Associativity Anomaly in String Field Theory

arXiv:hep-th/0202030v212Feb2002CITUSC/02-004hep-th/0202030UT-983AssociativityAnomalyinStringFieldTheoryItzhakBarsaandYutakaMatsuobaCIT-USCCenterforTheoreticalPhysics&Dept.ofPhysicsandAstronomyUniversityofSouthernCalifornia,LosAngeles,CA90089-2535,USAE-mail:[e-mail],UniversityofTokyoHongo7-3-1,Bunkyo-ku,Tokyo113-0033,JapanE-mail:[e-mail]dstudyoftheassociativityanomalyinopenstringfieldtheoryfromtheviewpointofthesplitstringandMoyalformalisms.Theoriginoftheanomalyisreducedtothepropertiesofthespecialinfinitesizematriceswhichrelatetheconventionalopenstringtothesplitstringvariables,andisintimatelyrelatedtomidpointissues.Wediscusstwostepstocopewiththeanomaly.Weidentifythefieldsubspacethatcausestheanomalywhichisrelatedtotheexistenceofclosedstringconfigurations,andindicateadecompositionofopen/closedstringsectors.Wethenproposeaconsistentcutoffmethodwithafinitenumberofstringmodesthatguaranteesassociativityateverystepofanycomputation.1IntroductionRecentdevelopmentsinvacuumstringfieldtheory[1]–[20]arepromisingforadescriptionofD-branesandclosedstringsfromtheviewpointofopenstringfieldtheory[21,22].Inparticular,asimplepictureofthestringysolitonsemergesasnoncommutativesolitonsofopenstringfields.Thealgebraicstructureofstringfieldtheoryisgreatlysimplifiedbydescribingtheopenstringintermsoftwohalvesseparatedbyamidpoint-thesplitstringformalism[23][1]-[4].Bydoingso,theopenstringfieldisregardedasaninfinitedimensionalmatrix.Furthermore,bytransformingtoaFourierspaceoftheoddfullstringmodesandusingsomespecialmatricesthatnaturallyemergedinthesplitstringformalism(theT,Rdiscussedbelow),Witten’sstarproductistranslatedintothestandardMoyalproductinvolvingthephasespaceoftheevenfullstringmodes[4].Thisestablishesanexplicitlinkbetweenopenstringfieldtheoryandnoncommutativegeometryinaformwhichisfamiliarinold[24]andrecentliterature[25].Inthiscontext,stringfieldtheorycomputations,includingtheconstructionofnoncommutativesolitonsbecomegreatlysimplified[26].Thereare,however,somesingularitiesinthesplitstringformalismthatrequiredeeperunderstanding.Inparticular,inthedescriptionofD-branessomeinfinitiesandzeroesareencountered[11,14].SoonemustlearnhowtoconsistentlyextractfinitequantitiesfrominfinitedimensionalmatrixcalculationsorMoyal-starcomputationsthathavesingularbehavior.Relatedphenomenawereobservedlongago[27,28,29],suchasthebreakdownofassociativityofthestarproductforcertainstringfieldconfigurations.Suchanomaliestypicallyappearforstringfieldsthatcorrespondtoclosedstringexcitations,suchasthosethatrepresentspace-timediffeomorphisms.Thebreakdownofassociativitywouldhaveahugeinfluenceontheverystructureofopenstringfieldtheory.Forexample,Witten’sactionwouldnotenjoyagaugesymmetryinthepresenceofanomalies.Itisthereforeimportanttoknowpreciselywhenandhowsuchananomalyoccursandhowitcanbetreated.Thepurposeofthisletteristopresentasystematicstudyofsuchanomalies.WewillshowthattheassociativityanomalyemergesfromtheverypropertiesoftheinfinitedimensionalmatricesT,Rthatrelatetheopenfullstringdegreesoffreedomtothesplitstringdegreesoffreedom,thusclarifyingtheoriginandthestructureoftheanomaly.Indeed,wewillseethattheHorowitz-StromingeranomalyishiddeninthematricesT,Rthemselves.Inordertotamesuchananomaly,wewilldiscusstwosteps:(1)separationoftheopen/closedstringsectors,(2)aconsistentcutoffmethod.Inthefirststep,westudythestructureoftheHilbertspaceforsplitstringsmorecarefully.WefindthattheHilbertspacecanbedecomposedintotwosectors.Thefirstsectoristhesubspaceinwhichassociativityismaintained.WemayregarditastheHilbertspaceofopenstringfields.Inthesecondsectorassociativityisexplicitlybroken.Thissubspaceischaracterizedbythefactthatunderstarproductswithsingularfieldsthelocationofthemidpointshifts(contrarytothedefinitionoftheoriginalstarproduct).Thus,weshowthesimplifiedoriginoftheanomaly,withadirectrelationtotheHorowitz-1Stromingeranomaly,throughitsrelationtothegaugevariationofclosedstringdegreesoffreedomthatarehiddenintheopenstringformalism.Itisnotclearhowtopreciselyseparatetheopen/closedsectorswhilemaintainingtheinfinitenumberofstringmodes.Therefore,inthesecondstep,weproposeaconsistentcutoffmethodusingafinitenumberofstringmodes,andsendingthenumberofmodestoinfinityattheendofcomputations.TheessenceofourcutoffmethodistomaintainallthecrucialalgebraicrelationssatisfiedbythematricesTandRforanynumberofmodes.ThiscutoffmethodisthenvalidinboththesplitstringandMoyalformalisms.Withafinitenumberofmodes,associativityismaintainedatallstagesofanycomputation.Whenthenumberofmodesissenttoinfinitytheoriginoftheanomalyemergesintheformof∞∞.Theambiguityinsuchquantitiesisseentobetheoriginoftheanomaly.Withtheconsistentcutoffmethodtheambiguityisresolvedandauniquevalueisobtainedinthelimit.Withthecutoffmethodallquantitiesoftheopenstringfieldtheory(off-shellvertex,integration,etc.)arereadilyexpressedintermsofafinitenumberofmodes,andcomputationsarecarriedoutinastraightforwardwaywithoutworryingabouttheassociativityanomaly.Weexpectthatourconsistentcut-offtheorywouldalsobequiteusefulinthenumericalstudy