Soft Heisenberg hair on black holes in three dimensions

TL;DR

This paper introduces a simple near horizon symmetry algebra for three-dimensional black holes, revealing soft hair as affine u(1) currents that do not affect black hole entropy, with implications for microstates and cosmological horizons.

Contribution

It proposes boundary conditions leading to a Heisenberg algebra of soft hair on 3D black holes, expanding understanding of horizon symmetries and their role in black hole microstates.

Findings

Soft hair is described by affine u(1) current algebras.

Soft hair does not contribute to the Bekenstein-Hawking entropy.

Boundary conditions ensure regularity and compatibility with horizon physics.

Abstract

Three-dimensional Einstein gravity with negative cosmological constant admits stationary black holes that are not necessarily spherically symmetric. We propose boundary conditions for the near horizon region of these black holes that lead to a surprisingly simple near horizon symmetry algebra consisting of two affine u(1) current algebras. The symmetry algebra is essentially equivalent to the Heisenberg algebra. The associated charges give a specific example of "soft hair" on the horizon, as defined by Hawking, Perry and Strominger. We show that soft hair does not contribute to the Bekenstein-Hawking entropy of Banados-Teitelboim-Zanelli black holes and "black flower" generalizations. From the near horizon perspective the conformal generators at asymptotic infinity appear as composite operators, which we interpret in the spirit of black hole complementarity. Another remarkable feature of our boundary conditions is that they are singled out by requiring that the whole spectrum is compatible with regularity at the horizon, regardless the value of the global charges like mass or angular momentum. Finally, we address black hole microstates and generalizations to cosmological horizons.