Partition Functions for Higher-Spin theories in AdS

TL;DR

The paper computes the one-loop partition function for massless higher-spin fields on AdS quotients, focusing on AdS$_5$ Vasiliev theory, by leveraging heat-kernel results. It derives a spin-$s$ determinant ratio for general $AdS_D$ and specializes to AdS$_5$ to obtain both the blind and refined partition functions, revealing a MacMahon-function–type structure and a nested, vacuum-character–like form. With nonzero chemical potentials for $SO(4)$ Cartans, the authors extract a refined partition function that factorizes into four sectors, consistent with the unrefined case when potentials are set to unity. The discussion highlights potential enhanced multiparticle symmetries and connections to known AdS$_3$ holography results, suggesting intriguing links to larger symmetry algebras and future directions in higher-spin holography.

Abstract

We calculate the one-loop partition function for a massless arbitrary-spin field on quotients of a general dimensional AdS background using the results of arXiv:1103.3627. We use these results to compute the one-loop partition function for a Vasiliev theory in AdS_5. An interesting form of the answer, suggestive of a vacuum character of an enhanced symmetry algebra is obtained. We also observe a close connection between the partition function for this Vasiliev theory and the d-dimensional MacMahon function.