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

Abstract

We evaluate the superconformal index of 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 SYM with gauge algebra so(2Nc + 1) in the Cardy-like limit. We then study the relation with the results obtained for the S-dual usp(2Nc), 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.