Higher spin AdS(d+1)/CFTd at one loop

Abstract

Following S. Giombi and I. R. Klebanov, [J. High Energy Phys. 12 (2013) 068], we carry out one-loop tests of higher spin AdS(d+1)/CFTd correspondences for d >= 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 Giombi and Klebanov, 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) similar to 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.