Virasoro hair and entropy for axisymmetric Killing horizons

We show that the gravitational phase space for the near-horizon region of a bifurcate, axisymmetric Killing horizon in any dimension admits a 2D conformal symmetry algebra with central charges proportional to the area. This extends the construction of [Haco et. al., JHEP 12, 098 (2018)] to generic Killing horizons appearing in solutions of Einstein's equations, and motivates a holographic description in terms of a 2D conformal field theory. The Cardy entropy in such a field theory agrees with the Bekenstein-Hawking entropy of the horizon, suggesting a microscopic interpretation.
Comment: 6+4 pages, 2 columns, 1 figure, v2: included a derivation of integrability of the horizon charges, added appendix on the near-horizon expansion for Kerr, references added; updated to match published version