Thermodynamics of two-dimensional conformal field theory dual to black holes

TL;DR

This paper explores the thermodynamics of 2D conformal field theories dual to black holes, deriving universal thermodynamic relations using the Cardy formula and supporting the black hole/CFT correspondence.

Contribution

It derives the thermodynamics of 2D CFTs dual to black holes from the Cardy formula under minimal assumptions, emphasizing universality and consistency with black hole physics.

Findings

CFT thermodynamics is universal and consistent with black hole thermodynamics.

The derivation uses both microcanonical and canonical ensembles.

Assumptions include mass-independent central charges and level matching.

Abstract

In this note we investigate the first law of thermodynamics of the two-dimensional conformal field theory (CFT) that is dual to black holes. We start from the Cardy formula and get the CFT thermodynamics with minimal reasonable assumptions. We use both the microcanonical ensemble and canonical ensemble versions of the Cardy formula. In the black hole/CFT correspondence the black hole is dual to a CFT with excitations, and the black hole mass $M$ and charge $N$ correspond to the energy and charge of the excited CFT. The CFT left- and right-moving central charges $c_{L,R}$ should be quantized, and so we assume that they are mass-independent. Also we assume the difference of the left- and right-moving sector levels $N_L-N_R$ is mass-independent dual to level matching condition. The thermodynamics of two-dimensional CFT we get is universal and supports the thermodynamics method of black hole/CFT correspondence.