Logarithmic Corrections to Black Hole Entropy from the Cardy Formula

TL;DR

The paper analyzes how logarithmic corrections to the Cardy formula yield the first-order quantum corrections to black hole entropy across diverse models. By deriving the leading $\ln$-term from modular invariance and applying it to the BTZ black hole, horizon CFTs, and string-theory countings, it demonstrates a universal $-3/2$ coefficient on the logarithmic correction with additional charge-dependent refinements. The results align with independent quantum geometry findings and suggest a form $S \sim {A\over4G} - {3\over2}\ln({A\over4G}) + \ln F(Q) + \text{const}$, indicating a potential universality of these corrections. The work also discusses how central-charge corrections and boundary conditions influence subleading terms and the extent to which a single underlying CFT could describe black-hole thermodynamics across theories and dimensions.

Abstract

Many recent attempts to calculate black hole entropy from first principles rely on conformal field theory techniques. By examining the logarithmic corrections to the Cardy formula, I compute the first-order quantum correction to the Bekenstein-Hawking entropy in several models, including those based on asymptotic symmetries, horizon symmetries, and certain string theories. Despite very different physical assumptions, these models all give a correction proportional to the logarithm of the horizon size, and agree qualitatively with recent results from ``quantum geometry'' in 3+1 dimensions. There are some indications that even the coefficient of the correction may be universal, up to differences that depend on the treatment of angular momentum and conserved charges.