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]