# Entropy function from toric geometry

## TL;DR

This paper explores the Cardy-like limit of superconformal indices in 4d $
abla=1$ toric quiver gauge theories, linking the entropy function to toric data and comparing Legendre transforms with existing results.

## Contribution

It extends the Cardy-like limit analysis from $
abla=4$ SYM to $
abla=1$ toric quiver theories, connecting entropy functions with toric geometry and providing new computations of Legendre transforms.

## Key findings

- Entropy function relates to toric data.
- Legendre transforms computed for specific models.
- Comparison with recent literature results.

## Abstract

It has recently been claimed that a Cardy-like limit of the superconformal index of 4d $\mathcal{N}=4$ SYM accounts for the entropy function, whose Legendre transform corresponds to the entropy of the holographic dual AdS$_5$ rotating black hole. Here we study this Cardy-like limit for $\mathcal{N}=1$ toric quiver gauge theories, observing that the corresponding entropy function can be interpreted in terms of the toric data. Furthermore, for some families of models, we compute the Legendre transform of the entropy function, comparing with similar results recently discussed in the literature.