W Symmetry and Integrability of Higher spin black holes

TL;DR

This paper explores the asymptotic symmetry algebra of higher spin black holes in sl(3,R) x sl(3,R) Chern-Simons theory, demonstrating the preservation of W-symmetries under various chemical potentials and linking to integrability structures.

Contribution

It shows that the asymptotic symmetry algebra remains unchanged under certain chemical potentials for different embeddings, connecting higher spin black hole symmetries with integrability methods.

Findings

W3 x W3 symmetry preserved with spin 3 chemical potentials

W3^(2) x W3^(2) symmetry preserved with spin 3/2 chemical potentials

Connections established between symmetry algebra and integrability hierarchies

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.