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

A bstract 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 T 2 with $${\mathbb{Z}}_{m}$$ -symmetries, m = 2, 3, 4, 6, and Poisson deformations of the orbifolds ( $${T}^{2}\times {\mathbb{C}})/{\mathbb{Z}}_{m}$$ . 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 E 6 , 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.