Boundary Conformal Field Theories, Limit Sets of K

arXiv:hep-th/9902167v424Jun1999TheDateBoundaryConformalFieldTheories,LimitSetsofKleinianGroupsandHolographyArkadyL.Kholodenko∗AbstractInthispaper,basedontheavailablemathematicalworksongeometryandtopologyofhyperbolicmanifoldsanddiscretegroups,someresultsofFreed-manetal(hep-th/9804058)arereproducedandbroadlygeneralized.AmongmanynewresultsthepossibilityofextensionofworkofBelavin,PolyakovandZamolodchikovtohigherdimensionsisinvestigated.Knowninphysicalliteratureobjectionsagainstsuchextensionareremovedandthepossibilityofanextensionisconvinsinglydemonstrated∗375H.L.HunterLaboratories,ClemsonUniversity,Clemson,SC29634-1905,USA.e-mail:[e-mail],therehadbeenattemptstoextendtheresultsoftwodimen-sionalconformalfieldtheories(CFT)tohigherdimensions[1.2].Sincepubli-cationofpapersbyWitten[3,4]ithadbecomeclearthatthereisaveryclosecorrespondencebetween2dphysicsofcriticalphenomenaand3dphysicsofknotsandlinks.Averydetailedstudyofthiscorrespondenceisdevel-opedbyMooreandSeiberg[5].AdditionalcontributionsmorerecentlyweremadeinRef[6],etc.AlltheseworksheavilyexploitthealgebraicaspectsofthiscorrespondencethroughuseofYang-Baxterequations,quantumgroups,etc.Muchlessereffortshadbeenspentondevelopmentofthesamecorre-spondencefromthetopologicalpointofviewthroughstudyof3-manifoldscomplementarytoknots(links)inS3=R3∪{∞}.Suchstudyispotentiallymorebeneficialsinceitisknown[7]thatinfourdimensionsallknotsaretriv-ial(i.e.unknotted)sothatthealgebraicmethodsusedsofararenecessarilylimitedto3dimensionsand,accordingly,tostudyoftwodimensionalCFTonly.Atthesametime,topologicalstudyofmanifoldsisnotlimitedtothreedimensions.ThereasonswhysuchstudiesareusefulcouldbeunderstoodfromthefollowingsimpleargumentstakenfromthebookbyMaskit[8].DefineaninclusionofRdintoRd+1throughRd={(x,t)|t=0}wherex∈Rd,−∞≤t≤∞.TheupperhalfspacePoincare′modelofhyperbolicspaceHd+1isdefinedbyHd+1={(x,t)|t0}(1.1)withx∈Rdsothat∂Hd+1=Rd.ConsideraspecialgroupGofmotionsG=Md+1ofRd+1={x,t}madeofa)translations:(x,t)→(x+a,t),a∈Rd−1;b)rotations:(x,t)→(r(x),t),r∈O(d−1);c)dilatations:x→λx,λ0,λ6=1andd)inversions:x→x|x|2.Itcanbeproven[8]thatthegroupGactsasagroupofisometriesofHd+1andiscalledddimensionalM¨obiusgroup.InitsactiononRd”Gactsasagroupofconformalmotionsbutnotasagroupofisometriesinanymetric”.Atthesametime,itiswellestablished[9]thatinanydimensionthephysicalsystematcriticalitypossessestheinvariancewhichisdescribedin3termsofthegroupG.Hence,theveryexistenceofcriticalityiscloselyassociatedwiththehyperbolicityoftheadjacentspace.Letx∈Hd+1andγ∈G.Consideramotion(anorbit)inHd+1bysucces-siveapplicationsofγtox.Itisofinteresttostudyifsuchamotionwilleverhit∂Hd+1=Rd.ThisproblemishighlynontrivialandwassolvedbyBeardonandMaskit[10](e.g.seesection5belowformoredetails)ford=2.Thenontrivialityofthisproblemcouldbebetterunderstoodif,insteadoftheupperhalfspaceHd+1model,wewouldconsidertheunitballBd+1modelofthehyperbolicspacewiththeunitsphereSd∞(sphereatinfinity)playingthesameroleasinthismodelas∂Hd+1=Rdintheupperhalfspacemodel.SincenotallsubgroupsofGwillallowhittingoftheboundary,itisclear,thatoneshouldbeinterestedonlyinthosesubgroupswhoseorbitsendupattheboundary.Thesesubgroups,inturn,couldbefurthersubdividedintothosewhoselimitpointsonSd∞willcovertheentiresphereandthosewhichwillcoveronlyapartofSd∞.ThispartweshalldenoteasΛ.ThelimitsetΛisactuallyafractal.ThefractaldimensionofΛisdirectlyrelatedtothecriticalindicesofthetwo-pointcorrelationfunctionsofthecorrespondingconformalmodelsatcriticality.DifferentsubgroupsofM¨obiusgroupGwillproducedifferentfractaldimensions.Inturn,thecorrespondinghyperbolicmanifoldsassociatedwiththesegroupscouldbeviewedascomplementsoftherelatedknots(links)inthecaseof2+1dimensionssothatdifferentconfor-malmodels,indeed,couldbeassociatedwithdifferenttypesofknots(links).Thisassociationbecomesunnecessarywhenoneisinterestedinconformalmodelsindimensions3andhigher.Onecouldstillconsidermotionsassoci-atedwithsubgroupsofM¨obiusgroupandthecorresponding,say,hyperbolic4-manifoldswithoutusingknots,braids,Yang-Baxterequations,etc.Althoughstatedindifferentform,recentresultsofMaldacena[11]andtheirsubsequentrefinementsinRefs[12-16](andmanyadditionalreferencesthereinandelsewherewhichwedonotinclude)areactuallydirectlycon-nectedwithideasjustdescribed.Inphysicsliteraturetheconnectionbe-tween”surface”and”bulk”fieldtheoriesisknownasholographicprinciple(holographichypothesis)[17,18].Insimpleterms[19],itcanbeformulatedasstatementthat”amacroscopicregionofspaceandeverythinginsideitcanberepresentedbyaboundarytheorylivingontheboundaryregion”.MathematicalsupportofthisprincipleinphysicsliteratureisattributedtoworksbyFeffermanandGraham[20]andGrahamandLee[21].TheseworksdiscussboundaryconditionsatinfinityforEinsteinmanifolds(spaces)and4initialvalueproblemforEinstein’sequations.Althoughourpreviousdiscus-siondidnotinvolvetheEinsteinmanifolds,actually,theresultsofRef.[21]areconsistentwiththosewhichfollowfromthehyperbolicgeometry.ThiscanbeunderstoodifonetakesintoaccountthatEinsteinspacesarecharac-terizedbythepropertythatthe