Permeable conformal walls and holography

arXiv:hep-th/0111210v220Feb2002PreprinttypesetinJHEPstyle-HYPERVERSIONCALT-68-2361CITUSC/01-045ITFA-2001-33LPTENS-01/42hep-th/0111210PermeableconformalwallsandholographyC.Bachas1,5,J.deBoer2,5,R.Dijkgraaf2,3,5andH.Ooguri4,51LaboratoiredePhysiqueTh´eoriquedel’EcoleNormaleSup´erieure∗24rueLhomond,75231ParisCedex05,France2InstituteforTheoreticalPhysics,UniversityofAmsterdamValckenierstraat65,1018XEAmsterdam,TheNetherlands3Korteweg-deVriesInstituteforMathematics,UniversityofAmsterdamPlantageMuidergracht24,1018TVAmsterdam,TheNetherlands4CaliforniaInstituteofTechnology452-48,Pasadena,CA91125,USA5InstituteforTheoreticalPhysics,UniversityofCaliforniaSantaBarbara,CA93106-4030,USAAbstract:Westudyconformalfieldtheoriesintwodimensionsseparatedbydo-mainwalls,whichpreserveatleastoneVirasoroalgebra.Wedeveloptoolstostudysuchdomainwalls,extendingandclarifyingtheconceptof‘folding’discussedinthecondensed-matterliterature.Weanalyzetheconditionsforunbrokensupersymme-try,anddiscusstheholographicdualsinAdS3whentheyexist.OneoftheinterestingobservablesistheCasimirenergybetweenawallandananti-wall.Whenthesesep-aratefreescalarfieldtheorieswithdifferenttarget-spaceradii,theCasimirenergyisgivenbythedilogarithmfunctionofthereflectionprobability.Thewallswithholo-graphicdualsinAdS3separatetwosigmamodels,whosetargetspacesaremodulispacesofYang-MillsinstantonsonT4orK3.Inthesupergravitylimit,theCasimirenergyiscomputableasclassicalenergyofabranethatconnectsthewallsthroughAdS3.Wecomparethisresultwithexpectationsfromthesigma-modelpointofview.∗Unit´emixteduCNRSetdel’EcoleNormaleSup´erieure,UMR8549.1.IntroductionStartingwiththepioneeringworkofCardy[1],boundaryconformalfieldtheory(BCFT)hasevolvedintoarichsubjectofgreatphysicalinterest.Thesubjectisofobviousrelevancetothestudyofcriticalphenomenainstatisticalmechanics.Fur-thermore,two-dimensionalconformalboundarystateshaveacquirednewimportanceinrecentyears,asbuildingblocksfortheD(irichlet)branesofstringtheory[2].TheinterplaybetweenthealgebraicapproachofConformalFieldTheory,andthecom-plementarygeometricviewpointofD-branes,hasbeenthethemeofmanyrecentinvestigations(seee.g.[3,4]andreferencestherein).TheusualsettingofBCFTisaspace(time)endingonaboundary.Inthissettingallincidentwavesarereflectedback,∗becausethereisnothingtheycantransmittoontheotherside.Onemay,however,alsoconsiderasituationinwhichtwo(ormore)non-trivialCFT’saregluedtogetheralongacommoninterface.Theinterfacecanbepermeable,meaningthatincidentwavesarepartlyreflectedandpartlytransmitted.Examplesofsuchboundaries(mostlybetweenidenticalCFT’s)havebeendiscussedinthecondensed-matterliterature,seeforinstance[5,6,7,8].Oneofourpurposesinthisworkwillbetoanalyzesuchpermeableinterfacesingeneral,andfromaratherdifferent,moregeometricperspective.Ourinterestinthesequestionswasmotivatedbyanissueinholography.StringtheoryinAdS3hasstaticsolutionsdescribinginfinitely-long(p,q)strings,whichstretchbetweentwopointsontheAdS3boundary[9].InthedualspacetimeCFT[10,11,12,13,14,15]theendpointofa(p,q)stringis,aswewillexplain,aninterfaceseparatingregionswithdifferentvaluesofthecentralchargeordifferentvaluesofthemoduli.Similarconfigurationshavebeenalsodiscussedinhigherdimensions[16,17].Theforceexertedbythestretchedstringonitsendpointstranslates,inthedualinterpretation,totheCasimirforcebetweentwo(ormore,ifoneconsidersstringnetworks)permeableinterfaces.InthispaperwewillcalculatethisCasimirforce,bothintheweak-andinthestrong-couplinglimits.Theresultswefindareinsomewaysreminiscentoftheheavyquark–antiquarkpotentialinfour-dimensionalN=4superYang–Mills[18,19].Fromatechnicalpointofview,aninterfacebetweentwoCFTsisdescribedbyaregularboundarystateinthetensor-producttheory.†Thisisintuitivelyobvioussinceonecan‘fold’spacealongtheinterface,sothatbothCFTsliveonthesameside[5].Permeablewalls,inparticular,aresimplyboundarystatesofthetensorproduct,thatcannotbeexpressedintermsofIshibashistatesofthefactortheories.Their∗Thelanguageissomewhatloose,becausestrictly-speakingaCFThasnoasymptoticparticlestates.Amoreaccuratephrasing,intwodimensions,isthattheboundarystatemapsholomorphicintoantiholomorphicfields,inawaythatcommuteswiththeactionoftheVirasoroalgebra.†Moreprecisely,thetensorproductofthetheoryononesideandofthe‘conjugate’theory,withleft-andright-moversinterchanged,ontheotherside.–1–studydoesnot,therefore,requiredrastically-newtechnology,butitleadstoahostofnovelquestionsandobservableswhicharenotusuallyconsideredinthestandardBCFTsetting.OneexampleofsuchanewobservableistheCasimirenergyofa‘CFTbubble’whichwecalculate.Theplanofthispaperisasfollows.Insection2weintroducethemainideasof‘conformalgluing’inthesimplestcontextofafreescalarfieldtheory,andexplainhowthisisrelatedtoconventionalconformalboundarystates.WecalculatetheCasimirenergyfortwoidenticalinterfaces,separatingregionswithdifferenttarget-circleradii,andshowthatitisgivenbythedilogarithmfunctionofthereflectionprobability.Insection3wegeneralizetheseconsiderationsinseveraldirections.Weshowhowsuperconformalinvarianceofthewallscanbeguaranteedbythecontinuityofappropriately-defin