Higher Spin Black Holes from CFT

TL;DR

The paper derives the thermodynamics of higher spin black holes in $AdS_3$ from the dual 2D CFT with $\mathcal{W}_\infty[\lambda]$ symmetry by expanding the CFT partition function $Z_{\rm CFT}(\hat{\tau},\alpha)$ in the spin-3 chemical potential $\alpha$ and performing a modular $S$-transformation to access the high-temperature regime. It expresses higher-spin corrections through torus correlators of the $\mathcal{W}$-currents and a generalized Zhu recursion for zero modes, showing that the leading contributions at large central charge $c$ reproduce the bulk free energy exactly up to $\mathcal{O}(\alpha^6)$ by computing the connected 2-, 4-, and 6-point functions. Crucially, nonlinear terms in the $\mathcal{W}_\infty[\lambda]$ algebra contribute starting at the 6-point level, and their inclusion is essential for matching the gravity result. This provides a universal, microstate-free check of higher-spin AdS$_3$/CFT$_2$ holography and links bulk holonomy data to boundary torus amplitudes.

Abstract

Higher spin gravity in three dimensions has explicit black holes solutions, carrying higher spin charge. We compute the free energy of a charged black hole from the holographic dual, a 2d CFT with extended conformal symmetry, and find exact agreement with the bulk thermodynamics. In the CFT, higher spin corrections to the free energy can be calculated at high temperature from correlation functions of W-algebra currents.

Figures (1)

• Figure 1: Contours of integration for the insertion of two zero modes, before and after the $S$ modular transformation. The fundamental region of the torus lies between the dashed lines, $|q| < |z| \leq 1$.