3d Chern-Simons theory from M5-branes

We study 5d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 2 maximally supersymmetric Yang-Mills theory with a gauge group G on S2  × M3, where M3 is a 3-manifold. By explicit localization computation we show that the path-integral of the 5d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ \mathcal{N} $\end{document} = 2 theory reduces to that of the 3d \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {G_{\mathbb{C}}} $\end{document} Chern-Simons theory on M3, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ {G_{\mathbb{C}}} $\end{document} is the complexification of G. This gives a direct derivation of the appearance of the Chern-Simons theory from the compactification of the 6d (2, 0) theory, confirms the predictions from the 3d/3d correspondence for G = SU(N), and suggests the generalization of the correspondence to more general gauge groups.