Marginal deformations in string field theory

arXiv:0704.2222v2[hep-th]28Sep2007PreprinttypesetinJHEPstyle-HYPERVERSIONarXiv:0704.2222AEI-2007-023MarginaldeformationsinstringfieldtheoryEhudFuchs,MichaelKroyterMax-Planck-Institutf¨urGravitationsphysikAlbert-Einstein-Institut14476Golm,[e-mail],[e-mail],DepartamentodeF´ısicaFaculdadedeCiˆenciaseTecnologia,UniversidadedoAlgarve8005-139Faro,[e-mail]:Wedescribeamethodforobtaininganalyticsolutionscorrespondingtoexactmarginaldeformationsinopenbosonicstringfieldtheory.Forthephotonmarginaldefor-mationwehaveanexplicitanalyticsolutiontoallorders.OurconstructionisbasedonapuregaugesolutionwherethegaugefieldisnotintheHilbertspace.Weshowthatthesolutionitselfisneverthelessperfectlyregular.Westudyitsgaugetransformationsandcal-culatesomecoefficientsexplicitly.Finally,wediscusshowourmethodcanbeimplementedforothermarginaldeformations.Keywords:StringFieldTheory.Contents1.Introduction12.Thephotonmarginaldeformation33.Gauge-choiceindependence73.1Ignoringcounterterms73.2Includingcounterterms104.Evaluatingcoefficientsintheoscillatorrepresentation135.Othermarginaldeformations166.Comparingwithformersolutions187.Conclusions19A.Provingtheequivalenceof(6.4)and(6.5)201.IntroductionTherecentanalyticconstructionbySchnabl[1](seealso[2–11])ofanexactsolutioninclassicalopenstringfieldtheory(OSFT)[12]correspondingtothetachyonvacuumhasgivenarenewedimpetusinusingOSFTasatoolintheanalysisofopenstringvacua.Inparticular,itallowedforaproofofthefirsttwoofSen’sconjectures[13,14].Aclassofrelatedsolutionsbasedongeneralprojectorshasbeendevelopedin[15].Thisadvanceraisedthehopethatothersolutionscanalsobefound,suchaslumpsolutionsandmarginaldeformations.ThatmarginaldeformationscanbedescribedwithintheframeworkofstringfieldtheorywasshownbySen[16–18].There,itwasshownthatboundarymarginaldeformationscanbedescribedwithinOSFT,whilebulkmarginalde-formationscanbedescribedusinganon-polynomialclosedstringfieldtheory,suchas[19].ThesesolutionswerefirstinvestigatedusingleveltruncationintheSiegelgauge[20].Otherstudiesofthesesolutionsappearedin[21–29].Morerecently,arecursiveprocedurehasbeendevelopedbyusingthetechniquesem-ployedinSchnabl’ssolution(inparticular,theB0gauge),yieldingexactlymarginaldefor-mationsorderbyorderinaparameterλparameterizingtheexactlyflatdirection[30,31].Thisapproachwasgeneralized[32,33]todescribealsothefirstanalyticalsolutionsofsuperstringfieldtheory[34].–1–Theapproachof[30,31]givesanexplicitsolutionformarginaldeformationgeneratedbycurrentoperatorswhichhavearegularOPEwiththemselves.Forthemoreinterestingcase,suchasthephotonmarginaldeformation,wheretheOPEofthevertexoperatorV(z)definingthemarginaldeformationwithitselfisV(0)V(z)∼1/z2,divergencesariseastheseparationsoftheboundaryinsertionsusedinconstructingthesolutiongotozero.Thismakesitnecessarytoaddcountertermsinordertocancelthesedivergences.However,theformofthesecountertermsisknownonlyuptothethirdorderanditisnotaprioriclearthatcountertermsforhigherordersexist.Inthisworkweproposeanalternativeapproachtowardtheanalyticalconstructionofexactlymarginaldeformations.Itisbasedonsolutionsthatareformallypuregauge,butnonethelessnontrivialasthegaugeparameterweemployisnotinthephysicalHilbertspace.Aswewillshow,thismethodallowsustoobtainanexplicitlydefinedsolution,perturbativeintheabove-mentionedparameterλ.Astheinsertionsofthevertexoperatorinourapproachremainatafinitedistance,thedivergencesencounteredin[30,31]donotarise.However,itturnsoutthatthesolutionsweobtainatfirstinstancearesingularinthesensethatthereisanon-normalizabledependenceonthecenterofmasscoordinatex0.Onlywithacarefullychosensetofcountertermsisitpossibletoregularizethisunwanteddependenceonx0,suchthatthesolutionsareinfactindependentofthecenterofmasscoordinate.Ourapproachstartsoffwiththepuregaugesolutionsforstringfieldtheoryoftheform[2,6]Ψ=(1−λφ)Q11−λφ=Qφλ1−λφ,(1.1)whichhavethestructureofapuregaugesolutiongeneratedbythegaugefieldΛ=−log(1−λφ).(1.2)Thisisthecase,becausethefinitegaugetransformationinstringfieldtheorytakestheformΨ→e−Λ(Ψ+Q)eΛ,(1.3)andwelookforagaugeequivalentofthetrivialsolutionΨ=0.Itseemsthatthisprocedurecannotgenerateanynon-trivialsolutions.However,thissolutioncanbecomeaphysicaloneinseveralways.Thefirstoptionistohaveafiniteradiusofconvergencewithrespecttoλ.Byarescalingofφthisvalueλcritcanbesettounity.Then,for|λ|1thesolutionisindeedagaugesolution,whilefor|λ|1itisnotwelldefined.Forλ=±1thesolutioncaneitherbegaugesolution,notbewelldefined,orbeaphysicalsolution.Indeed,Schnabl’ssolutionforthetachyonvacuum[1]isjustofthisform.There,thesolutionatλ=−1isnotwelldefined,whiletheλ=1caseisthedesiredsolution(afterproperregularization).Avariantofthismethodwouldbetohaveλcrit=∞suchthatthesolutioniswelldefined,butnon-gaugeatleastinoneofthelimitsλ→±∞.Anotheroptionwouldbetohavesomesortofasingularφ,suchthatthesolutionitselfisregular.Then,Qφisanexactsolutionina“largeHilbertspace”,butisanontrivialelementofthecohomologywhenconsideringthesmallerHilbertspace.Thisisthemethodthatwewanttoemploy.–2–Therestofthepaperisorganizedasfollows