Soft Heisenberg hair on black holes in three dimensions

TL;DR

The article demonstrates that non-spherically symmetric 3D AdS black holes admit near-horizon boundary conditions yielding a simple Heisenberg algebra of two affine u(1) currents, whose charges realize soft hair without modifying the Bekenstein–Hawking entropy. By expressing 3D gravity as a Chern–Simons theory, the authors explicitly derive the near-horizon charges and show that the Hamiltonian commutes with all soft hair modes, making them zero-energy excitations. They further connect the near-horizon symmetry to the standard asymptotic Virasoro algebra via a twisted Sugawara construction, illustrating black hole complementarity between horizon- and infinity-based descriptions. The work also outlines microstate counting approaches and discusses generalizations to cosmological horizons and other 3D gravity theories, suggesting that soft hair and horizon symmetries may have broad applicability.

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.