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 Z₌ -symmetries, m = 2, 3, 4, 6, and Poisson deformations of the orbifolds (T^2 C) /Z₌. 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.