S-duality in the Cardy-like limit of the superconformal index

TL;DR

This paper investigates the superconformal index of 4d $ ext{SO}(2N_c+1)$ SYM in the Cardy-like limit, exploring how S-duality relates to the $USp(2N_c)$ case and uncovering a non-trivial integral identity that preserves S-duality.

Contribution

It demonstrates the preservation of S-duality in the Cardy-like limit through a novel integral identity linking 3d Chern-Simons theories.

Findings

S-duality is preserved in the Cardy-like limit.

A non-trivial integral identity relates 3d Chern-Simons partition functions.

The relation between different gauge groups is clarified in specific charge regions.

Abstract

We evaluate the superconformal index of 4d $\mathcal{N}=4$ SYM with gauge algebra $so(2N_c+1)$ in the Cardy-like limit. We then study the relation with the results obtained for the S-dual $usp(2N_c)$, discussing the fate of S-duality in different regions of charges. We find that S-duality is preserved thanks to a non-trivial integral identity that relates the three sphere partition functions of pure 3d Chern-Simons gauge theories.