# A 4d N=1 Cardy Formula

## TL;DR

This paper analyzes the asymptotic behavior of the superconformal index in 4d N=1 theories, revealing connections to conformal anomalies and black hole entropy in the high-temperature limit.

## Contribution

It introduces a novel high-temperature limit of the superconformal index for 4d N=1 theories, linking it to conformal anomalies and black hole entropy calculations.

## Key findings

- Index asymptotics expressed in terms of conformal anomalies
- Macroscopic entropy derived matching black hole entropy
- Cardy limit accounts for Bekenstein-Hawking entropy

## Abstract

We study the asymptotic behavior of the (modified) superconformal index for 4d $\mathcal{N} = 1$ gauge theory. By considering complexified chemical potential, we find that the `high-temperature limit' of the index can be written in terms of the conformal anomalies $3c-2a$. We also find macroscopic entropy from our asymptotic free energy when the Hofman-Maldacena bound $1/2 < a/c < 3/2$ for the interacting SCFT is satisfied. We study $\mathcal{N} = 1$ theories that are dual to AdS$_5 \times Y^{p, p}$ and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.