4D Doubleton Conformal Theories, CPT and IIB String on AdS_5 X S^5

We study the unitary supermultiplets of the N=8, d=5 anti-de Sitter (AdS) superalgebra SU(2,2|4) which is the symmetry algebra of the IIB string theory on AdS_5 X S^5. We give a complete classification of the doubleton supermultiplets of SU(2,2|4) which do not have a Poincare limit and correspond to d=4 conformal field theories (CFT) living on the boundary of AdS_5. The CPT self-conjugate irreducible doubleton supermultiplet corresponds to d=4, N = 4 super Yang-Mills theory. The other irreducible doubleton supermultiplets come in CPT conjugate pairs. The maximum spin range of the general doubleton supermultiplets is 2. In particular, there exists a CPT conjugate pair of doubleton supermultiplets corresponding to the fields of N=4 conformal supergravity in d=4 which can be coupled to N=4 super Yang-Mills theory in d=4. We also study the "massless" supermultiplets of SU(2,2|4) which can be obtained by tensoring two doubleton supermultiplets. The CPT self-conjugate "massless" supermultiplet is the N=8 graviton supermultiplet in AdS_5. The other "massless" supermultiplets generally come in conjugate pairs and can have maximum spin range of 4. We discuss the implications of our results for the conjectured CFT/AdS dualities.