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

TL;DR

This paper explores rank 1 complex crystallographic reflection groups and their connection to Seiberg-Witten integrable systems, providing explicit geometric descriptions and quantum spectral curves for certain supersymmetric conformal field theories.

Contribution

It extends previous work by analyzing rank one cases with specific symmetries, linking them to known SCFTs and deriving explicit elliptic fibrations and quantum spectral curves.

Findings

Explicit description of elliptic fibrations for rank 1 groups

Derivation of quantum spectral curves as Fuchsian ODEs

Connection to Seiberg-Witten systems for Minahan-Nemeshansky SCFTs

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 $T^2$ 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--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.