A 4d N $$ \mathcal{N} $$ = 1 Cardy Formula

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 AdS5 × Y p,p and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.