Higher Spin Black Holes with Soft Hair

TL;DR

This work introduces boundary conditions for AdS$_3$ higher-spin gravity that yield two copies of affine $\hat{u}(1)$ currents as the asymptotic symmetry, providing a framework for higher-spin soft hair. Spin-3 (and spin-$N$) degrees of freedom are organized so that the $W_3$ (and $W_N$) currents arise as twisted-Sugawara composites of the affine currents, while the global charges remain in the $\hat{u}(1)$ sector. The authors construct and analyze stationary, non-spherically symmetric solutions called higher spin black flowers, establish Euclidean regularity, and compute entropy; on the BTZ-connected branch the entropy depends only on purely gravitational zero modes and matches known $W$-algebra expressions, both in diagonal and highest-weight gauges and in the metric formalism. They further extend the construction to $sl(N,\mathbb{R})$ theories, showing a universal entropy form and outlining connections to integrable hierarchies and holography, with multiple avenues for future exploration including microstate counting and hs$(\lambda)$ generalizations.

Abstract

We construct a new set of boundary conditions for higher spin gravity, inspired by a recent "soft Heisenberg hair"-proposal for General Relativity on three-dimensional Anti-de Sitter. The asymptotic symmetry algebra consists of a set of affine $\hat u(1)$ current algebras. Its associated canonical charges generate higher spin soft hair. We focus first on the spin-3 case and then extend some of our main results to spin-$N$, many of which resemble the spin-2 results: the generators of the asymptotic $W_3$ algebra naturally emerge from composite operators of the $\hat u(1)$ charges through a twisted Sugawara construction; our boundary conditions ensure regularity of the Euclidean solutions space independently of the values of the charges; solutions, which we call "higher spin black flowers", are stationary but not necessarily spherically symmetric. Finally, we derive the entropy of higher spin black flowers, and find that for the branch that is continuously connected to the BTZ black hole, it depends only on the affine purely gravitational zero modes. Using our map to $W$-algebra currents we recover well-known expressions for higher spin entropy. We also address higher spin black flowers in the metric formalism and achieve full consistency with previous results.