Entropy counting from a Schwarzschild/CFT correspondence and soft hair

Abstract

We revisit Carlip's approach to entropy counting. This analysis reemerged in a recently obtained Schwarzschild/conformal field theory (CFT) correspondence as the Sugawara-construction of a two-dimensional (2D) stress tensor. Here, for the example of a Schwarzschild black hole, we show how to single out diffeomorphisms forming in contrast to Carlip's analysis the full 2D local conformal algebra. We provide arguments as to why their Hamiltonian generators are expected to be the symmetry generators of a possible conformal field theory describing the part of phase space responsible for black hole microstates. Then, we can infer central charges and temperatures of this CFT by inspecting the algebra of these Hamiltonian generators. Using these data in the Cardy formula, precise agreement with the Bekenstein-Hawking entropy is found. Alternatively, we obtain the same CFT temperatures by thermodynamic considerations. Altogether, this suggests that the Hamiltonian generators need no corrections through possible noncanonical counterterms. We comment on the related recent work by Haco, Hawking, Perry and Strominger.