deformations in Closed String Field Theory

arXiv:hep-th/0502161v326Jul2005hep-th/0502161MIT-CTP-3587DilatonDeformationsinClosedStringFieldTheoryHaitangYangandBartonZwiebachCenterforTheoreticalPhysicsMassachusettsInstituteofTechnologyCambridge,MA02139,[e-mail],[e-mail]oremimpliesthatthecontributiontothedilatonpotentialfromcubicinteractionsofalllevelsmustbecancelledbytheelementaryquarticself-couplingofdilatons.Weusethisexpectationtotestthequarticstructureofclosedstringfieldtheoryandtostudytherulesforlevelexpansion.WeexplainhowtousetheresultsofMoellertocomputequarticinteractionsofstatesthat,justlikethedilaton,areneitherprimarynorhaveasimpleghostdependence.Ouranalysisofcancellationsismadericherbydiscussingsimultaneousdilatonandmarginaldeformations.Wefindevidencefortwofacts:asthelevelisincreasedquarticinteractionsbecomesuppressedandclosedstringfieldtheorymaybeabletodescribearbitrarilylargedilatondeformations.1Contents1IntroductionandSummary22Dilatonandmarginalfieldpotentialfromcubicinteractions42.1Directclosedstringcomputation..............................52.2Contributionstod4calculatedusingOSFT........................93Settingupelementaryquarticcomputations133.1Thequarticterminthestringaction............................133.2Strebeldifferentialandlocalcoordinates..........................153.3Quarticinteractionsforstateswithc¯cghostfactor....................174Theexplicitcomputationofquarticcouplings194.1Elementarycontributiontoa2d2..............................194.2Elementarycontributiontod4................................215Analysisandconclusions235.1Cancellationsandfits....................................235.2Quarticsuppressionandattemptstodefinelevel.....................265.3Openquestions........................................28ACubicpotentials29BTransformationlaws301IntroductionandSummaryLevel-expansioncomputationsinopenstringfieldtheoryhavebeenausefultoolinthestudyofopenstringtachyoncondensation[1].Theearlyattemptstocomputethepotentialforthe(bulk)closedstringtachyonofbosonicstrings[2,3]weredonebeforelevelexpansionwasunderstoodandtheresultswereinconclusive.Clearerresultswereobtainedrecentlyinthecomputationofpotentialsforthetwistedtachyons[4,5]thatliveonorbifoldcones.Aworkablelevelexpansionschemerequiresthatfinitenumberofcouplingsbeconsideredateachcomputationstage.Sinceclosedstringfieldtheoryisnonpolynomialitisnotobviousthatlevelexpansionworks.Ifaclassofclosed-stringcomputationscanbedoneinlevelexpansion,itisthennecessarytocomputehigher-ordercouplingsefficiently.TheresultsofMoeller[6]makethispossibleforthecaseoffour-pointcouplings.Moellerhasprovided2theRiemann-surfacedatanecessarytocomputearbitrarycouplingsoffourstringfields:aconcretedescriptionofthesubspaceV0,4⊂M0,4ofthemodulispaceoffour-puncturedspheresandthelocalcoordinatesaroundthefourpuncturesforeverypuncturedsphereinV0,4.Inapreviouspaper[7]weconsideredmarginaldeformationsinclosedstringfieldtheory.Themarginalparameter,calleda,wasthatassociatedwiththedimension-zeroprimaryoperatorc¯c∂X¯∂X.WhenthecoordinateXlivesonacircletheoperatorinducesachangeofradius.TheoperatorismarginalevenwhenthecoordinateXisnoncompact,butaddingittotheactiondoesnotchangethecorrelatorsoftheconformaltheory.Weusedthismarginaloperatortotestthequarticstructureofclosedstringfieldtheoryandthefeasibilityoflevelexpansion.Wecheckedthevanishingoftheeffectivepotentialfora.Inthelevelexpansionthequartictermsgeneratedbythecubicinteractions(toalllevels)mustbecancelledbytheelementaryquarticinteractionoffourmarginaloperators.Weconfirmedthisprediction,thusgivingevidencethatthesign,normalization,andregionofintegrationV0,4forthequarticvertexareallcorrect.Thiswasthefirstcalculationofanelementaryquarticamplitudeforwhichtherewasanexpectationthatcouldbechecked.Wealsoextendedthecalculationtothecaseofthefourmarginaloperatorsassociatedwithtwospacecoordinates.Inthispaperweconsideranontrivialextensionoftheaboveresults.Westudythepotentialforthezero-momentumdilaton,thefielddassociatedwiththeoperator12(c∂2c−¯c¯∂2¯c).Complicationsarisebecausethisoperatorisnotmarginal:ithasdimensionzerobutitisnotprimarysinceitfailstobeannihilatedbyL1andby¯L1.Thedilatontheorem[8]statesthatashiftintheexpectationvalueofdcorrespondstoachangeinthestringcouplingconstant.Aroundtheflatspacetimebackgroundthereisnopotentialforthedilaton,soitbehaveslikeamarginalfield.Therefore,apredictionsimilartothatforthefieldaexists:thequartictermsd4inducedbythecubicinteractions(toalllevels)mustbecancelledbytheelementaryquarticinteractionoffourdilatons.Wewillverifythisprediction.Inclosedstringfieldtheoryallquartictermsthathavebeencomputedtodateinvolvestatesthatareprimaryandhavec¯cghostdependence.Thesestatesareoff-shellonlybecausetheirdimensionisnotzero.Thecomputationsofelementaryfour-stringcouplingsinthispaperinvolvedilatonstates,whicharenonprimarystateswithnon-standardghostdependence.Theantighostinsertionsofthefour-stringinteractionbecomequitenontrivial:theyarenotoftheformb−1¯b−1actingonthemovingpuncture.ThestepsthatmustbetakenusingMoeller’sdatatocomputesuchgeneralfour-pointinterac