# Black hole entropy function for toric theories via Bethe Ansatz

## TL;DR

This paper computes the entropy functions of dual black holes in AdS5 by analyzing the superconformal indices of toric quiver gauge theories using the Bethe Ansatz, confirming recent predictions and exploring index structures.

## Contribution

It introduces a Bethe Ansatz approach to evaluate the large-N superconformal index of toric theories and derives black hole entropy functions, extending to infinite families of quiver theories.

## Key findings

- Large-N superconformal index behavior matches Cardy-like predictions.
- Derived entropy functions for various toric quiver theories.
- Analyzed index structure and Stokes lines in conifold theory.

## Abstract

We evaluate the large-$N$ behavior of the superconformal indices of toric quiver gauge theories, and use it to find the entropy functions of the dual electrically charged rotating $\mathrm{AdS}_5$ black holes. To this end, we employ the recently proposed Bethe Ansatz method, and find a certain set of solutions to the Bethe Ansatz Equations of toric theories. This, in turn, allows us to compute the large-$N$ behavior of the index for these theories, including the infinite families $Y^{pq}$, $X^{pq}$ and $L^{pqr}$ of quiver gauge theories. Our results are in perfect agreement with the predictions made recently using the Cardy-like limit of the superconformal index. We also explore the index structure in the space of chemical potentials and describe the pattern of Stokes lines arising in the conifold theory case.