Chemical potentials in three-dimensional higher spin anti-de Sitter gravity

TL;DR

This work shows how to introduce chemical potentials for higher-spin charges in three-dimensional AdS gravity without disturbing the asymptotic $W$-algebra structure. By implementing these potentials as temporal components of the Chern-Simons connection (i.e., as Lagrange multipliers), the canonical charges and the boundary symmetry algebra remain unchanged in the spin-2 and spin-3 settings. While nonzero spin-3 potentials can enlarge the full asymptotic symmetry to $W_3^2$ in general, the $W_3$ structure can be recovered perturbatively in the chemical potentials, preserving a robust holographic framework. The method offers a non-perturbative, boundary-condition-preserving approach that aligns with known higher-spin black hole solutions and holographic $W$-algebras across SL(N) extensions.

Abstract

We indicate how to introduce chemical potentials for higher spin charges in higher spin anti-de Sitter gravity in a manner that manifestly preserves the original asymptotic W-symmetry. This is done by switching on a non-vanishing component of the connection along the temporal (thermal) circles. We first recall the procedure in the pure gravity case (no higher spin) where the only "chemical potentials" are the temperature and the chemical potential associated with the angular momentum. We then generalize to the higher spin case. We find that there is no tension with the W(N) or W(infinity) asymptotic algebra, which is obviously unchanged by the introduction of the chemical potentials. Our argument is non-perturbative.