W symmetry and integrability of higher spin black holes

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 AdS(3) in the principal embedding, we show that the W-3 x W-3 symmetry is preserved upon turning on perturbatively spin 3 chemical potentials. On the other hand, starting from AdS(3) in the diagonal embedding, we show that the W-3((2)) x W-3((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 n = 3 KdV (Boussinesq) hierarchy.