# Cardy-like asymptotics of the 4d $\mathcal{N}=4$ index and AdS$_5$ blackholes

## TL;DR

This paper rigorously analyzes the Cardy-like limit of the 4d $
abla=4$ superconformal index using elliptic hypergeometric integrals, revealing new blackhole saddle points and bifurcation phenomena related to AdS$_5$ blackholes.

## Contribution

It provides a rigorous derivation of the index asymptotics, uncovers a new blackhole saddle point, and explores bifurcation phenomena in the index's behavior.

## Key findings

- Identification of a new blackhole saddle point.
- Discovery of bifurcation phenomena in index asymptotics.
- Clarification of the role of supersymmetric Casimir energy.

## Abstract

Choi, Kim, Kim, and Nahmgoong have recently pioneered analyzing a Cardy-like limit of the superconformal index of the 4d $\mathcal{N}=4$ theory with complexified fugacities which encodes the entropy of the dual supersymmetric AdS$_5$ blackholes. Here we study the Cardy-like asymptotics of the index within the rigorous framework of elliptic hypergeometric integrals, thereby filling a gap in their derivation of the blackhole entropy function, finding a new blackhole saddle-point, and demonstrating novel bifurcation phenomena in the asymptotics of the index as a function of fugacity phases. We also comment on the relevance of the supersymmetric Casimir energy to the blackhole entropy function in the present context.