Entropy Counting from Schwarzschild/CFT and Soft Hair

TL;DR

This paper refines Carlip's entropy counting method for Schwarzschild black holes by identifying a full 2D conformal algebra, deriving central charges and temperatures, and confirming the Bekenstein-Hawking entropy via the Cardy formula, supporting a CFT description of black hole microstates.

Contribution

It demonstrates how to isolate the full 2D local conformal algebra for Schwarzschild black holes and links it to a CFT microstate counting consistent with Bekenstein-Hawking entropy.

Findings

Identification of the full 2D conformal algebra for Schwarzschild black holes

Derivation of central charges and temperatures matching entropy calculations

Confirmation that Hamiltonian generators align with CFT symmetry generators

Abstract

We revisit Carlip's approach to entropy counting. This analysis reemerged in a recently obtained Schwarzschild/CFT-correspondence as Sugawara-construction of a 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, 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 this 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 non-canonical counterterms. We comment on the related recent work by Haco, Hawking, Perry, Strominger.