Character Integral Representation of Zeta function in AdS$_{d+1}$: II. Application to partially-massless higher-spin gravities

TL;DR

This paper computes one-loop free energies of higher-spin gravities in AdS space using a character integral method, extending previous results from massless to partially-massless theories across various dimensions.

Contribution

It applies a novel character integral representation of zeta functions to efficiently compute free energies of complex higher-spin theories, generalizing known results to partially-massless cases.

Findings

Successfully computed zeta functions for partially-massless higher-spin gravities.

Extended previous massless results to arbitrary $$ and dimensions.

Demonstrated the efficiency of the character integral method for complex field content.

Abstract

We compute the one-loop free energies of the type-A$_\ell$ and type-B$_\ell$ higher-spin gravities in $(d+1)$-dimensional anti-de Sitter (AdS$_{d+1}$) spacetime. For large $d$ and $\ell$, these theories have a complicated field content, and hence it is difficult to compute their zeta functions using the usual methods. Applying the character integral representation of zeta function developed in the companion paper arXiv:1805.05646 to these theories, we show how the computation of their zeta function can be shortened considerably. We find that the results previously obtained for the massless theories ($\ell=1$) generalize to their partially-massless counterparts (arbitrary $\ell$) in arbitrary dimensions.