The ABC of higher-spin AdS/CFT

In recent literature, one-loop tests of the higher-spin AdS _d + ₁ /CFT _d correspondences were carried out. Here, we extend these results to a more general set of theories in d > 2. First, we consider the Type B higher spin theories, which have been conjectured to be dual to CFTs consisting of the singlet sector of N free fermion fields. In addition to the case of N Dirac fermions, we carefully study the projections to Weyl, Majorana, symplectic and Majorana-Weyl fermions in the dimensions where they exist. Second, we explore theories involving elements of both Type A and Type B theories, which we call Type AB. Their spectrum includes fields of every half-integer spin, and they are expected to be related to the U(N)/O(N) singlet sector of the CFT of N free complex/real scalar and fermionic fields. Finally, we explore the Type C theories, which have been conjectured to be dual to the CFTs of p-form gauge fields, where p = ^d ₂ − 1. In most cases, we find that the free energies at O(N ⁰ ) either vanish or give contributions proportional to the free-energy of a single free field in the conjectured dual CFT. Interpreting these non-vanishing values as shifts of the bulk coupling constant G _N ∼ 1/(N − k), we find the values k = −1, −1/2, 0, 1/2, 1, 2. Exceptions to this rule are the Type B and AB theories in odd d; for them, we find a mismatch between the bulk and boundary free energies that has a simple structure, but does not follow from a simple shift of the bulk coupling constant.