Contents

Recently, the one-loop free energy of higher spin (HS) theories in Euclidean AdSd+1 was calcu-lated and matched with the order N0 term in the free energy of the large N “vectorial ” scalar CFT on the Sd boundary. Here we extend this matching to the boundary theory defined on S1×Sd−1, where the length of S1 may be interpreted as the inverse temperature. It has been shown that the large N limit of the partition function on S1 × S2 in the U(N) singlet sector of the CFT of N free complex scalars matches the one-loop thermal partition function of the Vasiliev theory in AdS4, while in the O(N) singlet sector of the CFT of N real scalars it matches the minimal theory containing even spins only. We extend this matching to all dimensions d. We also calculate partition functions for the singlet sectors of free fermion CFT’s in various dimensions and match them with appropriately defined higher spin theories, which for d> 3 contain massless gauge fields with mixed symmetry. In the zero-temperature case R × Sd−1 we calculate the Casimir energy in the scalar or fermionic CFT and match it with the one-loop correction in the global AdSd+1. For any odd-dimensional CFT the Casimir energy must vanish on general grounds, and we show that the HS duals obey this. In the U(N) symmetric case, we exhibit the vanishing of the regularized 1-loop Casimir energy of the dual HS theory in AdSd+1. In the minimal HS theory the vacuum energy vanishes for odd d while for even d it is equal to the Casimir energy of a single conformal scalar in R×Sd−1 which is again consistent with AdS/CFT, provided the minimal HS coupling constant is ∼ 1/(N−1). We demonstrate analogous results for singlet sectors of theories of N Dirac or Majorana fermions. We also discuss extensions to CFT’s containing Nf flavors in the fundamental representation of U(N) or O(N)