W Symmetry and Integrability of Higher spin black holes

TL;DR

This work analyzes the $SL(3,\mathbb{R})\times SL(3,\mathbb{R})$ Chern-Simons description of higher-spin gravity in AdS$_3$ with Dirichlet boundary conditions at fixed chemical potentials $\mu$, for both the principal and diagonal $sl(2,\mathbb{R})$ embeddings. The authors show that the asymptotic symmetry algebras are preserved under turning on $\mu$: $\mathcal{W}_3\times\mathcal{W}_3$ for the principal embedding and $\mathcal{W}^{(2)}_3\times\mathcal{W}^{(2)}_3$ for the diagonal embedding, with the algebras realized by field-dependent generators and a well-defined initial data formulation. They connect the bulk dynamics to integrable systems by identifying a Boussinesq/KdV hierarchy structure, establishing bi-Hamiltonian frameworks on the time slice, and constructing infinite towers of commuting charges that map to KdV charges. The results reinforce the robustness of higher-spin asymptotic symmetries under chemical potential deformations and illuminate the holographic interplay between $\mathcal{W}$-algebras, integrable hierarchies, and black hole boundary data.

Abstract

We obtain the asymptotic symmetry algebra of sl(3,R) x sl(3,R) Chern-Simons theory with Dirichlet boundary conditions for fixed chemical potential. These boundary conditions are obeyed by higher spin black holes. For each embedding of sl(2,R) into sl(3,R), we show that the asymptotic symmetry group is independent of the chemical potential. On the one hand, starting from AdS3 in the principal embedding, we show that the W3 x W3 symmetry is preserved upon turning on perturbatively spin 3 chemical potentials. On the other hand, starting from AdS3 in the diagonal embedding, we show that the W3^(2) x W3^(2) symmetry is preserved upon turning on finite spin 3/2 chemical potentials. We also make connections between the canonical Lagrangian formalism and integrability methods based on the third KdV (Boussinesq) hierarchy.