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

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 $(T^2\times \mathbb{C})\mathrm{/}{\mathbb{Z}}_m$. The $m=2$ case was studied in [2], while $m=3,4,6$ correspond to Seiberg–Witten integrable systems for the rank 1 Minahan–Nemeshansky 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.