Higher Spin Black Holes with Soft Hair

TL;DR

This paper introduces new boundary conditions for higher spin gravity in AdS3, revealing an affine u(1) symmetry algebra and constructing higher spin black hole solutions with soft hair, extending known results to higher spins.

Contribution

It develops a novel set of boundary conditions for higher spin gravity, demonstrating the emergence of affine u(1) symmetries and constructing higher spin black flower solutions with soft hair.

Findings

Asymptotic symmetry algebra is affine u(1) currents.

Higher spin black flowers are stationary, non-spherical solutions.

Entropy depends only on affine zero modes, matching known formulas.

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.