A note on vectorial AdS₅/CFT₄ duality for spin-j boundary theory

The vectorial holographic correspondences between higher-spin theories in AdS₅ and free vector models on the boundary are extended to the cases where the latter is described by free massless spin-j field. The dual higher-spin theory in the bulk does not include gravity and can only be defined on rigid AdS₅ background with S⁴ boundary. We discuss various properties of these rather special higher-spin theories and calculate their one-loop free energies. We show that the result is proportional to the same quantity for spin-j doubleton treated as if it is a AdS₅ field. Finally, we consider even more special case where the boundary theory itself is given by an infinite tower of massless higher-spin fields.