Partition Functions and Casimir Energies in Higher Spin AdS_{d+1}/CFT_d

TL;DR

The paper extends higher-spin AdS/CFT tests to boundary theories on $S^1 \times S^{d-1}$, showing that large-$N$ singlet partition functions of free scalars and fermions reproduce the HS one-loop partition functions in AdS$_{d+1}$ across dimensions. It provides explicit formulas for the temperature-dependent piece $F_\beta$ and the Casimir energy $E_c$, proving that $E_c=0$ for odd $d$ and that the minimal HS theory yields a Casimir energy equal to that of a single conformal scalar in $R \times S^{d-1}$, consistent with a bulk coupling $G_N\sim 1/(N-1)$. The analysis covers complex and real scalars, Dirac and Majorana fermions, and extends to theories with $N_f$ flavors, including sectors with mixed-symmetry HS fields in higher dimensions, thereby giving a comprehensive, dimension-spanning set of checks of vectorial AdS/CFT. These results sharpen the dictionary between singlet-sector CFTs and their HS duals, clarifying the role of holonomy averaging, singlet constraints, and bulk coupling constants in the finite-temperature and Casimir-energy sectors.

Abstract

Recently, the one-loop free energy of higher spin (HS) theories in Euclidean AdS_{d+1} was calculated and matched with the order N^0 term in the free energy of the large N "vectorial" scalar CFT on the S^d boundary. Here we extend this matching to the boundary theory defined on S^1 x S^{d-1}, where the length of S^1 may be interpreted as the inverse temperature. It has been shown that the large N limit of the partition function on S^1 x S^2 in the U(N) singlet sector of the CFT of N free complex scalars matches the one-loop thermal partition function of the Vasiliev theory in AdS_4, while in the O(N) singlet sector of the CFT of N real scalars it matches the minimal theory containing even spins only. We extend this matching to all dimensions d. We also calculate partition functions for the singlet sectors of free fermion CFT's in various dimensions and match them with appropriately defined higher spin theories, which for d>3 contain massless gauge fields with mixed symmetry. In the zero-temperature case R x S^{d-1} we calculate the Casimir energy in the scalar or fermionic CFT and match it with the one-loop correction in the global AdS_{d+1}. For any odd-dimensional CFT the Casimir energy must vanish on general grounds, and we show that the HS duals obey this. In the U(N) symmetric case, we exhibit the vanishing of the regularized 1-loop Casimir energy of the dual HS theory in AdS_{d+1}. In the minimal HS theory the vacuum energy vanishes for odd d while for even d it is equal to the Casimir energy of a single conformal scalar in R x S^{d-1} which is again consistent with AdS/CFT, provided the minimal HS coupling constant is ~ 1/(N-1). We demonstrate analogous results for singlet sectors of theories of N Dirac or Majorana fermions. We also discuss extensions to CFT's containing N_f flavors in the fundamental representation of U(N) or O(N).