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

We evaluate the superconformal index of 4d $\mathcal{N}$ \( \mathcal{N} \) = 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.