On the exact entropy of $\mathcal{N}=2$ black holes

TL;DR

This paper computes the exact entropy of four-dimensional $ =2$ black holes in M-theory by connecting microscopic degeneracies from elliptic genus Fourier coefficients to macroscopic localization calculations, confirming the leading Bessel function result.

Contribution

It establishes a precise link between the microscopic Rademacher expansion of elliptic genus coefficients and the macroscopic localization approach for $ =2$ black hole entropy, including subleading corrections.

Findings

Leading entropy matches a Bessel function from microscopic and macroscopic calculations.

Subleading corrections are explained by instanton contributions.

The measure for the localization integral is related to Chern-Simons theory on AdS$_2 imes$S$^1$.

Abstract

We study the exact entropy of four-dimensional $\mathcal{N}=2$ black holes in M-theory both from the brane and supergravity points of view. On the microscopic side the degeneracy is given by a Fourier coefficient of the elliptic genus of the dual two-dimensional $\mathcal{N}=(0,4)$ SCFT and can be extracted via a Rademacher expansion. We show how this expansion is mapped to a modified OSV formula derived by Denef and Moore. On the macroscopic side the degeneracy is computed by applying localization techniques to Sen's quantum entropy functional reducing it to a finite number of integrals. The measure for this finite dimensional integral is determined using a connection with Chern-Simons theory on AdS$_2 \times$S$^1$. The leading answer is a Bessel function in agreement with the microscopic answer. Other subleading corrections can be explained in terms of instanton contributions.