Higher Spin Superalgebras in any Dimension and their Representations

Fock module realization for the unitary singleton representations of the d − 1 dimensional conformal algebra o(d − 1, 2), which correspond to the spaces of single-particle states of massless scalar and spinor in d− 1 dimensions, is given. The pattern of the tensor product of a pair of singletons is analyzed in any dimension. It is shown that for d > 3 the tensor product of two boson singletons decomposes into a sum of all integer spin totally symmetric massless representations in AdSd, the tensor product of boson and fermion singletons gives a sum of all half-integer spin symmetric massless representations in AdSd, and the tensor product of two fermion singletons in d > 4 gives rise to massless fields of mixed symmetry types inAdSd depicted by Young tableaux with one row and one column together with certain totally antisymmetric massive fields. In the special case of o(2, 2), tensor products of 2d massless scalar and/or spinor modules contain infinite sets of 2d massless conformal fields of different spins. The obtained results extend the 4d result of Flato and Fronsdal [1] to any dimension and provide a nontrivial consistency check for the recently proposed higher spin model in AdSd [2]. We define a class of higher spin superalgebras which act on the supersingleton and higher spin states in any dimension. For the cases of AdS3, AdS4, and AdS5 the isomorphisms with the higher spin superalgebras defined earlier in terms of spinor generating elements are established.