One loop tests of higher spin AdS/CFT

Vasiliev's type A higher spin theories in AdS₄ have been conjectured to be dual to the U(N) or O (N⁰) singlet sectors in 3-d conformal field theories with N -component scalar fields. We compare the O (N⁰) correction to the 3-sphere free energy F in the CFTs with corresponding calculations in the higher spin theories. This requires evaluating a regularized sum over one loop vacuum energies of an infinite set of massless higher spin gauge fields in Euclidean AdS₄. For the Vasiliev theory including fields of all integer spin and a scalar with Δ = 1 boundary condition, we show that the regularized sum vanishes. This is in perfect agreement with the vanishing of subleading corrections to F in the U(N) singlet sector of the theory of N free complex scalar fields. For the minimal Vasiliev theory including fields of only even spin, the regularized sum remarkably equals the value of F for one free real scalar field. This result may agree with the O(N) singlet sector of the theory of N real scalar fields, provided the coupling constant in the Vasiliev theory is identified as G_N ∼ 1/(N - 1). Similarly, consideration of the USp(N) singlet sector for N complex scalar fields, which we conjecture to be dual to the husp(2; 0|4) Vasiliev theory, requires G_N ∼ 1/(N + 1). We also test the higher spin AdS ₃/CFT₂ conjectures by calculating the regularized sum over one loop vacuum energies of higher spin fields in AdS₃. We match the result with the O (N⁰) term in the central charge of the W _N minimal models; this requires a certain truncation of the CFT operator spectrum so that the bulk theory contains two real scalar fields with the same boundary conditions.