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arXiv · 2309.12760

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

Philip C. Argyres·Oleg Chalykh·Yongchao Lü
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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.

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Philip C. Argyres, Oleg Chalykh, Yongchao Lü. 2023-09-22. Complex crystallographic reflection groups and Seiberg-Witten integrable systems: rank 1 case. https://doi.org/10.1007/jhep12(2025)084

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