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Title: Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case

Abstract: We consider generalisations of the elliptic Calogero-Moser systems associated to complex crystallographic groups in accordance to [1]. In our previous work [2], we proposed these systems as candidates for Seiberg-Witten integrable systems of certain SCFTs. Here we examine that proposal for complex crystallographic groups of rank one. Geometrically, this means considering elliptic curves T2 with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb{Z}}_{m}$$\end{document}-symmetries, m = 2, 3, 4, 6, and Poisson deformations of the orbifolds (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${T}<^>{2}\times {\mathbb{C}})/{\mathbb{Z}}_{m}$$\end{document}. The m = 2 case was studied in [2], while m = 3, 4, 6 correspond to Seiberg-Witten integrable systems for the rank 1 Minahan-Nemeschansky SCFTs of type E6,7,8. This allows us to describe the corresponding elliptic fibrations and the Seiberg-Witten differential in a compact elegant form. This approach also produces quantum spectral curves for these SCFTs, which are given by Fuchsian ODEs with special properties.

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