$ \mathcal{W} $ symmetry and integrability of higher spin black holes

Abstract

We obtain the asymptotic symmetry algebra of sl(3, R) × 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 × 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 (2) 3 ×W (2) 3 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.

Boundary conditions and partition functions in higher spin AdS3/CFT2

“…Note that (4.1) requires one to identify precisely what sources one chooses, and different choices of sources or thermodynamic variables will correspond to different boundary terms and boundary conditions, as recently discussed in [25]. A class of boundary conditions was studied in [59][60][61][62][63][64][65] that lead to the so-called "canonical thermodynamics", consistent with canonical definitions of conserved charges and thermodynamics in gravitational theories, with a perturbative application of Wald-like formulae for the entropy and energy [66], and with the thermal limit of entanglement entropy calculations in higher spin theories [21,22]. A feature of the canonical approach is that, once sources for the currents are switched on, quantities such as the energy and higher spin charges, for example, acquire an explicit dependence on the chemical potentials and differ from their undeformed counterparts.…”

“…A different set of "canonical" boundary conditions in the presence of sources was proposed in [62][63][64][65] from a bulk perspective, with the flatness condition on the connection resulting in Ward identities of the form (2.53). We have shown that these boundary conditions correspond to deformations of the CFT Hamiltonian of the form (2.5).…”

Conformal perturbation theory and higher spin entanglement entropy on the torus

“…3 On the other hand, the holographic duals to such states which are the spin-three black holes of [13,14] appear to display an RG flow from an IR AdS3 geometry with W∞[λ] symmetry [38][39][40] to a UV AdS3 with different (non-principal) asymptotic W-algebra symmetry [41]. Several works have pointed out both the intriguing consequences of this picture [42,43], and possibly the complete absence of an RG flow interpretation [44,45].…”

About the phase space of SL(3) black holes

“…The name deriving from the fact that they can be identified with the chemical potentials of conserved higher spin currents in a 2D CFT. Recently, attention has been paid to the fixed time canonical bracket structure of these families [1,18] (studies for the highest weight boundary conditions can be found at [5,6,21]). One main point of interest regards the classification of charges of generalised black hole solutions.…”

“…Higher spin theories [2][3][4][5][6] in 3D, have been of great interest recently, and specifically, the study of higher spin black holes in the Chern-Simons formulation has been one of the most active lines of research [1,[7][8][9][10][11][12][13][14][15][16][17][18].…”