# The asymptotic growth of states of the 4d N=1 superconformal index

## TL;DR

This paper demonstrates that the superconformal index of 4d N=1 theories grows exponentially with charges in a large-charge limit, matching black hole entropy predictions in AdS/CFT correspondence.

## Contribution

It provides a detailed analysis of the asymptotic growth of the superconformal index in a Cardy-like limit, establishing a precise match with black hole entropy.

## Key findings

- Asymptotic growth of the index is exponential in charges.
- Saddle point analysis confirms the growth matches black hole entropy.
- Results hold for both finite and large N regimes.

## Abstract

We show that the superconformal index of N=1 superconformal field theories in four dimensions has an asymptotic growth of states which is exponential in the charges. Our analysis holds in a Cardy-like limit of large charges, for which the index is dominated by small values of chemical potentials. In this limit we find the saddle points of the integral that defines the superconformal index using two different methods. One method, valid for finite N, is to first take the Cardy-like limit and then find the saddle points. The other method is to analyze the saddle points at large N and then take the Cardy-like limit. The result of both analyses is that the asymptotic growth of states of the superconformal index exactly agrees with the Bekenstein-Hawking entropy of supersymmetric black holes in the dual AdS$_5$ theory.