Quantum W-symmetry in AdS_3

TL;DR

The authors compute the one-loop quantum corrections for massless higher-spin fields on thermal AdS$_3$ and show the determinants factor holomorphically into left- and right-moving pieces that assemble into vacuum characters of ${\cal W}_N$ (or ${\cal W}_{\infty}$) algebras. A general spin-$s$ analysis yields a compact determinant form $Z^{(s)}=[\det(-\Delta + s(s-3)/\ell^{2})^{TT}_{(s)}]^{-1/2}[\det(-\Delta + s(s-1)/\ell^{2})^{TT}_{(s-1)}]^{1/2}$, with cancellations removing non-TT contributions; summing over spins reproduces the ${\cal W}_N$ vacuum character and, in the hs$(1,1)$ limit, the MacMahon function. This supports the view that the quantum Hilbert space of these theories is organized by boundary ${\cal W}$-symmetry and suggests perturbative exactness of the one-loop result. The MacMahon function connection hints at deeper ties to topological strings and ${\cal W}_{\infty}$ symmetry, and the analysis lays groundwork for supersymmetric extensions and nonperturbative questions.

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.