Abstract:We examine the thermodynamic properties of recently constructed black hole solutions in SL(3, R) × SL(3, R) Chern-Simons theory in the presence of a chemical potential for spin-3 charge, which acts as an irrelevant deformation of the dual CFT with W 3 × W 3 symmetry. The smoothness or holonomy conditions admit four branches of solutions describing a flow between two AdS 3 backgrounds corresponding to two different CFTs. The dominant branch at low temperatures, connected to the BTZ black hole, merges smoothly with a thermodynamically unstable branch and disappears at higher temperatures. We confirm that the UV region of the flow satisfies the Ward identities of a CFT with W (2) 3 × W (2) 3 symmetry deformed by a spin-3 2 current. This allows to identify the precise map between UV and IR thermodynamic variables. We find that the high temperature regime is dominated by a black hole branch whose thermodynamics can only be consistently inferred with reference to this W (2) 3 × W (2) 3 CFT.

We consider free fermion and free boson CFTs in two dimensions, deformed by a chemical potential µ for the spin-three current. For the CFT on the infinite spatial line, we calculate the finite temperature entanglement entropy of a single interval perturbatively to second order in µ in each of the theories. We find that the result in each case is given by the same non-trivial function of temperature and interval length. Remarkably, we further obtain the same formula using a recent Wilson line proposal for the holographic entanglement entropy, in holomorphically factorized form, associated to the spin-three black hole in SL(3, R) × SL(3, R) Chern-Simons theory. Our result suggests that the order µ 2 correction to the entanglement entropy may be universal for W-algebra CFTs with spinthree chemical potential, and constitutes a check of the holographic entanglement entropy proposal for higher spin theories of gravity in AdS 3 .

We consider conformal field theories in 1+1 dimensions with W-algebra symmetries, deformed by a chemical potential µ for the spin-three current. We show that the order µ 2 correction to the Rényi and entanglement entropies of a single interval in the deformed theory, on the infinite spatial line and at finite temperature, is universal. The correction is completely determined by the operator product expansion of two spin-three currents, and by the expectation values of the stress tensor, its descendants and its composites, evaluated on the n-sheeted Riemann surface branched along the interval. This explains the recently found agreement of the order µ 2 correction across distinct free field CFTs and higher spin black hole solutions holographically dual to CFTs with W-symmetry.

We study black holes carrying higher spin charge in AdS 3 within the framework of SL(N, R) × SL(N, R) Chern-Simons theory. Focussing attention on the N = 4 case, we explicitly analyze the asymptotic symmetry algebra of black hole solutions with a chemical potential for spin-four charge. We demonstrate that the background describes an RG flow between an IR fixed point with W 4 symmetry and a UV fixed point with W-symmetry associated to a non-principal embedding of sl(2) in sl(4). Matching Chern-Simons equations with Ward identities of the deformed CFT, we show that the UV stress tensor is twisted by a certain U(1) current, and the flow is triggered by an operator with dimension 4/3 at the UV fixed point. We find independent confirmation of this picture via a consistent formulation of thermodynamics with respect to this UV fixed point. We further analyze the thermodynamics of multiple branches of black hole solutions for N = 4, 5 and find that the BTZ-branch, dominant at low temperatures, ceases to exist at higher temperatures following a merger with a thermodynamically unstable branch. We also point out an interesting connection between the RG flows and generalized KdV hierarchies.