One-loop tests of the supersymmetric higher spin AdS4/CFT3 correspondence

We compute one-loop free energy for D = 4 Vasiliev higher spin gravities based on Konstein-Vasiliev algebras hu(m; n|4), ho(m; n|4), or husp(m; n|4) and subject to higher spin-preserving boundary conditions, which are conjectured to be dual to the U(N), O(N) or USp(N) singlet sectors, respectively, of free conformal field theories (CFTs) on the boundary of AdS(4). Ordinary supersymmetric higher spin theories appear as special cases of Konstein-Vasiliev theories, when the corresponding higher spin algebra contains OSp(N|4) as a subalgebra. In AdS(4) with an S-3 boundary, we use a regularization scheme for individual spins that employs their character such that the subsequent sum over all spins is finite, thereby avoiding the need for additional regularization. We find that the contribution of the infinite tower of bulk fermions vanishes. As a result, the free energy is the sum of those which arise in type A and type B models with internal symmetries, the known mismatch between the bulk and boundary free energies for type B model persists, and ordinary supersymmetric higher spin theories exhibit the mismatch as well. The only models that have a match are type A models with internal symmetries, corresponding to n = 0. The matching requires identification of the inverse Newton constant G(N)(-1) with N plus a proper integer as was found previously for special cases. In AdS4 with an S-1 x S-2 boundary, the bulk one-loop free energies match those of the dual free CFTs for arbitrary m and n. We also show that a supersymmetric double-trace deformation of free CFT based on OSp(1|4) does not contribute to the O(N-0) free energy, as expected from the bulk.