Higher Spin AdS$_{d+1}$/CFT$_d$ at One Loop

Following arXiv:1308.2337, we carry out one loop tests of higher spin AdS$_{d+1}$/CFT$_d$ correspondences for $d\geq 2$. The Vasiliev theories in AdS$_{d+1}$, which contain each integer spin once, are related to the $U(N)$ singlet sector of the $d$-dimensional CFT of $N$ free complex scalar fields; the minimal theories containing each even spin once -- to the $O(N)$ singlet sector of the CFT of $N$ free real scalar fields. Using analytic continuation of higher spin zeta functions, which naturally regulate the spin sums, we calculate one loop vacuum energies in Euclidean AdS$_{d+1}$. In even $d$ we compare the result with the $O(N^0)$ correction to the $a$-coefficient of the Weyl anomaly; in odd $d$ -- with the $O(N^0)$ correction to the free energy $F$ on the $d$-dimensional sphere. For the theories of integer spins, the correction vanishes in agreement with the CFT of $N$ free complex scalars. For the minimal theories, the correction always equals the contribution of one real conformal scalar field in $d$ dimensions. As explained in arXiv:1308.2337, this result may agree with the $O(N)$ singlet sector of the theory of $N$ real scalar fields, provided the coupling constant in the higher spin theory is identified as $G_N\sim 1/(N-1)$. Our calculations in even $d$ are closely related to finding the regularized $a$-anomalies of conformal higher spin theories. In each even $d$ we identify two such theories with vanishing $a$-anomaly: a theory of all integer spins, and a theory of all even spins coupled to a complex conformal scalar. We also discuss an interacting UV fixed point in $d=5$ obtained from the free scalar theory via an irrelevant double-trace quartic interaction. This interacting large $N$ theory is dual to the Vasiliev theory in AdS$_6$ where the bulk scalar is quantized with the alternate boundary condition.