On S-duality of 5d super Yang-Mills on S1

We study a duality of 5d maximally supersymmetric Yang-Mills on S1, which exchanges the tower of Kaluza-Klein W-bosons and the tower of instantonic monopoles. This duality maps a non-simply-laced gauge theory to a simply-laced gauge theory twisted by an outer automorphism around S1, and is closely related to the Langlands dual of affine Lie algebras. We also discuss how this S-duality is implemented in terms of 6d \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} = (2, 0) theory. This is straightforward except for the 6d theory of type SU(2n + 1) with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{Z}_2}$\end{document} outer-automorphism twist, for which a few new properties are deduced. For example, this 6d theory, when reduced on an S1 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\mathbb{Z}_2}$\end{document} twist, gives 5d USp(2n) theory with nontrivial discrete 5d theta angle.