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

Philip C. Argyres, Oleg Chalykh, Yongchao Lü

TL;DR

This work develops a rank-one theory of complex crystallographic reflection groups via elliptic Cherednik algebras, producing Poisson deformations of $T^*\mathcal{E}/\mathbb{Z}_m$ and a single Hamiltonian $h$ (with dual $h^\vee$) that together furnish an elliptic fibration encoding SW data for rank-one Minahan–Nemeschansky theories. By constructing elliptic Dunkl operators and a Lax matrix, the authors derive explicit spectral curves and elliptic pencils, revealing a SW differential whose residues correspond to geometric mass parameters. The quantisation yields a family of Fuchsian quantum curves with semisimple monodromy, connecting to opers, Hitchin systems on orbifolds, and de Rahm moduli spaces, and tying the rank-one cases to broader frameworks of Hitchin fibrations, quiver varieties, and 5d/4d SCFT correspondences. The results provide concrete, geometry-driven realizations of Seiberg–Witten integrability for $m=2,3,4,6$, and lay groundwork for higher-rank generalisations of MN theories. Overall, the paper links elliptic Cherednik algebras, Lax integrability, spectral geometry, and SW theory in a coherent rank-one setting with rich mathematical-physical structure.

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.

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

TL;DR

This work develops a rank-one theory of complex crystallographic reflection groups via elliptic Cherednik algebras, producing Poisson deformations of and a single Hamiltonian (with dual ) that together furnish an elliptic fibration encoding SW data for rank-one Minahan–Nemeschansky theories. By constructing elliptic Dunkl operators and a Lax matrix, the authors derive explicit spectral curves and elliptic pencils, revealing a SW differential whose residues correspond to geometric mass parameters. The quantisation yields a family of Fuchsian quantum curves with semisimple monodromy, connecting to opers, Hitchin systems on orbifolds, and de Rahm moduli spaces, and tying the rank-one cases to broader frameworks of Hitchin fibrations, quiver varieties, and 5d/4d SCFT correspondences. The results provide concrete, geometry-driven realizations of Seiberg–Witten integrability for , and lay groundwork for higher-rank generalisations of MN theories. Overall, the paper links elliptic Cherednik algebras, Lax integrability, spectral geometry, and SW theory in a coherent rank-one setting with rich mathematical-physical structure.

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 with -symmetries, , and Poisson deformations of the orbifolds . The case was studied in [2], while correspond to Seiberg--Witten integrable systems for the rank 1 Minahan--Nemeshansky SCFTs of type . 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.
Paper Structure (54 sections, 8 theorems, 240 equations, 4 figures, 2 tables)

This paper contains 54 sections, 8 theorems, 240 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Complex crystallographic groups and Cherednik algebras in rank one
  3. Elliptic curves with symmetries
  4. Rational Cherednik algebra for $W=\mathbb{Z}_m$
  5. Elliptic Cherednik algebra of rank one
  6. Case of a punctured elliptic curve
  7. Hamiltonians
  8. Explicit formulas
  9. Case $m=2$:
  10. Case $m=3$:
  11. Case $m=4$:
  12. Case $m = 6$:
  13. Rational form
  14. Case $m=2$:
  15. Case $m=3$:
  16. ...and 39 more sections

Key Result

Theorem 2.4

The hamiltonians $\widehat{h}_i$ represent global sections of the sheaf $B_{\hbar, c}(\mathbb{C}^n/\Gamma, W)$ of spherical Cherednik algebras. Furthermore, they generate the full algebra of global sections, that is, any global section of $B_{\hbar, c}(\mathbb{C}^n/\Gamma, W)$ is a polynomial in $\w

Figures (4)

  • Figure 1: Fundamental domain, group action, and fixed points.
  • Figure 2: Three and four punctured spheres corresponding to $(T^2\times\mathbb{C})/\mathbb{Z}_m$ with fixed points ("punctures") labeled by Young diagrams representing partitions of $m$.
  • Figure 3: Shown above are the five-brane webs corresponding to 5d Seiberg's theories. For all cases, the internal part of the diagram is represented by a large black circle, while only the external legs are illustrated in detail. Black dots are used to represent seven-branes and lines represent five-branes.
  • Figure 4: Affine Dynkin diagrams of type ${D}_4$ and ${E}_{6,7,8}$.

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Theorem 2.4: EFMV11ecm
  • Proposition 2.5
  • Proposition 2.6
  • Proposition 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • ...and 14 more