Iterating free-field AdS/CFT: higher spin partition function relations

We find a simple relation between a free higher spin partition function on the thermal quotient of AdS(d+1) and the partition function of the associated d-dimensional conformal higher spin field defined on the thermal quotient of AdS(d). Starting with a conformal higher spin field defined in AdS(d), one may also associate to with another conformal field in d - 1 dimensions, thus iterating AdS/CFT. We observe that in the case of d = 4, this iteration leads to a trivial 3d higher spin conformal theory with parity-even non-local action: it describes a zero total number of dynamical degrees of freedom and the corresponding partition function is equal to 1.