Quantum Anti-de Sitter space and sphere at roots of unity

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

9

9

9

1

t

c

O

1

2

v

7

3

1

9

/9h

t

-

p

e

:hv

i

X

r

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

1

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

2r+12r4r 14r 1obtainstheproductsAdSq×S2r+1/ΓandAdSq×Sχ/Γ,whereΓ=(Z2)r,andSχisacertain“chiral”sectorofS4r 1.Thequotientsoftheclassicalspacesareactuallytwisted

sectorsoforbifolds.Itshouldbeemphasizedthatnospeci cassumptionshavebeenmadehere,itissimplyaconsequenceoftheremarkablestructuresthatappearatrootsofunity.Ofcourse,thisisquiteintriguinginthecontextoftheAdS–CFTcorrespondencementioned

3547above,sinceweobtainAdSq×S3,AdSq×S5andAdSq×Sχ,whicharepreciselycasesof

interestthere(apartfromthe“chiralsector”ofS7,whosemeaningisnotentirelyclear).TheseandotherphysicalaspectswillbediscussedfurtherinSection7.

Thispaperisorganizedasfollows.InSection2,somebasicfactsaboutquantumgroupsandspacesarereviewed,includingaspectsoftherepresentationtheoryatrootsofunitywhichwillbeneeded.InSection3,wediscussindetailthemeaningofrealitystructures,anddeterminetherealformofthequantumAdSgroupUq(so(2,D 2)).Section4isdevotedtoacloseranalysisofthestructureofpolynomialfunctionsonthecomplexquantumspacesatrootsofunity.InSection5,weidentifydi erentnoncompactsectors,whichleadstothede nitionofHilbertspacesofscalar elds.TheirproductstructurewithclassicalspheresisanalyzedinSection5.2.Sections5.3and5.4aremathematicalinterludes,andwillallowustowritedownexplicitlythestarstructureoftherealquantumspacesinTheorems5.3and

5.4,whicharesomeofthemainresultsofthiswork.InSection6,wecommentonfurtherdevelopmentstowardsformulatingphysicalmodels,andproposeanon–shellconditionwhichissomewhatreminiscentofstringtheory.SomephysicalaspectsarediscussedinSection7.TheAppendicesincludeseveralproofsthatwereomittedinthetext,aswellanexpositionofthevectorrepresentationsofso(D)forconvenience.

Someadvicetothereader:InSections5.3and5.4,thestarstructureisde nedinseveralsteps,andconsiderablee ortismadetogivetheprecisemathematicalde nitionsandtoexplainwhyitisthecorrectone.Howeverthe nalresult,Theorems5.3and5.4canbestatedverybrie y.Thusthereaderwhoisnotinterestedinthemathematicaldetailsmayskipmuchofthesesectionsandsimplyaccepttheresults.

2

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

2Thebasicalgebras

We rstrecalltheclassicalAnti–deSitterspaceAdSD 1,whichisa(D 1)–dimensionalmanifoldwithconstantnegativecurvatureandsignature(+, ,..., ).ItcanbeembeddedinaD–dimensional atspacewithsignature(+,+, ,..., )by

22222 z2 ... zDz1+zD 1=R,(2.1)

whereRwillbecalledthe”radius”oftheAdSspace.ThegroupofisometriesofthisspaceisSO(2,D 2),whichplaystheroleofthe(D 1)–dimensionalPoincar´egroup.

Thisspacehassomeratherpeculiarfeatures.Itstime–likegeodesicsare niteandclosed,andthetime”translations”istheU(1)subgroupofrotationsinthe(z1,zD)–plane.Thespace–likegeodesicsareunbounded.Thereexistniceunitarypositive–energyrepresentationsofSO(2,D 2)whichcorrespondtoelementaryparticleswitharbitraryspin.ItisalsoworthrecallingthatSO(2,D 2)istheconformalgroupinD 2dimensionsactingon(D 2)–dimensionalMinkowskispace,whichcanbeinterpretedastheboundaryofAdSD 1.Tode nethenoncommutativeversion,we rstreviewsomebasicfactsabouttheq–deformedorthogonalgroupandEuclideanspace[8];foramoredetaileddiscussionseee.g.

[9,35].ThealgebraoffunctionsFunq(SO(D,C))ontheorthogonalquantumgroupisgeneratedbymatrixelementsAijwithrelations

iknik nm mnRAmjAl=AnAmRjl,(2.2)

ikisexplainedbelow.Funq(SO(D,C))istheHopfalgebradualtothewherethematrixRmnquantizeduniversalenvelopingalgebraUq(so(D,C)),whichiseasiertoworkwithinpractice.

GivenarootsystemofasimpleLiegroupgwithKillingmetric(,)andCartanmatrixAij,Uq:=Uq(g)istheHopfalgebrawithgenerators{Xi±,Hi;i=1,...,r}andrelations[16,7,8]

[Hi,Hj]=0, ±±Hi,Xj=±AjiXj,

+ qdiHi q diHiXi,Xj=δi,j

1qi qi(2.3)(2.4)approachesnasq→1.Thecomultiplicationis

(Hi)=Hi 1+1 Hi

(Xi±)=Xi± q diHi/2+qdiHi/2 Xi±,

3(2.6)

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

antipodeandcounitare

S(Hi)= Hi,

S(Xi+)= qdiXi+,S(Xi )= q diXi ,

ε(Hi)=ε(Xi±)=0.(2.7)

Theclassicalcaseisobtainedbytakingq=1.Theconsistencyofthisde nitioncanbecheckedexplicitly.

TheCartan–Weylinvolutionisde nedas

θ(Xi±)=Xi ,θ(Hi)=Hi,(2.8)

extendedasalinearanti–algebramap;inparticular,θ(q)=qforanyq∈C.Itisobviouslyconsistentwiththealgebra,andonecancheckthat

(θ θ) (x)= (θ(x)),

S(θ(x))=θ(S 1(x)).(2.9)(2.10)

±,0BorelsubalgebrasUqcanbede nedintheobviousway.Thisde nesaquasitriangular

Hopfalgebra,whichmeansthatthereexistsaspecialelementR∈Uq Uqwhichsatis es

′(x)=R (x)R 1(2.11)

foranyx∈Uq,andotherpropertieswhichwillnotbeusedexplicitly.Here ′(x)=τ (x)isthe ippedcoproduct.ThereareexplicitformulasforR,oftheform[20,19]

1αijdiHi djHj+res resR=q1 1+Uq Uq(2.12)

whereαij=(αi,αj).Inthispaper,weconsiderg=so(2r+1)=Brandg=so(2r)=Dr.Aswasshownin[7],thefollowingremarkableelement

ρv=(SR2)R1q 2 (2.13)

isinthecenterofUq,andwillbecalledDrinfeld–Casimir.Hereρ isdualtotheWeylvectorρ=1

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

ConsiderthematrixRij

kl=πik

S=qfor πjl(R)whereπikistheD–dimensional(“vector”)

sentationofUq,andletqso(2r),and2qS=q2forso(2r+1).ThenR repre-

ij=Rji

decomposesasR ij=(qSP+ q 1P +q1 DP0)ij

kl,whereP+,P andP0klkl

aretheinvariant

projectorsonthekltracelesssymmetric,StheSantisymmetric,andthesingletcomponentinthetensorproductof2vectorrepresentations,respectively.Theinvarianttensorgijisgiven(P0)ijq2by 1

kl=S

1+qDqρi′x2,i<i′(2.20)

S 2

Bothalgebrasarecovariantundertherightcoactionxi→xj AjiofFunq(SO(D,C)),whichisequivalenttoaleftaction

xi→u·xi=xjπji(u)(2.21)

foru∈Uq(so(D,C)).Wewillusuallyworkwiththelatter,whichismorefamiliarfromtheclassicalLiealgebras;thenπij(uu′)=πik(u)πkj(u′).Wecanusethisforaquickcheckofthe rstrelationin(2.20):

X+1/2

1·(x1x2 qSx2x1)=qSx1x1 qSq 1/2

Sx1x1=0(2.22)

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

using(2.6)andAppendixA,asitmustbe.Theotherrelationscanbecheckedsimilarly.ThealgebraoffunctionsAdSqD 1de ningthequantumAnti–deSitterspacewillbede nedasarealformofthiscomplexquantumsphere,witha(co)actionofthequantumAnti-deSittergroup.Thereforeasanalgebra,itisagainde nedby(P )ij2Onecouldintroduceaphysicalscalebysettingkltitj=0,t=1.

yi:=tiR(2.23)

foraconstantR>0,sothaty2=R2.Wewillsimplyusetheunitsde nedbyR=1;physicallyspeaking,thescalewillbesetbythe“radius”ofAdSspace.

Sofar,allthesespacesarecomplex.Thecrucialissueisto ndthecorrectde nitionofthecorrespondingrealquantumspaces.Thisisnotobviousespeciallyifqisaphase,andinfactdi erentpossibilitieshavebeenproposedintheliterature[4,13].Thiswillbediscussedindetailinlatersections.

2.1RootsofUnityandRepresentations

Sinceweareprimarilyinterestedinthecasewhereqisarootofunity,wewillconsideramorepowerfulversionoftheabove,theso–called“restrictedspecialization”Uqres:=Uqres(so(D,C))

[22]withgeneratorsX±(k)(X±

i)k

i=

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

Wewillonlyconsider nite–dimensionalrepresentationsinthispaper.TheCartangen-eratorscanthenbediagonalized,witheigenvalues

<Hi,λ>=

∨satisfy(Λi,αj)=δi,j,therefore(α,α)(αi,λ)isthecorootofα.ThefundamentalweightsΛi

(2.27)<Hi,Λj>=δij,

andspanthelatticeofintegralweights.Theirreduciblehighest–weightrepresentationswithhighestweightλwillbedenotedbyLres(λ).

ThevectorrepresentationVDofUq(so(D,C))istherepresentationLres(λ1)withbasisxi(orti)fori=1,...,D.TheirweightsλiaregivenexplicitlyinAppendixA.IfD>4,thenthehighestweightλ1isequaltoΛ1.Wealsode ne

dS=(λ1,λ1),MS=M/dS,(2.28)

sothatqS=qdSwhichwasusedabove.

Itiswell–known[31]thatforgenericq,therepresentationtheoryisessentiallythesameasintheclassicalcase.Inparticular,all nite–dimensionalrepresentations(=modules)ofresUqaredirectsumsofsomeLres(λ).Theircharacter

resλχ(L(λ))=edimLres(λ)ηe η=:χ(λ)(2.29)

η>0

isgivenbyWeylsformula.HereLres(λ)ηistheweightspaceofLres(λ)withweightλ η.

finTheirreduciblehighestweightrepresentationsofUqwillbedenotedbyLfin(λ).

ThevalueoftheDrinfeld–Casimirv(2.13)onLres(λ)(andonanyhighest–weightmodulewithhighestweightλ)was rstdeterminedin[30]:

v·w=q cλwforw∈Lres(λ),(2.30)

wherecλ=(λ,λ+2ρ)isthevalueoftheclassicalquadraticCasimironL(λ).Inparticularforhighestweightsoftheformλ=kλ1,theclassicalCasimirforso(2r+1)is

ckλ1=2k2+2k(D 2),

andforso(2r)itis

ckλ1=k2+k(D 2).(2.31)(2.32)

Finally,wequoteafewimportantfactsaboutirreduciblerepresentationsatrootsofunity.The rstone[2,5]statesthatthestructureofLreswith“small”highestweightλisthesameasclassically:

7

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

Theorem2.1Assumethatλisadominantintegralweightwith(λ+ρ,α)≤Mforallpositiverootsα.ThenthehighestweightrepresentationLres(λ)hasthesamecharacterχasintheclassicalcase,givenbyWeyl’scharacterformula.

Thisfollowsfromthestronglinkageprinciple,whichwas rstshownin[2];foramoreelementaryapproach,see[38].Moreover,Lfin(λ)=Lres(λ)fortheseweightsλ,sincethe±(M)Xiiacttrivially.

Forgeneralλ,thestructureofLres(λ)isdi culttoanalyze.Howeverforthe“specialweights” λz=MiziΛiforzi∈Z(2.33)itcanbeunderstoodeasily,andthiswillbethekeyformuchofthefollowing.Therelation

res(2.5)togetherwith(2.25)impliesthatforanyhighestweightmoduleUq·wλzwithhighest

weightλz,

(2.34) i:=Xi ·wλz

isahighestweightvector(possiblyzero)foranyi,i.e.Xi+· j=0foralli,j.BecauseLres(λz)isirreduciblebyde nition,itfollowsthat

Xi ·wλz=0foralli(2.35)

fin(so(D))areone–inLres(λz).Inparticular,theirreduciblerepresentationsLfin(λz)ofUq±(M)resdimensional.Howeverthe“large”generatorsXii∈UqdoactnontriviallyonLres(λz),

aswewillseenext.

resUsingthecommutationrelations,anyelementofUq(so(D))canbewrittenasasum

fin1.ItfollowsthatallweightsofLres(λz)haveoftermsoftheformX...XikkUqi1 theformλz′=λz iniMiαiwithni∈N.Inotherwords,Lres(λz)isadirectsumof

fin,sinceallλz′arespecialpoints.Infact,one–dimensionalrepresentationsLfin(λz′)ofUqtheweightsλzhavethestructureofaweightlatticewith“fundamentalweights”MiΛi.This ij=Aji ,withCartanmatrixAturnsouttobetherescaledlatticeofadualLiealgebrag

=so(D)ifDiseven,andg =sp(D 1)ifprovided(2.24)holds[37].Inthepresentcase,g

res“contains”acorrespondingclassicalLiealgebraasaquotient.ThisisDisodd.Infact,UqtheessenceofaremarkableresultofLusztig[23],andcanbemadeexplicitasfollows([37],

Theorem4.2):

Letai∈{0,1}suchthatai+aj=1ifAij=0andi=j;thisisalwayspossible.De ne i=qdiMiHi,andK (Mi) (Mi)

+=X+(Mi)K ai,Xiii

2 =X (Mi)K 1 aiqMi,Xiiii i=[X +,X ]Hii

8(2.36)

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

Thenonecanshowthefollowing:

Theorem2.2Forallspecialweightsλz,Lres(λz)isanirreduciblehighest–weightrepresen- ±andH i.If tationoftheclassicaluniversalenvelopingalgebraU(g),withgeneratorsXi ′ i·vz′=z′vz′.vz′∈Lres(λz)hasweightjzjMjΛj,thenHi

Inparticularifλzisadominantweight,thenthecharacterofLres(λz)isinvariantundertheWeylgroup,andcanbeobtainedfromWeyl’ingthis,thestructureofLres(λ)with“large”dominantintegralλcanbedescribedasfollows[5]:∨Theorem2.3Letλzasin(2.33)andλ0beanintegralweightswith0≤(λ0,αi)<Mifor

alli.Then

Lres(λ0+λz)=Lres(λ0) Lres(λz).(2.37)

Thisisnothardtoprove,see[5]or[37].Moreover,thegenerators(2.36)essentiallyacton

finthesecondfactorin(2.37)andUqonthe rst,butwithacertain“twisting”[37].

3Realitystructuresandsymmetryalgebra

Sincetheproperchoiceoftherealitystructureiscrucialinthefollowing,we rstdiscusstherelationoftherealstructuresonthespaceswiththeirsymmetryalgebras.

ThealgebraoffunctionsonbothclassicalandquantumD–dimensionalcomplexEu-clideanspaceisgeneratedbycoordinatefunctionsxi,whichtransforminthevectorrepre-sentationVDofUq(so(D,C)).Thetensorproductof2suchrepresentationscontainsauniquetrivialrepresentation;inotherwords,thereisaninvariantbilinearform<,>:VD×VD→C.InHopf–algebralanguage,invariancemeans<f,g>=<u(1)·f,u(2)·g>forf,g∈VD,where (u)=u 1+1 u=u(1) u(2)denotesthecoproductofu∈U(so(D,C))orUq.Thisextendsimmediatelytopolynomialfunctions.

Intheclassicalcase,thealgebraofcomplexfunctionsonrealEuclideanspace(oranyrealmanifold)isequippedwitha –structure,i.e.anantilinearmapwhosesquareistheidentity,de nedbythecomplexconjugation.Theaboveinvariantbilinearformtheninducesahermitianinnerproductby

(f,g)=<f ,g>,(3.1)

whichsatis es(f,g) =(g,f).Thesignature(p,D p)ofthisinnerproductisthesignatureofthe(pseudo)Euclideanrealspace,anditispositivede niteonlyintheEuclideancase.

9

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

forallu∈U(so(D,C)).Thisde nestherealformU(so(p,D p)),andonecanthenstudyitsunitaryrepresentations,whicharein nite–dimensionalexceptintheeuclideancase.Intheq–deformedcase,weusethisconnectionbetweentherealformofafunctionspaceanditssymmetryalgebraintheotherdirection:therewillbeaclearchoiceoftherealformofUq,whichthendeterminestherealformofthefunctionalgebras.Arealformor –structureofUqisanantilinearanti–algebramap onUqwhosesquareistheidentity.AninvariantinnerproductonarepresentationofUqisahermitianinnerproductwhichsatis es(3.2)forallu∈Uq;inotherwords,thestaronUqisimplementedbytheadjoint,whichiswell–de nedfornondegenerateinnerproducts.ThisisparticularlyintuitiveforelementsofUqoftheformg=exp(itu)withu =u,sincetheng =g 1.

ArepresentationofUqisunitarywithrespecttoarealformofUqifithasapositivede niteinvariantinnerproduct.Then

i (π(u)ji)=πj(u)Butinanycase,itde nesa –structureonthealgebraU(so(D,C)),i.e.anantilinearanti–algebramap,by(f,u·g)=(u ·f,g)(3.2)(3.3)

foranyu∈Uq,inanorthonormalbasis.

Ourguidingprincipleto ndtheappropriateq–deformedrealspacesisthatthestarstructureonUqshouldadmitasu cientlylargeclassofunitaryrepresentationsofthequan-tumAdSgroupinordertodescribeelementaryparticles,inthespiritofWigner.TheAdSgroupisparticularlywellsuitedforsuchanapproach,becauseithasunitaryrepresenta-tionsforanyhalf–integerspincorrespondingtomassiveaswellasmasslessparticleswithpositiveenergy,inanydimension.Infact,onecanchoosetheCartansubalgebrasuchthattheenergyisoneofitsgenerators,andtheunitaryrepresentationsarethenlowest–weightrepresentationswithpositiveenergyanddiscretespectrum.TheunitaryrepresentationsofthePoincar´egrouparerecoveredinthe atlimit.Wewanttomaintainthesefeaturesintheq–deformedcase.Thiswilluniquelyselecttherealstructure.

3.1StarstructuresonUq

Xi±=siXi , Thereareessentially2typesofstarstructures3onUq[37],the rstoftheformHi =Hi,(3.4)

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

andthesecondoftheform

Xi±=siXi±, Hi = Hi,(3.5)

withsi=±1.Theyde neconsistentstaralgebrasforbothrealqandqarootofunity.Thecompatibilityconditionswiththecoproductaredi erentforrealqandqaphase,whichwillshowupindi erentrealitystructuresoftheassociatedquantumspaces.

Itisknownthatthereexist nite–dimensionalunitaryrepresentationsofthe rsttype(3.4)ifqisarootofunity[38,37],whichhavethedesiredpropertiesincludingahigh–energycuto intheAdScase.Forq∈R,unitaryrepresentationsofnoncompactformsalsoexist,buttheyhavenocuto .Therearealsocertainunitaryrepresentationsofthesecondtypeforqaphase,e.g.forUq(so(2,1))[32],buttheyarenothighest–weightrepresentations;inparticular,theCartansubalgebracannotbediagonalized,andtheenergyintheAdScaseisnotpositivede nite(noticethattheCartansubalgebraisdistinguishedforq=1,unlikeintheclassicalcase).Finite–dimensionalunitaryrepresentationsofthesecondtypecannotexist,sincethenHicouldbediagonalizedwithpurelyimaginaryspectrum,whichisincontradictionwiththecommutationrelation(2.4);ThereforeweonlyconsiderstarstructuresonUqofthe rsttype(3.4)fromnowon,withqarootofunity.

ConsiderthevectorrepresentationVDofUq,withbasisxiandweightsλifori=1,...,D.VDisunitarywithrespecttothecompactform

Xi±=Xi , Hi =Hi,(3.6)

whichde nesUq(so(D))forbothrealqandqaphase.Ingeneral,thereisauniqueinvariantinnerproductonhighestweightmodulessatisfying(3.2)forstarstructuresofthetype(3.4).Theweightvectorsxiarethenorthogonalfordi erentweights,andtheycande nedtobeorthonormal,i.e.

(xi,xj)=δij.(3.7)

Thisisthestandardconventionintheliterature.

Nowitiseasyto ndthede nitionofUq(so(2,D 2)).LetEbetheelementoftheCartansubalgebrawhichisdualtoλ1/dS,sothat<E,λ>=(λ1,λ)/dS.ThentheeigenvaluesofEonVDareEi:=(λ1,λi)/dS=(1,0,...,0, 1);Ewillturnouttobetheenergy.Weclaimthatthestarstructurede ningUq(so(2,D 2))is

Xi±=( 1)Eθ(Xi±)( 1)E=siXi , Hi =Hiwithsi=( 1)<E,αi>,(3.8)wheresi=( 1,1...,1)forD=4,andsi=( 1, 1)forD=4.Thisisastaralgebraofthe rsttype(3.4)whichcanbeconsideredforbothq∈Randqarootofunity,andthereexistunitaryrepresentationsinbothcases.ThemaximalcompactsubalgebraisUq(so(D 2))×±±Uq(so(2))whereUq(so(D 2))generatedbyX2,...,Xr,H2,...,Hr(forD=4),andUq(so(2))

11

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

byE;thesesubalgebrascommute.Theunique(uptonormalization)correspondinginvariantinnerproductonVDwhichsatis es(3.2)is

(xi,xj)=( 1)Eiδij(3.9)

intheabovebasis,andhasthecorrectsignaturefortheAdScase.Wewillalso ndthedesiredunitaryrepresentationsofU

de nethe“physical”quantumAnti–deqfin(so(2,D 2)),provided(2.24)holds.Thereforewe

Sittergroupatrootsofunitytobetherealform

(3.8)ofUqfin(so(2,D 2)).

3.2QuantumEuclideanandAnti–deSitterspaceforrealqForrealq,thereisastandardstarstructureonthealgebraCDq.ThealgebraRDqontherealquantumEuclideanspaceisde nedby[8]

x i=xjgji.(3.10)

Thisisaconsistentstaralgebra,whichindeedleadstotheinvariantinnerproduct(xi,xj)=δij,andby(3.1)correspondstotherealform(3.6)onUq.Itisalsoconsistentwiththeconstraintxixjgij=1,therebyde ningthequantumEuclideansphereSD 1forrealq.ThealgebraAdSqD 1onquantumAnti–deSitterspaceforrealqissimilarlyqde nedby

t i= ( 1)Eitjgji,(3.11)

togetherwith

t2=titjgij=1.(3.12)

Thisisconsistentwith(2.17)becauseEisintheCartansubalgebra,andhasthecorrectclassicallimit.By(3.1),thisleadstotheinvariantinnerproduct(ti,tj)=( 1)Eiδij.Ifqisaphase,(3.10)doesnotextendasastaronthealgebraCD

q.Inthiscase,star

structuresdi erentfrom(3.10)havebeenproposed,ofthetypexD

becomesastaralgebra,theyleadtostarstructuresonUi=±xi[8,4].WhileCthenthesecondtype(3.5),q

qof

whichwehavediscardedabove.However,wewillseebelowthatthereisastaronthesemidirectproductalgebraUq

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

situationforq∈Riscompletelyanalogous.Atrootsofunityhowever,thestructureofpolynomialschangescompletely,andwewillseethattheanaloguesoftheclassicalscalar eldscaninfactbewrittenaspolynomialsintheti.

Toseethis,wehavetostudyCDqinmoredetail.ConsiderthesetofhomogeneouspolynomialsMqk CDqwithdegreekinthexi,whichformsasubmoduleofthek–foldtensorproductrepresentationV k

xDofUqres.Mqkisnotirreducible,becausethemetricprojectormay

benontrivial.Clearly1...x1=(x1)k∈Mqkisahighestweightvectorwithweightkλ1.Itgeneratesthehighestweightmodule

F(k):=Uqres·(x1)k Mqk.(4.1)

Ifqisgeneric,thenF(k)isairreduciblerepresentationwithhighestweightkλ1correspondingtoatotally(q–)symmetrictracelesstensor,andMqk=F(k)⊕x2

Ifqisarootofunity,thenMqkisnotcompletelyreducibleanyF(more,k 2)which⊕...asisclassically.atypicalphenomenonatrootsofunity.ThefullstructureofMk

discussedelsewhere.Hereweonlyconsiderthosemodesqisquitecomplicated,andwillbe

whichwillbeneededfortheHilbert

spaceofscalar eldsonAdSq.

First,weidentifythepolynomialsF(k)inCDqwhichhaveessentiallythesamestructureasclassically,whichmeansthecharacterofLres(kλ1)ing(2.24),Theorem2.1impliesthatχ(Lres(kλ1))=χ(kλ1)fork≤

indeed,anypositiverootαcanthenbewrittenasα=

3)d1 MSniα i(withD n3)i,whichis≤M≤(assuming=ai,whereDai≥are4;

theCoxeterlabels.Hence(kλ1+ρ,α)≤(k+Dd1M1providedk≤M1

However, (Dthis 3).boundForDcan=3,betheimprovedboundisusingk≤MS

the strong1/2).linkageprinciple,seee.g.[38].

Wecanassumethatk<MS;thenitimpliesthatthecharacterofLres(kλ1)=Lfin(kλ1)isthesameasclassically,unlessitcontainsadominantintegralweightµwhichisintheorbitofkλ1undertheWeylgroupactingwithcenterMSλ1 ρ(sinceMsλ1isaspecialweight).Since(λ1

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

(ForoddD,thereisasimilarphenomenon).Thisisrelatedtothescalarsingleton eldontheAdSspace,aswewillseeinthenextsection.Onecancheckthate.g.theCasimirv(2.13)indeedbecomesdegeneratefortheseweights.kTosummarize,thestructureofMqisthesameasclassicallyforsmallk:

Theorem4.1Forn0≤kSandrootsofunityoftheform(2.24),

F(n0)～=Lres(n0λ1)=Lfin(n0λ1)

n0hasthesamecharacterχ(n0λ1)asclassically.Moreover,Mqisthedirectsum

n0Mq(4.5)=

0≤k≤n0/2 (x2)kF(n0 2k).(4.6)

TheproofiscompletedinAppendixC.2

NextconsiderF(kMS),whichisofcentralimportancetous.ItshighestweightkMSλ1isaspecialweight(2.33).Using(2.6)andthecommutationrelations(2.20),onehas

X1·(x1)n=qS (n 1)/2

(n 1)/2

(n 1)/2x2(x1)n 1+qS (n 3)/2x1x2(x1)n 2+...+qS)x2(x1)n 1(n 1)/2(x1)n 1x2(4.7)=qS2(1+qS+...+qS2(n 1)

=qS[n]qSx2(x1)n 1.

Since[kMS]qS=0,itfollowsthatXi ·(x1)kMS=0foralli,hence(x1)kMSisaone–

findimensionalrepresentationofUq.Asdiscussed inSection2.1below(2.35),thisimplies

thatallweightsofF(kMS)havetheformλz=ziMiΛi,andF(kMS)isarepresentation ±andH i.Moreover,itis )withgeneratorsXoftheclassicaluniversalenvelopingalgebraU(giahighestweightmodulewithhighestweightvector(x1)kMS.Bytheclassicalrepresentationtheory,itfollowsthatitisirreducible,hence

F(kMS)～=Lres(kMSλ1),(4.8)

(exceptforD=3,whereitisL(kΛ1)).whichisessentiallyL(kλ1)ofg

Nowconsidermoregenerallyn=n0+kMSwith0≤n0<kSandk∈N.ThenbothF(n)andF(n0) F(kMS)arehighestweightmoduleswithhighestweightnλ1.ClearlyF(n) F(n0)F(kMS),by(4.1)and(2.6).Ontheotherhand,F(n0)F(kMS) F(n0) F(kMS),whichisisomorphictoLres(nλ1)byTheorem2.3.Hencewehaveshownthat

14

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

Theorem4.2Forn=n0+kMSwith0≤n0<kSandk∈N,

F(n)=F(n0)F(kMS)～=Lres(nλ1).=F(n0) F(kMS)～(4.9)

Inparticular,itisessentiallythetensorproductofF(n0)withtheirreduciblerepresentation

.L(kλ1)(orL(kΛ1)forD=3)oftheclassicalalgebrag

5

5.1RealformsandHilbertspacerepresentationsofthequantumspacesD 1D 1HilbertspacesforSqandAdSq

Nowwearereadytodiscusstherealitystructureatrootsofunity.Aswaspointedoutbefore,(3.10)and(3.11)arenotconsistentwiththealgebraforqaphase.To ndthecorrectde nition,we rstconstructtheHilbertspaces,andthensimplycalculatetheadjointoftheoperatorsofinterest.Wewanttoemphasizethattheinnerproductsonirreduciblerepresentationsaredetermineduniquelyby(3.2).Indeedonanyhighestweightmodule,thereexistsaunique(uptonormalization)invariantinnerproductforagivenstarstructureoftheform(3.4).Thisisbecauseoncetheinnerproductisde nedonthehighestweightvector,itcanbecalculatedforalldescendantvectorsusing(3.2);ingeneral,itisnotunitary.Theresultinginvariantinnerproductisnon–degenerateiftherepresentationisirreducible.

resWe rstdiscussthequantumsphere.AsrepresentationofUq,wecanconsider

resD 1F(n)=Uq·(t1)n Sq,C(5.1)

insteadof(4.1).However,notalltheseF(n)shouldbeconsideredas eldsonthe“real”quantumsphere,onlythosewhichareunitaryrepresentationsofthecompactform

Xi±=Xi , Hi =Hi,(5.2)

fin(so(D)),withthenaturalinnerproductdiscussedabove.Itwasprovedin[37]thatofUqthiscertainlyholdsforthoseLres(kλ1)=Lfin(kλ1)withk≤kS(4.2).Heretheassumption(2.24)isimportant.

Hencewecouldde neaHilbertspaceoffunctionsontherealquantumspheretobethedirectsumofallF(k)withk≤kS.Inordertoobtainasimplede nitionofposition

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

operatorsinSection5.5,weimposetheslightlystrongerboundk<kS,andde netheHilbertspaceoffunctionsontherealquantumspheretobe

finD 1Uq·(t1)n0.(5.3)F(n0)=Sq:=

0≤n0<kS0≤n0<kS

ThepositionoperatorswillbediscussedinSection5.5.ItsgeneratorsareessentiallythetiasinSection2;theywillhavetobeslightlymodi edinordertoaccountforthecuto .TheirstarstructureisdetermineduniquelybytheadjointontheHilbertspace,andwillbe

D 1givenexplicitlyinSection5.5.SqisatruncatedversionoftheclassicalsphereSD 1=

⊕n≥0L(nλ1),whichisrecoveredinthelimitM→∞,orq→1.

Nowconsidermoregenerallyn=n0+kMSwith0≤n0<kS,sothatF(n)=F(n0) F(kMS)accordingto(4.9).SinceF(kMS)～=Lres(kMSλ1)isa nite–dimensional

,ithasastandardpositive–de niteinvariantinnerirreduciblerepresentationoftheclassicalg

product.WejustdiscussedtheinnerproductonF(n0).Thissuggestsanaturalpositivede niteinnerproductonF(n)asthetensorproduct,sothat

(fρ(k),f′ρ′(k)):=(f,f′)(ρ(k),ρ′(k))(5.4)

wheref,f′∈F(n0)andρ(k),ρ′(k)∈F(kMS).Thisleadstoaninterestingphysicalinterpre-tation,aswewillsee.Wewillalwaysuseanorthonormalbasiscorrespondingtothisinnerproductfromnowon.

resWehaveseenintheprevioussectionthatallweightsofL(kMSλ1)havetheformλz= fin(n0λ1+λz)iziMiΛi.ThereforeF(n)isthedirectsumofirreduciblerepresentationsL

finofUqforvariousλz;ifthemultiplicityofacertainLfin(n0λ1+λz)inF(n)islargerthanone,thenthecorrespondingsubspacecanbedecomposedasanorthogonalsumofirreduciblecomponents.HenceF(n)isadirectorthogonalsumofHilbertspacesoftypeLfin(n0λ1+λz),withtheinducedinnerproduct(5.4),andthedi erentcomponentsarerelatedbytheaction ±(2.36).OnecannowcalculatetheadjointofthegeneratorsofoftheclassicalgeneratorsXifinfinUqonL(n0λ1+λz)withrespecttothisinnerproduct.Theresultis(see[37],Theorem

5.1) Hi =Hi,Xi±=siXi ,wheresi=( 1)zi.(5.5)

jfinIfπi(u)istherepresentationofu∈UqonLfin(n0λ1+λz)withrespecttoanorthonormalbasis,thismeansthatji (π(u) )j(5.6)i=πj(u)=πi(u)

asin(3.3);here reallydependsonzasin(5.5),whichwillhoweverbesuppressedinthefollowing.HenceLfin(n0λ1+λz)isaunitaryrepresentationofacertainrealformofthetype

fin(3.4)ofUq(so(D,C)),andthe“sectors”withdi erentλzbutthesamesiareunitarywith

16

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

paringwithSection3.1,weconcludethatifallziareeven,thenLfin(n0λ1+λz)isaunitaryrepresentationofthecompactform(3.6)andhence

D 1ascalar eldonSq.Iftheziaresuchthatsi=( 1)ziisasin(3.8),thenitisaunitary

finrepresentationoftheAnti–deSittergroupUq(so(2,D 2)),andshouldinfactbeviewed

D 1asascalar eldonAdSq,correspondingtoasquare–integrablescalar eldintheclassical

case.Tounderstandthis,considertheLfin((2MS k)λ1)aslowest–weightrepresentations

finLfin(kλ1)ofUqwithlowestweightkλ1.Forlowenergies,theyhavethesamestructure

asthescalar eldsontheclassicalAdSspace,whichareirreducibleunitarylowest–weightrepresentationsofSO(2,D 2)realizedintermsofsquare–integrablefunctions.Itisveryremarkablethattheyarerealizedhereintermsofpolynomialsinthecoordinatefunctions.Thereforewede netheHilbertspaceoffunctionsontherealquantumAnti–deSitterspacetobe

D 1finnD 1Lfin(nλ1)=Sq(t1)MS.(5.7)Uq·(t1)=AdSq:=

MS≤n<MS+kSMS≤n<MS+kS

MS nMSfinfinIntermsoflowest–weightrepresentationsLfin(nλ1)=Uq·(tDt1)ofUq,thiscanbe

writtenas D 1Lfin(nλ1).(5.8)AdSq=

(D 2)/2<n≤MS

ForenergieslessthanMS,thestatesintheHilbertspacesarethesameasclassically,and

fintheactionofUqapproachestheclassicaloneforanygivenweightasM→∞.Theenergy

ofallstatesislessthan2MS.Theprecisede nitionofthepositionoperatorswillbegiveninSection5.5.Intheclassicalcase,thelowerbound(D 2)/2canbeseenbyasimpledimensionalargument;howeveritcanbeviolatedslightly.M (D 4)/2MSfinForevenD>4,theirreduciblequotientofUq·(tDSt1)isthescalarsingletonLfin((D 4)/2λ1),withlowestweight(D 4)/2λ1;wewillnotconsideritanymorehere.

5.2Productspaces

Fromtheabovediscussion,itwouldseemmuchmorenaturaltoconsiderallpolynomials

fininthetiinsteadofjustcertainUq·(t1)n.Forsimplicity,wewillrestrictourselvestothe

respolynomialsoftheformF(n)=Uq·(t1)nforallkMS<n<kMS+kSandallk∈N,and

1=λ1forD>3,andλ 1=λ1/2=Λ1forD=3.Thenstudytheir eldcontent.Letλ

F(n)=F(n0)F(kMS)using(4.9),wherethesecondfactorisessentiallytherepresentation 1)oftheclassicalsymmetryalgebrag ,whichconnectsthevariouscomponentswithL(kλ

17

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

di erentλz′.Hence

⊕F(n)= F(n0) k∈N0≤n0<kS 1),L(kλ(5.9)

wherecertainmodeswereomittedasinSection5.1forsimplicity.Forthemomentweignoretherealitystructure. 1)areveryparticularrepresentationsoftheclassical ObservethattheL(kλg,whichhaveaniceinterpretation.Consider rstD=2r.Thenthedualalgebraisso (2r)=so(2r)as 1)canbeviewedasafunctionontheclassicalsphereSD 1.Henceshownin[37],andL(kλ 1)～⊕k∈NL(kλ=Fun(SD 1),whichisthespaceofpolynomialfunctionsontheclassicalsphereSD 1. 1=Λ1,andF(kMS)isNext,considerD=2r+1,whichislessobvious.Thenλ

thehighest–weightrepresentationL(kΛ1)ofso (2r+1)=sp(2r)withhighestweightkΛ1.Observethatthe2r–dimensionalrepresentationL(Λ1)isnotreal,inthesensethatthe2rvariablesziarenecessarilycomplex,andcanbeconsideredas4rrealvariablesxi.Thecompactform USp(2r)actsbymultiplyingtheziwithaunitarymatrix.Thereforethe

2radiusx=izi

本文来源：https://www.bwwdw.com/article/ng2q.html