Quantum W-symmetry in AdS_3

TL;DR

This paper provides evidence that quantum massless higher spin theories in AdS_3 exhibit an enlarged W_N-symmetry, confirmed through one-loop calculations and heat kernel techniques, with results expressed via vacuum characters and the MacMahon function.

Contribution

It demonstrates the realization of W_N-symmetry at the quantum level in AdS_3 higher spin theories through explicit one-loop computations.

Findings

One-loop determinants factor holomorphically.

Contributions organize into vacuum characters of W_N.

Results for hs(1,1) relate to the MacMahon function.

Abstract

It has recently been argued that, classically, massless higher spin theories in AdS_3 have an enlarged W_N-symmetry as the algebra of asymptotic isometries. In this note we provide evidence that this symmetry is realised (perturbatively) in the quantum theory. We perform a one loop computation of the fluctuations for a massless spin $s$ field around a thermal AdS_3 background. The resulting determinants are evaluated using the heat kernel techniques of arXiv:0911.5085. The answer factorises holomorphically, and the contributions from the various spin $s$ fields organise themselves into vacuum characters of the W_N symmetry. For the case of the hs(1,1) theory consisting of an infinite tower of massless higher spin particles, the resulting answer can be simply expressed in terms of (two copies of) the MacMahon function.