An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
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aQuantumAnti–deSitterspaceandsphereatrootsofunityH.Steinacker1SektionPhysikderLudwig–Maximilians–Universit¨atTheresienstr.37,D-80333M¨unchenAbstractAnalgebraoffunctionsonq–deformedAnti–deSitterspaceAdSDisde nedwhichiscovariantunderU(so(2,D 1)),forqarootofunity.Thestar–structureqqisstudiedindetail.Thescalar eldshaveanintrinsichigh–energycuto ,andarisemostnaturallyas eldsonorbifoldsAdSDa niteabeliangroup,×SD/ΓifDisodd,andAdSDandSχisacertain“chiral×sector”S2D 1/ΓifDiseven.HereΓisqqχoftheclassicalsphere.Hilbertspacesofsquareintegrablefunctionsarediscussed.Analogousresultsarefoundfortheq–deformedsphereSDq.
LMU-TPW99-15
An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
1Introduction
TheD–dimensionalAnti–deSitterspaceAdSDisahomogeneousspacewithconstantneg-ativecurvatureandcosmologicalconstant.ItssymmetrygroupSO(2,D 1)playstheroleoftheD–dimensionalPoincar´egroup,whichisrecoveredinthe atlimitbyacontraction.Itisofconsiderableinterestintheoreticalphysicsforseveralreasons.Forexample,itcanbeusedasasimplemodelfor eldtheoryoncurvedspaces[11],anditarisesnaturallyinthecontextofsupergravity[40].Recently,aninterestingconjecturerelatingstringorMtheoryonAdSD×Wwith(super)conformal eldtheoriesontheboundaryhasbeenproposed[26],whereWisacertainsphereoraproductspacecontainingasphere.Moreover,thereissomeevidencethatafullquantumtreatmentwouldleadtosomenon–classicalversionofthemanifolds.Thisincludestheappearanceofa“stringyexclusionprinciple”[27]inthespectrumof eldsonAdSspace.
Inthispaper,westudyanon–commutativeversionoftheAdSspace,whichiscovariantunderthestandardDrinfeld–JimboquantumgroupSOq(2,D 1).ItcanbeunderstoodasaquantizationofacertainPoissonstructureontheclassicalAdSspace,whereq 1isadeformationparameterwhichplaystheroleofthePlanckconstant.Inprinciple,thisiscanbedoneforrealqandqaphase.Forrealq,thequalitativefeaturesofquantumgroupsandspacesaretypicallysimilartotheclassicalcase;inparticular,nocuto isexpected.
Hereweconsiderthecasewhereqisarootofunity.Itiswell–knownthatthenquantumgroupsshowcompletelynew,“non–perturbative”features;roughlyspeaking,phenomenawhicharetypicalforin nite–dimensionalrepresentationsofclassicalnon–compactgroupsoccuralreadywith nite–dimensionalrepresentations.Inparticular,ithasbeenshownthatthereexist nite–dimensionalunitaryrepresentationsofthequantumAdSgroupsatrootsofunity[37,6],whereallthefeaturesoftheclassicalcaseareconsistentlycombinedwithacuto .
Thecorrectde nitionofquantum–AdSspaceforqaphaseisnotobvious;di erentver-sionshavebeenproposedintheliterature[4,13],whicharenotverysatisfactoryorincom-plete.The rstgoalofthispaperistoclarifythissituation,andtogiveaprecisede nitionintermsofoperatorsonHilbertspaces.To ndtheproperde nition,wemake2basicassump-tions:1)covarianceundertheq–deformeduniversalenvelopingalgebraUq(so(2,D 1)),and2)allowingonly nite–dimensionalrepresentations,henceinsistingonafullregular-izationandavoiding“q–analysis”.Itisveryremarkablethatthisisindeedpossible,whilemaintainingthecorrectlow–energylimit.
Aswewillshow,theseassumptionsleadtoanalgebraoffunctionsonthecomplexi edquantumsphere,whichdecomposesintodi erentsectorscorrespondingtodi erentreal
Dforms.TheydescribethecompactsphereSqandcertainnoncompactforms,inparticular
DthequantumAnti–deSitterspaceAdSq.Thiswillprovideuswithscalar eldswhichare
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An algebra of functions on q-deformed Anti-de Sitter space AdS_q^D is defined which is covariant under U_q(so(2,D-1)), for q a root of unity. The star-structure is studied in detail. The scalar fields have an intrinsic high-energy cutoff, and arise most na
unitaryrepresentationsofUq(so(2,D 1)),andcorrespondtotheclassicalsquare–integrablescalar eldsonAdSspace,describingspin0elementaryparticles.Theremarkabledi erencetotheclassicalcaseisthatallthishappenswithintheframeworkofpolynomialfunctions,whosepropertiesarecompletelydi erentfromtheclassicalcase.Nevertheless,theclassical eldsarerecoveredinthelimitofqapproaching1.Inparticular,thisallowstostudyquestionsoffunctionalanalysisintheclassicalcasewithpurelyalgebraicmethods.
DMoreover,itwillturnoutthatthede nitionofAdSqimpliesanumberofadditional,
unexpectedfeatures.TheyincludetheappearanceofanadditionalundeformedsymmetrygroupSO(D+1)ifDisoddandSp(D)ifDiseven,whichareinsomesensespontaneouslybroken[37].Moreover,itturnsoutthatthequantumspacesareobtainedmostnaturallyasproductofthequantumAdSspace(orsphere)withaclassicalsphere.Moreprecisely,one