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

Abstract

We study the asymptotic behavior of the (modified) superconformal index for 4d $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N = 1 theories that are dual to AdS5 × Yp,p and find that the Cardy limit of our index accounts for the Bekenstein-Hawking entropy of large black holes.

6d superconformal Cardy formulas

“…There are only gauge and mixed CS terms on S 3 . After integrating over S 3 , we obtain the following results, It is exactly the same with the Cardy formula given in [37,38] JHEP02(2021)092…”

“…In this section, we reproduce a 4d N = 1 Cardy formula found in [37,38] by following our anomaly-based approach. We consider the following superconformal index of a 4d N = 1 theory,…”

“…In [34], the Cardy formula of 4d SCFTs was studied with the strictly real chemical potentials. It was further studied in [26,[35][36][37][38] at the complex chemical potentials. It turns out that the free energy of the index captures N 2 growth, which accounts for the entropy of AdS 5 black holes.…”

“…The modified index turns out to be more 'refined' in the sense that it can capture more entropy than the original index in the Cardy limit. Similar modification of the index in 4d was studied in [37]. In order to compute the index (1.2), we consider the following thermal partition function on S 5…”

Large N gauge theories with a dense spectrum and the weak gravity conjecture

“…Recently it was demonstrated that the index can indeed detect the phase structure, thereby accounting for the entropy of large AdS black holes [27][28][29]. The Cardy-like formula for the 4d index [30,31] describing the high-energy behavior suggests that our model is unlikely to exhibit such a phase transition. However, it would be desirable to study detailed phase structure of our theory using the superconformal index, as was done in [28,32,33].…”