Four-dimensional black hole entropy from F-theory

TL;DR

This paper calculates the entropy of four-dimensional BPS black holes in F-theory by analyzing a 2D superconformal field theory, emphasizing the importance of Kaluza-Klein modes in matching microscopic results.

Contribution

It introduces a method to compute black hole entropy in F-theory using 2D SCFT central charges, highlighting the role of Kaluza-Klein states in the calculation.

Findings

Correctly reproduces microscopic entropy from supergravity calculations.

Shows the necessity of including infinite Kaluza-Klein towers for accurate results.

Provides explicit central charges and levels for the SCFT.

Abstract

We study the central charges and levels of a two-dimensional $N=(0,4)$ superconformal field theory describing four-dimensional BPS black holes in F-theory. These arise from D3-branes wrapping a curve in the base of an elliptically fibered Calabi-Yau threefold times a circle, and probe a transverse Taub-NUT space. The near horizon geometry of these D3-branes is AdS$_3 \times \text{S}^3/\mathbb{Z}_m$, where $m$ is the NUT charge. Starting from a six-dimensional supergravity effective action we compute three-dimensional Chern-Simons terms to deduce the central charges and levels. We find that it is crucial to integrate out an infinite tower of massive Kaluza-Klein states on S$^3/\mathbb{Z}_m$ to match the expected microscopic results. The induced corrections turn out to contribute at leading order to the central charges and levels, which in turn determine the black hole entropy.