JHEP05(2014)052 PublishedforSISSAbySpringerReceived:December31,2013Revised:March19,2014epted:April11,2014Published:May13,2014 CommentsonR´enyientropyinAdS3/CFT2 EricPerlmutterDAMTP,CentreforMathematicalSciences,UniversityofCambridge,Cambridge,CB30WA,
U.K.E-mail:
E.Perlmutter@damtp.cam.ac.uk Abstract:WeextendandrefinerecentresultsonR´enyientropyintwo-dimensionalconformalfieldtheoriesatlargecentralcharge.Todoso,weexaminetheeffectsofhigherspinsymmetryandofallowingunequalleftandrightcentralcharges,atleadingandsubleadingorderinlargetotalcentralcharge.Theresultisastraightforwardgeneralizationofpreviouslyderivedformulae,supportedbybothgravityandCFTarguments.TheprecedingstatementspertaintoCFTsinthegroundstate,oronacircleatunequalleft-andright-movingtemperatures.ForthecaseoftwoshortintervalsinaCFTgroundstate,wederivecertainuniversalcontributionstoR´enyiandentanglemententropyfromVirasoroprimariesofarbitraryscalingweights,toleadingandnext-to-leadingorderintheintervalsize;thisresultappliestoanyCFT.Whentheseprimariesarehigherspincurrents,suchtermsareplacedinone-to-onecorrespondencewithtermsinthebulk1-loopdeterminantsforhigherspingaugefieldspropagatingonhandlebodygeometries. Keywords:AdS-CFTCorrespondence,ConformalandWSymmetry,Holographyandcondensedmatterphysics(AdS/CMT) ArXivePrint:1312.5740v2 Openess,cTheAuthors.ArticlefundedbySCOAP3. doi:10.1007/JHEP05(2014)052 JHEP05(2014)052 Contents 1Introduction
2 2Reviewofrecentprogressin2dCFTR´enyientropy
5 2.1CFTatlargec
6 2.2GravityatsmallGN
9 2.3Quantumcorrections
9 3R´enyientropyinhigherspintheories 10 3.1Rudimentsofhigherspinholography 10 3.2Bulkarguments 12 3.3CFTarguments 14 3.4Lookingahead 17 4TheshortintervalexpansionforspinningVirasoroprimaries 17 4.1Generalities 18 4.2Spinningprimariesrevisited 19 4.3Summaryofresultsandtheirentanglemententropylimit 22 5R´enyientropyinhigherspintheoriesII:shortintervalexpansion 22 5.1Higherspinfluctuationsonhandlebodygeometries 23 5.2Comments 25 6ChiralasymmetryandcL=cRCFTs 25 6.1Chirallyasymmetricstates 25 6.2Chirallyasymmetrictheories 27 6.3Spinningtwistfieldsfromgravitationalanomalies 29 6.4Aholographicperspective 29 7Discussion 31 ATheW3algebra 34 BQuasiprimariesintensorproducthigherspinCFTs 34 CR´enyientropyfromW-conformalblocks?
35 –1– JHEP05(2014)052 1Introduction Two-dimensionalconformalfieldtheoriesareamongthemostwell-understoodquantumfieldtheories.Likewise,theirasionaldualitywiththeoriesofgravityinthree-dimensionalanti-deSitterspacesubjectstheAdS/CFTcorrespondencetosomeofitsmoststringentanddetailedchecks.Nevertheless,wecontinuetodiscovernewfeaturesofthesetheories,evenintheirgroundstatesorinholographiclimits.Ifwecouldunderstandallphysicsofthegroundstateof2dCFTsatlargecentralcharge,thiswouldconstituteamajoradvancement. TherehasbeenarecentsurgeofinterestinentanglemententropyandtheassociatedR´enyientropy,partlyduetoitspromiseinhelpingusachievethatgoal.TheseareexamplesofquantitiesthatcontainalldataaboutagivenCFT,processedinawaythatemphasizesthewayinformationanized.EveninthegroundstateofaCFT,R´enyientropiesarehighlynontrivial;theirstudyprovidesaquantumentanglementalternativetotheconformalbootstrap. WecandefinethegroundstateR´enyientropy,inanyspacetimedimension,asfollows.Atsomefixedtime,wesplitspaceintotwosubspaces.Uponformingareduceddensitymatrixρbytracingoveronesubspace,onedefinestheR´enyientropySnas Sn=1logTrρn1−n (1.1) wheren=1isapositive,realparameter.Inthelimitn→1,thisreducestothevonNeumannentropy,otherwiseknownastheentanglemententropy,SEE≡limn→1Sn.Intwodimensions,thesespatialregionsareunionsofNdisjointintervals,withendpointszi,wherei=1,...,2N.TheR´enyientropycanputedeitherasthepartitionfunctiononahighergenusRiemannsurfaceusingthereplicatrick,orasacorrelationfunctionof2NtwistoperatorsΦ±locatedattheendpointszi.(Seee.g.[1]foranoverview.) Itwasconvincinglyarguedin[2,3],buildingon[4,5],thatgroundstateR´enyientropiesexhibituniversalbehaviorincertainclassesofCFTsatlargecentralcharge.Thisisastrongstatement,akintotheCardyformulafortheasymptoticgrowthofstatesinany2dCFT.Subjecttocertainassumptionsthatwewilldescribeindetailbelow,localconformalsymmetryispowerfulenoughpletelyfixtheR´enyientropiesforanynumberofdisjointintervals,atleadingorderinlargecentralcharge.TakingtheentanglementlimitamountstoarigorousderivationoftheRyu-Takayanagiformula[4,6]forentanglemententropyinthedual3dgravitytheories,asappliedtopureAdS3.SimilarconclusionsapplytoCFTsatfinitetemperatureoronacircle. Itisnaturaltowonderhowfaronecanpushtheseconclusions.Theresultsof[2,3]applytochirallysymmetricstatesofCFTswithasparsespectrumatlowdimensions,andequallylargeleftandrightcentralcharges,cL=cR.ArethereotherfamiliesofCFTsinwhichR´enyientropybehavesuniversally?
Doweknowthegravitydualsofsuchtheories?
Ifwedo,canweformulatetheargumentsingravitationallanguage?
IsR´enyientropyalsouniversalinstateswithasymmetrybetweenleft-andright-movers?
Furthermore,suchgeneralizationswouldbeusefulinhoningtheholographicentanglementdictionary:ifweeedingeneralizing,wecanunderstandwhichfeaturesoftheseCFTsandtheir –2– JHEP05(2014)052 dualsparticipateintheuniversality,andwhichdonot.Inquiriesofasimilarspiritwereinvestigatedin[5,7,8]. Thegoalofthispaperistoshowthatonecananswereachofthesequestionsaffirmatively.WewillconsiderCFTswithhigherspinsymmetry;withcL=cR;andonthetoruswithunequalleft-andright-movingtemperatures.Holographically,wewillcontendwithhigherspingravity,ologicallymassivegravityandrotatingBTZblackholes,respectively.Thepunchlineisthatatleadingorderinlargetotalcentralcharge,theR´enyientropieschangeinaratherstraightforwardway,ifatall,andthatthesubleadingcorrectionsarecalculable. TherearevariousmotivationstoconsiderCFTswithhigherspincurrents.HigherspinholographyhasprovidedvaluableperspectiveonwhatkindsofCFTscanbeexpectedtohavegravityduals;toboot,thereareconsistenttheoriesof3dhigherspinsthatlieonthegravitysideofspecificdualityproposals[9–11].Thesetheoriesarewell-knowntopresentunfamiliarpuzzlesregardingtherolesofspacetimegeometry,thermodynamicsandgaugeinvariance.Likewise,withrespectputingR´enyientropy,theypresentaninterestingcase[12,13].Ontheonehand,fromtheVirasoroperspectivethecurrentsarenotparticularlyunusual;theoriginalapproachof[2]neitherruledoutnoraddressedthepossiblepresenceofhigherspincurrents.Ontheother,thefactthattheyfurnishanextendedchiralalgebraimpliesthattheproblemmaybecastinthelanguageofalargersymmetry,andthebulksideofthestoryisfarlessunderstoodthanordinarygravity.ItturnsoutthatifweinquireabouthigherspinCFTR´enyientropyformultipleintervalsinthegroundstate,theresultisindependentoftheexistenceofhigherspincurrentstoleadingorderinlargec.OntheCFTside,provingthisstatementboilsdowntoaslightlynontrivialVirasoroprimarycountingproblem;onthehigherspingravityside,thisstatementfollowsfromshowingthatputationisidenticaltotheoneinpuregravity,inwhichtheR´enyientropyputedasapartitionfunctiononahandlebodymanifold. Ageneral2dCFThascL=cR.TheuniversaldynamicsofthecurrentsectorofsuchtheoriesatlargecL+cRisdescribedologicallymassivegravity(TMG)[14,15],anotherfairlyexotictheory.Thistime,wefindthatatleadingorderinlargecL+cR,theR´enyientropyonlyreflectsthegravitationalanomaly[16]forchirallyasymmetricstates,suchasapactCFTwithunequalleft-andright-movingtemperatures.Themannerinwhichitdoessoisquiteclean,isconsistentwiththeCardyformula[17],andimpliesaformulafortheon-shellTMGactionevaluatedonhandlebodygeometries.Thereasonforthissimpleresultisthattheconformalblockpositionofa2dCFTcorrelationfunctionholomorphicallyfactorizesforeachindividualconformalfamily.Thenewcontributioncanbechalkeduptothefactthatthetwistfieldshavespin,proportionaltocL−cR. Letusgiveasnapshotoftheseresultsinequations,deferringfullexplanationtothebodyofthetext.1WemainlyhaveinmindCFTsintheirgroundstate(ontheplane),atfinitetemperatureoronacircle(onthecylinder),orboth(onthetorus).Atlarge(cL,cR), 1Weusetheconventionsof[2],althoughwecallessoryparametersγi,asin[18],ratherthanci. –3– JHEP05(2014)052 inatheorywithorwithouthigherspincurrents,theR´enyientropySnis,toleadingorder, ∂Sn= n(cLγL,i+cRγR,i) ∂zi6(n−1) (1.2) The(γL,i,γR,i)areso-calledessoryparameters,whichpartlydeterminethebehaviorof thestressponents(T(z),T(z)),respectively,inthepresenceofthetwistoperator insertions.Theyaredeterminedbythesametrivialmonodromyproblemgivenin[2,3]— oneontheleft,andoneontheright.Forfixedzi,thechoiceoftrivialmonodromycycles isdeterminedbyaminimizationcondition,justasin[2,3].Theessoryparametersare alsorelatedtotheVirasorovacuumblocksatlarge(cL,cR),whichwecallf0,Landf0,
R, respectively: ∂f0,L=γL,i,∂f0,R=γR,i ∂zi ∂zi (1.3) Whenthestateischirallysymmetric(e.g.inthegroundstate),γL,i=γR,i,onlythetotalcentralchargecontributes,andwereturntotheresultof[2,3].Afamiliarexampleofachirallyasymmetricstatetowhich(1.2)appliesisthecaseofaCFTatfinitetemperatureandchemicalpotentialforangularmomentum.Whenonly,say,cLislarge,theleadingorderresultisagain(1.2),butwithonlythecLpiece.2 Atleadingorderinlargecentralcharges,(1.2)isthefullanswer;whataboutatnextto-leadingorder?
Forthehigherspincase,wewillfocusontheR´enyientropyoftwointervalsinthegroundstate,andputecorrectionsatnext-to-leadingorderinlargec=cL=cR.WorkinginbothCFTandgravity,ourmethodologyisasfollows. First,inCFT,weemploytheperturbativeshortintervalexpansionof[5,19,20].Weuseitputecertainuniversalcontributionsofapairofholomorphicandantiholomorphicspin-scurrentstoR´enyiandentanglemententropy,workingtoleadingandnext-to-leadingorderintheintervalsize.ThesecontributionsaremanifestlyofO(c0).By“universal,”wemeanthatthesetermsaregeneratedinanyCFTthatpossessesthesecurrents,althoughinanyparticularCFTtheremaybeother“non-universal”termsthatareparableorder;thispossibilitydependsonthedetailsofthespectrumandOPEdata,aswedescribeinsubsection4.2. Ingravity,AdS/binedwiththedefinitionofR´enyientropytellsusthatthe 1-loopfreeenergyofhigherspingaugefieldsontheappropriatehandlebodygeometriesMshouldbeproportionaltotheO(c0)contributiontotheR´enyientropyfromtheirdual currents.BecausethehigherspincurrentsdonotcontributeatO(c)asdescribedearlier, thisisinfacttheirleadingcontribution.Following[18],putethese1-loopdeter- minantsinthesmallintervalexpansionusingknownformulasforhandlebodies[21]andspin-sgaugefields[22],andfipleteagreementwithCFT.Thatis,ifSn(s)isthecontributiontotheCFTR´enyientropyfromthepairofspin-scurrents,andSn(s)(M)istheholographiccontributiontotheR´enyientropyobtainedfromlinearizedspin-sgauge 2Theanalogof(1.2)wasonlyprovenin[2,3]toholdforapactCFTinitsgroundstateand,implicitly,forallstatesrelatedbyconformaltransformations.Thesameistrueofourproofof(1.2).Asin[18],wetaketheperspectivethatitholdsmoregenerally,e.g.foraCFTonthetorus.However,itshouldbenotedthatthelatterstatementhasnotbeenproven,eitherfromaCFTorgravitationalpointofview. –4– JHEP05(2014)052 fieldfluctuationsaboutM,thentotheordertowhichpute, Sn(s)x1=Sn(s)(M)1−loop. (1.4) Thevariablexparameterizestheintervalsize. Weemphasizethatthisequalityholdsterm-by-term:thatis,wewillassociateagiventerminthebulkdeterminanttothecontributionofaspecificCFToperatortotheR´enyientropy.Inall,(1.4)isasatisfyingcheckofourproposalthatisquitesensitivetotheexistence,andchoice,ofthebulksaddlepoint:uponpickingtherightone,theonlyfurtheringredientisthemostbasicoftheAdS/CFTcorrespondence,ZAdS=ZCFT. Finally,asapreludetotheO(c0)resultsjustdescribed,wewillderivearesultapplicabletotheshortintervalexpansioninanyCFT:forthecaseoftwoshortintervalsintheCFTgroundstate,wederivetheuniversalcontributionstotheR´enyiandentanglemententropiesfromVirasoroprimarieswitharbitraryscalingweights(h,h),atleadingandnext-to-leadingorderintheintervalsize.Theexplicitexpressionscanbefoundinequations(4.24)and(4.25)below.IfaCFTcontainsnoVirasoroprimarieswithconformaldimension∆=h+h≤1,theseexpressionsreceiveno“non-universal”correctionstotheordertowhichwework.Ourresultinvolvesaproposedcorrectiontoaresultin[20]regardingtheroleofspinning(notnecessarilychiral)primaryoperators.Inparticular,ourresultisindependentoftheoperatorspin,s=h−h. TothoseacquaintedwithcalculationsinvolvingR´enyientropies,higherspintheories,orTMG,theseconclusionsmaynotbesurprising.heless,webelievethatthey,andtheirimplications,areuseful. Thepaperanizedasfollows.Insection2wereviewthenecessarybackgroundmaterial.Section3isdevotedtoR´enyientropiesinhigherspintheoriesatlargecL=cR.Section4zoomsoutandrevisitstheshortintervalexpansionasappliedtogenericspinningprimaryfields;theseresultsareappliedtothehigherspinarenainsection5,whereweverify(1.4).Insection6,weconsiderR´enyientropyinchirallyasymmetrictheorieswithcL=cRandinchirallyasymmetricstates,anditsimplicationsforTMG.AlloftheseargumentsaresupportedbygravityandCFTcalculations.Weconcludeinsection7withadiscussion.Threeappendicesactasthecaboose. ThedaythisworkappearedonthearXiv,sodid[23],whichoverlapswithsections4and5below. 2Reviewofrecentprogressin2dCFTR´enyientropy WhatfollowsisareviewoftheimpressiveprogressmadeinthelastyearputingR´enyientropyin2dCFTsatlargecentralcharge.InsubsequentsectionswewillneedmoredetailsoftheCFTargumentsthanoftheirgravitycounterparts;ordingly,ourtreatmentislopsided.Wereferthereadertotheoriginalpapers[2,3]forfurtherdetailsandtotheirpredecessors[4,5,24],amongmanyothers. –5– JHEP05(2014)052 2.1CFTatlargec WestartontheCFTside,following[2]insomedetail.WeworkwithfamiliesofCFTswhich,basicallyspeaking,are“holographic.”Considerafamilyofunitary,pact,modularinvariantCFTs—callthisfamilyC—withcL=cR=cthatadmitsalargeclimitwiththefollowingtwokeyproperties.First,itobeysclusterposition(andhencehasauniquevacuum),andcorrelationfunctionsaresmoothinafiniteneighborhoodofcoincidentpoints.ThisrestrictstheOPEcoefficientsofthetheorytoscale,atmost,exponentiallyinc.Second,thetheoryhasagap,inthesensethatthedensityofstatesofdimension∆O(c)growspolynomiallywithc,atmost.3DualstoEinsteingravitywithafinitenumberoflightfieldshaveanO(c0)densityofstatesbelowthegap,buttheseareamerecornerofthegeneralspaceofholographicCFTs[25,26]. WenowputingtheR´enyientropyinsomestatethatadmitsuseofthereplicatrick,viatwistfieldcorrelationfunctions.Forconcreteness,considerthecaseoftwodisjointspatialintervalsinthegroundstate,whereby Trρn=Φ+
(0)Φ−(z)Φ+
(1)Φ−(∞)Cn/Zn (2.1) wherez=z.(Wehaveusedconformalinvariancetofixthreepositions.)Thesetwistfields havedimensions c
1 hΦ=hΦ=24n−n (2.2) ThecalculationisperformedintheorbifoldtheoryCn/Zn,whichinheritsaVirasoro×VirasorosymmetryfromC,sowecanexpandthisinVirasoroconformalblocks.Inthez→0channel,say, Φ+
(0)Φ−(z)Φ+
(1)Φ−(∞)Cn/Zn=p(Cp)2F(hp,hΦ,nc,z)F(hp,hΦ,nc,z)(2.3) ThesumisoverVirasoroprimariesintheuntwistedsectorofCn/Znlabeledbyp,withΦ+Φ−OPEcoefficientsCp.Theorbifoldtheoryhascentralchargenc. Unitarity,pactness,modularinvarianceandclusterpositiontellusthatthissumisoveraconsistent,discretesetofpositiveenergystateswithauniquevacuumandgoodbehaviorasoperatorscollide.Tounderstandtheroleofourotherassumptions,wetakethelargeclimit.ThislimitisnontrivialifweholdhΦ/ncandhp/ncfixed.Thereisstrongevidence[27,28]thatinthislimit,theVirasoroblocksexponentiate: limF(hp,hΦ,nc,z)≈exp−ncfhp,hΦ,z c→∞ 6ncnc (2.4) Wewillsaymoreaboutfshortly.Fornow,wenoteonlythatforhpincreasingfunctionofhpatfixedz1. O(nc),itisan 3Actually,[2]makesastrongerclaim,butthisrelaxationisallowed,andwewillrelaxevenfurtherinthenextsection.WethankTomHartmanforvaluablediscussionsonthesematters. –6– JHEP05(2014)052 Inthelargeclimit,wecanapproximatethesumoverpin(2.3)byanintegral.Definingδp≡hp/ncandδp≡hp/nc, Φ+
(0)Φ−(z)Φ+
(1)Φ−(∞)Cn/Zn ≈∞dδp∞dδpC2(δp,δp,nc)exp−ncfδp,hΦ,z+fδp,hΦ,z
0 0
6 nc nc (2.5) Thisisourkeyequation.ThefunctionC2(δp,δp,nc)isthesumofallsquaredOPEcoefficientsofinternalprimarieswithdimensions(δp,δp).Thisdefinitionountsforanydegeneracyd(p)ofsuchoperators: d(p) C2(δp,δp,nc)=(Cip)2 i=
1 (2.6) Asourdefinitionmakesclear,C2(δp,δp,nc)dependsonthecentralcharge.Note,however,thatC2(0,0,nc)isoforderone. Havingmassagedthecorrelatorintotheform(2.5),whatcanweconclude?
Thisdependsstronglyonthegrowthofthemeasure,C2(δp,δp,nc).[2]madethefollowingobservations.First,atanyfixedzintheneighborhoodofz=0,theheavyoperators withδp+δp>O(c)areexponentiallysuppressed.Thisisrequirediftheidentityistobethedominantcontributionforafiniteregionaroundz=
0.Hencetheintegrationin(2.5) isboundedfromabovebyδp∼δp∼O
(1).Second,(2.5)isinfactexponentiallydominatedbythevacuumcontribution,δp=δp= 0,inafiniteregionaroundz=
0.Denotingthevacuumblockasf0hnΦc,z≡f0,hnΦc,z,thismeansthat nc hΦ hΦ Φ+
(0)Φ−(z)Φ+
(1)Φ−(∞)Cn/Zn≈exp −
6 f0 nc,z+f0 ,znc ,(2.7) timesnon-exponentialcontributionsinc.Thereasonisthatourassumptionsabouttheboundedgrowthofstates,andoftheOPEcoefficients,implythatC2(δp,δp,nc)growsatmostexponentiallyinc.4 FirsttreatingthemoremanageablecasethatC2(δp,δp,nc)growspolynomiallywithc—asitdoesforatheorywithanEinsteingravitydual–(2.5)isexponentiallydominated bytheδp=δp=0blocks,onountofthefactthatfisanincreasingfunctionofδpforfixedz.Wecanalsoexpand(2.5)inthet-channel,whereuponz→1−z,withthesame conclusions,thistimeinaregionaroundz=
1.Themostoptimisticperspectiveisthat theintegralsarealwaysdominatedbythevacuumsaddle,δp=δp=
0.Then(2.7)holdsfor01 3(n−1)nc
2 (2.8) 4Wewillexaminethisstatementmorecloselyinsection3. –7– JHEP05(2014)052 Aparallelanalysisholdsinthet-channel,andthefinalanswerfortheR´enyientropy,toleadingorderinlargec,isthen Sn≈ncminf0hΦ,z,f0hΦ,1−z 3(n−1) nc nc (2.9) ThephysicalstatementisthatonlythevacuumanditsVirasorodescendantscontribute totheR´enyientropy. Inthemoreextremescenariothatthemeasurefactorin(2.5)growslikeexp[cβ(δp,δp)] whereβ(δp,δp)isac-independentfunction,theseconclusionsstillholdonountofcluster position,butinasmallerregionaroundz=0orz=
1.Theprecisesizeofthis regiondependsonthetheory. Toputetheformofvacuumblockf0connectsontothegeometricinter- pretationofR´enyientropy.Twistfieldsaside,theR´enyientropycanbedefinedinterms ofapartitionfunctiononasingulargenusgRiemannsurface,Σ.Onecanalwayswrite anyRiemannsurfaceasaquotientofplexplane,Σ=C/Γ,whereΓisanorder gdiscretesubgroupofPSL(
2,C),theM¨obiusgroup.ΓiscalledtheSchottkygroup,and theprocedureofwritingΣassuchaquotientisknownasSchottkyuniformization;more detailsinthiscontextcanbefoundin[2,3,18]. Ifvisplexcoordinateonthecutsurface,theequationthatuniformizesthe surfaceis ψ(v)+T(v)ψ(v)=
0 (2.10) T(v)canberegardedastheexpectationvalueofthestresstensorinthepresenceofheavybackgroundfields(towit,thetwistfields).Writingthetwolinearlyindependentsolutionsasψ1(v)andψ2(v),thecoordinatew=ψ1(v)/ψ2(v)issingle-valuedonΣ. SpecifyingtoputationofR´enyientropyfortwointervalsinthegroundstate,theintervalsareboundedbyendpoints{z1,z2,z3,z4}.Σhasgenusg=n−
1.InthiscaseT(v)takestheform[27]
5 4 T(v)= i=
1 6hΦ
1 γi c(v−zi)2−v−zi (2.11) Theparametersγiarecalledessoryparameters,andtheyaredeterminedbydemand- ingthatψ(v)hastrivialmonodromyaroundgcontractiblecyclesofΣ.Thespecific monodromyconditiononψ1,2(v)isdiscussedindetailin[2,3];weonlywishtoemphasize thatthereisachoiceoftrivialmonodromycycles.Theseessoryparameters,finally,are relatedtotheVirasorovacuumblockf0as ∂f0=γi∂zi (2.12) Theupshotisthatwecanrewrite(2.9)intermsoftheessoryparametersas ∂Sn≈ncminkγ(k)∂z3(n−1) (2.13) whereweminimizeoverthes-andt-channelmonodromies,indexedbyk.(2.13)generalizestoanynumberofintervals[2]. 5ThisequationassumesthattheuniformizationpreservestheZnreplicasymmetry. –8– JHEP05(2014)052 2.2GravityatsmallGN Onthebulkside,thegoalisinprinciplestraightforward.Wewantputethehigher genuspartitionfunctionholographically,byperformingthebulkgravitationalpathintegral overmetricsasymptotictoΣwithappropriateboundaryconditions.Thiswasdonein[3]. Allsolutionsofpure3dgravitycanbewrittenasquotientsofAdS3(inEuclideansignature,H3).Inparticular,solutionsoftheformM=H3/Γasymptoteatconformalinfinity toquotientsoftheplane.SoamongthecontributionstothebulkpathintegralwithΣ ologyaretheso-calledhandlebodysolutionsM,whichrealizetheboundary quotientΣ=C/Γ.HandlebodiesarenottheonlybulksolutionswithΣontheboundary; but,whileunproven,[3]motivatedtheproposalthattheholographicR´enyientropyshould bedeterminedbysaddlepointcontributionsofhandlebodies.Furthermore,[3]assumed thatthedominantsaddlesrespecttheZnreplicasymmetry.Thesemiclassicalapproxima- tionofthepathintegralasasumoversaddlepointsleadsustoidentifytherenormalized, on-shellbulkaction,Sgrav,withminusthelogoftheCFTpartitionfunction;thisimplies theholographicrelation 1Sn=−1−nSgrav(M) (2.14) ToactuallyevaluateSgrav(M)isafairlytechnicalprocedure.Theresultisthatitis fixedbytherelation ∂Sgrav(M)≈ncγi ∂zi
3 (2.15) wheretheγiaretheessoryparametersdefiningtheSchottkyuniformizationofΣasin(2.11).Thatis,Sgrav(M)isproportionaltothelargecVirasorovacuumblock.Fromthisperspective,theenforcementoftrivialmonodromyaroundspecificpairsofpointsontheboundaryisequivalenttodemandingthesmoothcontractionofaspecificsetofgcyclesinthebulkinterior.TheminimizationconditionthatdeterminestheR´enyientropyfollowsfromtheusualHawking-Page-typetransitionpetingsaddlesoffixedgenus. Buildingon[2,3,18]generalizedthisconstructiontothecaseofasingleintervalinaCFToffinitesizeandatfinitetemperatureT,undertheassumptionthatthesingleintervalversionof(2.9)stillholdsonthetorus.Amongotherresults,thisshowedthatfinitesizeeffectsareinvisibleatO(c). 2.3Quantumcorrections WhataboutbeyondO(c)?
ThisiswherethedetailsoftheparticularCFTinquestioneapparent.Inthebulk,theideaisstraightforward:AdS/CFTinstructsuspute1-loopdeterminantsonthehandlebodygeometries,forwhateverbulkfieldsarepresent.Theformulaforsuchdeterminantsisknownforagenerichandlebodysolution[21]: ZH3/Γ= ZH3/Z(qγ)1/2 γ∈
P (2.16) Theqγare(squared)eigenvaluesofacertainsetofelementsγbelongingtotheSchottkygroupΓ;wedefermoretechnicalexplanationtosection5.Thefullbulkanswerisaproduct –9– JHEP05(2014)052 of(2.16)foralllinearizedexcitations,butthegravitoncontributionisuniversal.UnlessMisthesolidtorus,thisexpressionisgenerallynot1-loopexact. Thistechnologywasputtousein[18]forthegraviton,withphysicallyinterestingresults.First,[18]providedasystematicalgorithmputingtheeigenvaluesqγforthecaseoftwointervalsinthegroundstate,inanexpansioninshortintervals.(“Short”meansparametricallysmallerthananyotherlengthscaleintheproblem;inthegroundstateofaCFTonaline,theonlyscaleistheintervalseparation.)Thisrevealed,amongotherthings,anonvanishingmutualinformationforwidelyseparatedintervals,incontrasttotheclassicalresultofRyu-Takayanagi.[18]alsoshowedthatforasingleintervalonthetorus,finitesizeeffectsdoappearat1-loop.Similarcalculationswerecarriedoutin[29]. TheCFTsideofthisstoryisalsostraightforwardinprinciple:onemustcorrecttheleadingsaddlepointapproximationto(2.5).Usingresultsof[20,30]essfullycarriedthisoutforgroundstate,twointervalR´enyientropyduetothevacuumblockalone,i.e.theexchangeofthestresstensoranditsdescendants.Thisissufficienttomatchtothebulkgravitondeterminant. 3R´enyientropyinhigherspintheories Wenowturnourattentiontotheorieswithhigherspinsymmetry.BeforeaddressingR´enyientropy,werecallsomebasicingredients. 3.1Rudimentsofhigherspinholography LetusfirstdefinewhatwemeanbyahigherspinCFT.Thetheoriesweconsiderobeyallassumptionslaidoutinsection2andcontain,inadditiontothestresstensor,extraholomorphicandanti-holomorphicconservedcurrents(J(s),J(s))ofintegerspins>
2.Theholomorphiccurrentshavedimension(s,0),andaplanarmodeexpansion (s) Jn(s) J=zs+n n∈
Z (3.1) andsimilarlyfortheanti-holomorphiccurrents.ThesecurrentsareVirasoroprimaryfields, (s) sJ(s)
(0)∂J(s)
(0) T(z)J
(0)∼z2+z+... (3.2) Thefullsetofcurrentsfurnishesanextendedconformalalgebra(i.e.aW-algebra)withcentralextension.ThereexistsazooofW-algebras;forthesakeofclarity,wewillfocusexclusivelyonCFTswithnomorethanonecurrentateachspin.InparticularwewillmakeuseoftheWNalgebra,whichcontainsonecurrentateachintegerspin2≤s≤N,aswellastheW∞[λ]algebra[31],whichcontainsaninfinitetowerofintegerspincurrentss≥2andafreeparameter,λ.ThesimplestsuchalgebrabeyondVirasoro(≡W2)istheW3algebra,mutationrelationsweprovideinappendixAforhandyreference;see[32]foranextensivereviewofW-algebras. –10– JHEP05(2014)052 ItisconvenienttopresenttherepresentationcontentofagivenCFTusingcharacters.ForWN,theVermamodulecharacter,whichwedenoteχh,N,is ∞h−c/24
1 h−c/24 N−
1 χh,N=q (1−qn)N−1≡q F(q) n=
1 (3.3) wherehistheL0eigenvalueoftheprimaryfield.ThefunctionF(q)issimplythegenerating functionforpartitions.TheCFTvacuumstate|0isinvariantunderaso-called“wedge algebra,”whichactsastheanalogofsl(
2,R)intheVirasorocase.Inparticular,itis annihilatedbymodes J−(sn)|0=
0,|n|N,R).ordinglywemustprojectthese statesoutofthevacuumrepresentation.Aftermoddingoutthenullstatesfromthe vacuumVermamoduleχ
0,N,itiseasytoseethatthevacuumcharacteroftheWNalgebra, χN≡χ
0,N/(nullstates),is N∞−c/24
1 χN=q 1−qn s=2n=s (3.5) AtcertainvaluesofcatwhichWNisthechiralalgebraofarationalCFT(e.g.the3statePottsmodelwithW3symmetry),thevacuumrepresentationcontainsevenmorenullstates;atgenericc>N−1,however,allstatescountedby(3.5)havepositivenorm. ForW∞[λ],thevacuumcharacter,χ∞,istheN→∞limitof(3.5),andthewedgealgebraishs[λ],ahigherspinLiealgebra.ForallW-algebras,thewedgealgebrasepropersubalgebrasonlyupontakingc→∞dueto(1/csuppressed)nonlinearities.(SeeappendixAforthecaseofW3.) GiventhatourCFTisassumedtonothaveanexponential(inc)numberdensityoflightVirasoroprimariesinalargeclimit,thelimittheorycanbeexpectedtohaveaclassicalgravitydual.TheAdS/CFTcorrespondencetellsusthateachpairofspin-scurrentsisdualtoaspin-sgaugefieldinAdS3,denotedϕ(s).LikeordinaryAdSgravity,theoriesofpurehigherspinscanbeefficientlywrittenusingtwocopiesofChern-SimonstheorywithgaugealgebraG×
G, kSbulk=4πSCS[A]−SCS[A] (3.6) with2 SCS[A]=TrA∧dA+3A∧A∧
A (3.7) whereTristheinvariantquadraticbilinearformofG.AlsolikeordinaryAdSgravity,thesefieldsologicalandformaconsistent(classical)theorybythemselves.Thistheorycanalsobesupersymmetrized,non-Abelianized,and/orconsistentlycoupledtopropagatingmatter.ThelatterdefinestheVasilievtheoriesofhigherspins[9,33]. Therealizationofsymmetriesinthisformulationisquiteelegant:G×GisidentifiedwiththewedgealgebraoftheCFT,andtheasymptoticsymmetryofthetheorywithsuitablydefinedAdSboundaryconditionsistheextensionofG×Gbeyondthewedgeto –11– JHEP05(2014)052 thefullhigherspinconformalalgebra[34–36].ThegeneratorsofG,whichwelabelVns,areidentifiedwiththewedgemodesJn(s),with|n|N,R)×sl(
N,R)theory[37]. Onecanmakecontactwithafamiliarmetric-likeformulationintermsoftensorsasfollows.Expandingtheconnectionsalongtheinternaldirections, A= A(ns)(xµ)Vns,A= A(ns)(xµ)Vns s|n|A e= ,ω=
2 2 (3.9) whereuponthemetric-likefieldsϕ(s)arebinationsofproductsoftraceinvariants,ofordersinthevielbein.(Theprecisemapbeyondlowspinsissubjecttosomeambiguitiesthathaveyettoberesolved[38].) Animportantpointisthatwealwaysdemandthatourtheorycontainametric.ThuswerequirethatG⊃sl(
2,R);thelattercorrespondstometricdegreesoffreedom,andthetheory(3.6)admitsaconsistenttruncationtothepuregravitysector.Thisfeatureisnotsharedbythe4dVasilievtheory,whichexplainswhyAdS-Schwarzchildisnotasolutionofthe4dtheory,butallsolutionsofpure3dgravityaresolutionsof(3.6).Inparticular,thislaststatementappliestothehandlebodysolutionsusedputeholographicR´enyientropiesinpuregravity. 3.2Bulkarguments WenowconsiderthehigherspinputationofgroundstateR´enyientropiesformultipledisjointintervals.ThinkingoftheR´enyientropyasproportionaltothefreeenergyoftheCFTonaprescribedRiemannsurfaceΣ,weaimtoestablishthefollowingthreefacts:
1.TheR´enyientropycanputedusingasaddlepointapproximationtothehigherspingravitationalpathintegral.
2.Therelevantsaddlepointsarethesameonesasinpuregravity,namely,thehandlebodysolutionsconsideredin[3].
3.Thefreeenergyitself,andthereforetheholographicR´enyientropy,isthesameasinpuregravity. Westartbyrecallingwhatexactlyitisthatwearetryingpute.Aswenoted,ΣcanalwaysbedescribedasaquotientoftheplanebyasubgroupΓ∈PSL(
2,C),theglobalconformalgroup.ThisstatementisindependentofwhethertherearehigherspincurrentsintheCFT.Wearesittinginthegroundstate,sothereisnohigherspincharge. –12– JHEP05(2014)052 SoourgoalisputethebulkpathintegraloverfieldconfigurationsthatincludeametricwhichasymptotestoΣ=C/Γatconformalinfinity,andhigherspinfieldswhichvanishthere.Inordinarygravity,thesemetricsarethequotientsM=H3/Γ,whichremainsolutionsofthehigherspintheory.Arethereotherconfigurationsinthehigherspintheorythatsatisfytherequisiteboundaryconditions?
Inparticular,thehigherspintheoryadmitsmanymoresolutionsthatarequotientsofAdS3,andweneedtoaddresswhetheranyoftheseentersputationofR´enyientropy. ThenullvectorsintheCFTvacuumVermamodulemaptoaG×GsymmetryofAdS3.Thisbulksymmetryactioninducesaboundaryactionofthewedgealgebra.Thelatterincludesthesubalgebraofglobalconformaltransformations,whosegeneratorsarerepresentedasdifferentialbulkoperators.ButGislargerthansl(
2,R),andtheremainderofthesymmetrycannotbedescribedaspurecoordinatetransformations.Thisisaglobalversionofthestatementthathigherspingaugesymmetriesmixdiffeomorphismswithother,non-geometricsymmetries.6GiventhattheconstructionofaRiemannsurfaceispurelygeometric,noneoftheGsymmetriessittingoutsideofthesl(
2,R)subalgebraisrelevant—thesearebonafidehigherspinsymmetries.Putanotherway,bulkquotientsH3/ΓwithΓ∈/sl(
2,R)willturnonhigherspinfieldsand/orcharge.Therefore,weonlyfocusonEinsteinquotientsofH3. Havinglocalizedtheproblemtothepuregravitysector,itisplaintoseethefinalresult.First,wefollow[3]inassumingthatthereplica-symmetrichandlebodiesMdominatethepathintegral.Soourgoalistoevaluatetheon-shellactionofthehigherspintheoryonthehandlebodiesM.Butthismusttakethesamevalueasinpuregravity,duetotheallowedconsistenttruncationofthehigherspinfields,ϕ(s)=
0. Perhapsitishelpfultowriteanequation.In[40],theauthorsconvertedtheChernSimonsaction(3.6)forG=sl(
3,R)tometric-likelanguage,workingtoquadraticorderinthespin-3field,ϕ≡ϕ
(3),andtoallordersinthemetric.Assuminginvertibilityofthegravitationalvielbein,theresultis 1S= 16π
G d3x√−g(R+2Λ)+ϕϕ+O(ϕ4) (3.10) whereisaparticulartwo-derivativeoperatorderivedin[40].ThepuremetricpartoftheactionissimplyEinstein-Hilbert,asfollowsfromtheexistenceofapuregravitysubsector.SincetheasymptoticchiralW3algebrahascentralchargec=3/2G,thecoefficientofthegravityactionisalsothesameasinpuregravity.Itfollowsthattherenormalizedon-shellactionforthehandlebodygeometriesMisthesameasinpuregravity.Thisaction-basedargumentisnotuniquetoG=sl(
3,R). Weemphasizethattheseconclusionsholdevenforabulktheorywithaninfinitetowerofhigherspins,suchastheG=hs[λ]theory.7 6See[39]foradiscussionofhowtothinkoftheactionofsl(
3,R)transformationsonCFTobservables.7WhenthesearefurthercoupledtomatteralaVasiliev,thesituationislessclear.Inparticular,the Vasilievpathintegralisnotexpectedtoadmitacleandescriptionintermsofsaddlepoints.Inthediscussion section,wewillremarkonthisinthecontextoftheproposeddualityof3dVasilievtheorywithWNminimalmodelsatlargec. –13– JHEP05(2014)052 Insection5,wewillcheckthisproposalatthequantumlevelputing1-loopdeterminantsofhigherspinfieldsonMandshowingthattheymatchadualCFTcalculation.Hadwechosenthewrongsaddle—oriftherewasnodominantsaddlepointatall—thatagreementwoulddisappear. 3.3CFTarguments AnefficientstrategyhereistophrasetheprobleminVirasorolanguage,ratherthanWlanguage.8InaW-algebra,theWcurrentsareVirasoroprimary(seeappendixAfortheW3algebra,forinstance).SoaremanyoftheirW-descendants.IfwecanshowthattherearenottoomanylightVirasoroprimariesasafunctionofc,thentheargumentsof[2]applywithoutmodification. Amoreprecisestatementisasfollows:aslongasthegrowthoflightVirasoroprimariesatsomefixeddimensionintheorbifoldtheoryCn/Znislessthanexp(βc)forac-independentconstantβ,themeasurefactorin(2.5)doesnotgrowfastenoughtospoiltheconclusionsofsection2,andtheR´enyientropyisstillgivenbytheoriginalresultsobtainedintheabsenceofhigherspinsymmetry.ThetwistfieldcorrelatorsinvolvedintheR´enyiputationsareperformedintheorbifoldtheoryCn/Zn.ThistheoryhasmanymoreVirasoroprimariesthanC,anditisthesestatesthatruninthevirtualchannelswhosegrowthweneedtocount.WenowshowthattheaboveconditionismetbyCFTsobeyingourassumptionsthathaveessentiallyanyW-symmetry. LetusfirstphrasetheproblemforageneralW-algebra.ThespectrumofaCFTCwith(twocopiesof)aW-symmetrycananizedintoirreduciblerepresentationsoftheW-algebra.ordinglythepartitionfunctioncanbewrittenasasumoveritshighestweightcharacters.Highestweightrepresentationsmaycontainnullstates,buttheseareabsentatlargecforagenericrepresentation,andweremainindifferentabouttheirpossibleappearancefornow.Wethuswriteagenericpartitionfunctionas ZC=TrqL0−c/24qL0−c/24 =|χ0|2+Nhhχhχh h,h (3.11) ThetraceisovertheentirespectrumofW-primaries.χ0isthevacuumcharacter,andχhisthecharacterofanirreduciblehighestweightrepresentationlabeledbydimensionh(andotherW-quantumnumbersthatdonotfeatureinthistrace).Asforthemultiplicities Nhh,werequireonlythatfordimensionspopulatingtheO(c)gap,theygrowslowerthanexp(βc)aswetakec→∞,whereβisac-independentconstant. WeareinterestedincountinglightVirasoroprimarieswithhO(c)intheorbifoldtheoryCn/Znatlargec.WeignoretheorbifoldforthetimebeingandfocusonthetensorproducttheoryCn,whichhaspartitionfunctionZCn=(ZC)n.Using(3.11),thiscanbeexpandedasasumovertermsoftheform k (χ0χ0)n−kNhihiχhiχhi×(Constant), i=
1 0≤k≤n (3.12) 8OrganizingtheproblemusingthefullWsymmetryofthetheoryturnsouttobeunproductive,onountplicationsofW-algebrarepresentationtheory.WebrieflydiscusssuchanapproachinappendixC. –14– JHEP05(2014)052 (hi,hi)cantakeanyallowedvaluesinthespectrumofC.Ifwecontinuetoignoretheorbifold,wecanplaceacrudeupperboundonthegrowthoflightVirasoroprimariesbybranching(3.12)intoVirasorohighestweightcharacters,andestimatingtheirgrowthatlargedimension.Theconstantisbinatoricfactorwewillnotneed,andourassumptionaboutNhihiallowsustoignorethattoo. Ingeneral,itisasimplemattertobranchagivencharacterχintogenericVirasorohighestweightcharacters,whicharegivenin(3.3)withN=
2, χh≡χh,2=qh−c/24F(q) (3.13) ToexecutethebranchingofχwithoutincludingtheVirasorovacuumcharacterinthe position,wewriteχ=primariesisthen hd(h)χh.Using(3.13),thegeneratingfunctionforVirasoro χ/F(q)=q−c/24d(h)qh (3.14) h Ifwewishtoincludethevacuuminthisposition,onesimplysubtracts1−qfrom theleft-handsideof(3.14).Ofcourse,inestimatingthegrowthofd(h)atlargeh,this makesnodifference. Applying(3.14)forpresentpurposes,wetakeχtobetheholomorphicpartof(3.12), k χ≡(χ0)n−kχhi, i=
1 0≤k≤n (3.15) andestimatethegrowthofd(h)atasymptoticallylargehusingsaddlepointmethods. DespitedoublyovercountingputingthegrowthofVirasoroprimariesintheproducttheoryCnatasymptoticallylargeh,thiswillbesufficientinawiderangeofcasestoshowthatthegrowthofVirasoroprimariesinCn/ZnwithhO(c)isslowenoughtoleavethelargecR´enyientropyunaltered. i)WNCFT: HavingpresentedouralgorithmforgeneralhigherspinCFTs,weapplyittoaCFTwithWNsymmetry.TheWNVermamodulecharacter(χh,N)andvacuumcharacter(χN)weregivenin(3.3)and(3.5),respectively;notethatχNcanbewrittenconvenientlyas χN=q−c/24F(q)N−1·
V (3.16) whereVistheVandermondedeterminant, Ni−
1 V= (1−qi−j) i=2j=
1 (3.17) Furthermore,letusassumethattheonlynullstateslieinthevacuumrepresentation(thisonlyservestoincreasethetotalnumberofstates),sothehighestweightcharactersareindeedgivenbyχh,
N.OurCFTpartitionfunctionisthentheWNversionof(3.11), ZC=|χN|2+Nhhχh,Nχh,N h,h (3.18) –15– JHEP05(2014)052 Consequently,wewanttoapplyouralgorithmwiththeWNsubstitutionsχ0→χNandχhi→χhi,
N. Applyingthesesubstitutionsto(3.14)and(3.15),wewanttoestimatetheasymptoticgrowthofdN,n(h)asdefinedby qxF(q)(N−1)n−1Vn−k=q−c/24dN,n(h)qh h (3.19) wherex=−nc/24+ ki=
1 hi, and
0 ≤ k ≤ n. We have labeled the degeneracy dN,n(h) to reflectitsdependenceonthen-foldproductandourchoiceofWN.Theqxfactorisnot relevantforwhatfollows. Weseenowthatweareintheclear:sinceVisafinitepolynomialinq,theleading growthisgenericallycontrolledbytheF(q)factor,whichisnothingbutacoloredpartition functionwithafinitenumber(N−1)n−1ofcolors.Coloredpartitionsgrowliked(h)≈ h−α exp(2π √β h) with color-dependent constants (α,β). Cruciallyforus,thisgrowthis slowerthanexp(βh). Amorecarefulanalysisconfirmsthisexpectation.Averysimilarcalculationwasdone in[41]usingtraditionalsaddlepointmethods,andweutilizetheirapproachhere.The leadingorderresultfortheasymptoticgrowthofdN,n(h)in(3.19)isfoundtobe dN,n(h)∼h−αexp2πβh, (3.20) where 31α=+ (N2−1)n+N(N−1)k−
1 , (N−1)n−1β= 44
6 (3.21) Weareinterestedinfinite(N,n)only,whereupon(3.20)isvalid(forhnN3)andthe constantsαandβarefinite.Therefore,despiteovercountinginseveralways,wecan concludethataWNCFTsubjecttoourassumptionsdoesnotleadtoaneptablenumberofVirasoroprimarystatesintheorbifoldtheoryCn/Zn. ThisconclusionextendstoanyW-algebrawithafinitenumberofcurrents.Actually, wenoteinpassingthatthisargumentisnecessaryfortheoriginalconclusionsof[2]tohold intheVirasorocase,whichcorrespondstoN=2intheabove. ii)W∞[λ]CFT: WecanevenextendthistypeofargumenttothecaseofW∞[λ]forgenericλ.ThisisslightlytrickierbecausetheVermamodulehasaninfinitenumberofstatesatalllevelsabovethegroundstate;someexamplesofW∞[λ]highestweightcharactersofdegeneraterepresentationsaregivenin[42,43].RatherthanspecifyingpreciselywhichoftheserepresentationsweallowinourCFT,letusjustwriteitspartitionfunctionas ZC=|χ∞|2+Nh,hχhχh h,h (3.22) Thefirsttermisthevacuumcharacter,givenin(3.5)withN→∞.WehaveimplicitlybranchedallW∞[λ]highestweightrepresentationsofourtheoryintoVirasororepresentations,whichthesecondtermsumsover.Indoingso,weonlyrequirethatNhhdoesnot –16– JHEP05(2014)052 growexponentiallywithcfordimensionspopulatingthegap(whichwewillfurtherjustify inamoment). Using(3.22)andourpreviousbranchingformulas,thistimewewishtoestimatethe growthof qxF(q)k−1(χ∞)n−k=q−c/24d∞,n(h)qh (3.23) h atlargeh,wherex=−kc/24+theasymptoticdegeneracyis ki=
1 hi. Again using saddle point techniques as in [41], d∞,n(h)∼h−αexph2/3β, (3.24) where 11 27ζ
(3)(n−k)1/3 α=+(11k−5n),β= 236
4 (3.25) WhilefasterthantheWNgrowthofprimaries,thisisnotfastenoughtospoiltheconclusions,despitetheinfinitetowerofspinsandourovercounting.9 ThissamelogicappliestoallW-algebraswithaninfinitetowerofcurrents,withany finitenumberofcurrentsateachspin.Thelattergroupincludesalgebrasthatshouldariseasasymptoticsymmetriesinnon-AbelianVasilievtheories,forexample,thesW˜∞
(4)[γ]algebrawithlargeN=4supersymmetrythathasrecentlyfeaturedinanattemptto connect3dhigherspinstostringtheory[44]. 3.4Lookingahead WehavearguedthattheleadingorderR´enyientropyisunaffectedbythepresenceofhigherspincurrentsintheCFTspectrum.Thesecurrentsdo,however,affecttheR´enyientropyatO(c0)andbeyond,andwewouldlikeputetheircontributionsandmatchthemtoagravitycalculationat1-loop.Ourstrategywillbetofocusonthecaseoftwointervalsinthegroundstateandperformashortintervalexpansion. 4TheshortintervalexpansionforspinningVirasoroprimaries TheshortintervalexpansionofR´enyientropieswasinitializedin[5,19]andpresentedinfullgeneralityby[20].(Seealso[45].)Wewillprovideastreamlined,andslightlydifferent,versionoftheirpresentationhere.Thiswillleadustoproposeacorrectiontoaresultin[20]regardingthecontributionsofspinning(notnecessarilychiral)Virasoroprimaryfields. 9ThiscalculationalsopartiallyjustifiesourassumptionaboutthegrowthofNh,hin(3.22).CharactersofasimpleclassofdegeneraterepresentationsofW∞[λ]canbewrittenasχ∞timesthelargeNlimitofu(N)characters(i.e.charactersofhs[λ])[42]Givenourresult(3.24),theserepresentationsdonotcontainmorethanexp(βc)Virasoroprimariesbelowthegap.Thisessentiallyfollowsfromthefactthatthehs[λ]characterscanbeviewedasrestrictedpartitionsintoafiniteset,whichhaveonlypolynomialasymptoticgrowth.ThusifthespectrumofCcontainsfewerthanexp(βc)suchdegeneraterepresentationsofW∞[λ],itsbranchingintoVirasororepresentationsissufficientlybounded. –17– JHEP05(2014)052 4.1Generalities Considerareduceddensitymatrix,ρ,obtainedbytracingoversomedegreesoffreedomofaCFTweagainlabelC.Itwasshownthatintheshortintervalexpansion,TrρncanbewrittenintermsofcorrelationfunctionsofoperatorsinthespectrumofCn.LetusquicklyestablishsomenotationpertainingtoCn.WerefertooperatorsinthemotherCFT,C,asO,andtooperatorsofCnasO.IfChasanextendedchiralalgebra,sodoesCn.Themodesofthelatter,callthemWn,arewrittenintermsofthemodesoftheformer,Wn,asasumoverncopiesofC, Wn=Wn⊗1⊗...⊗1+perms (4.1) Here,1denotesthevacuum,andtheproductisovernsheetsofCn.ThesemodesWnincludetheprivilegedVirasoromodes,Ln.WewillbemostlyinterestedinquasiprimariesofCn,whichobeyL1O=
0. Henceforthwespecifytothetwo-intervalcaseinthegroundstateofContheinfinite line.Asin(2.1),wewantputetheorbifoldtheorytwistfieldfour-pointfunction, Trρn=Φ+(z1)Φ−(z2)Φ+(z3)Φ−(z4)Cn/Zn (4.2) Usingconformalsymmetrytofixtwoofthepositionsbytakingz1=0,z3=1andrelabelingz2=1,z4=1+2,thecrossratioxis x=x=z12z34=12z13z241−(1−2) (4.3) whereiaretheintervallengthsobeying0<1<1and2>
0.Theessentialfeaturesof whatfollowsdependonlyonthecross-ratio,sotomakelifesimplerwetake1=2=;thenthesmallintervalexpansionisinasmallcross-ratio,x=21. Inthisregime,thecalculationcanbeeffianizedasasumovercorrelatorsof quasiprimariesofCnandtheirdescendants.Thisamountstoderivingthefusionrulefor thetwistoperators, [Φ+]×[Φ−]=[1]+[OK] (4.4)
K whereKindexesquasiprimaries{OK}ofCn,andthebracketdenotestheentireSO(2,1)×SO(2,1)conformalfamily.WetaketheoperatorsOKtohavedimension∆KandnormNK=OK|OK,andwewritetheirOPEcoefficientsappearinginthefusion(4.4)asdK.ThenwecanwriteTrρnveryneatlya