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Quantum Geometry, Integrability, and Opers

Peter Koroteev

TL;DR

This work surveys a unifying framework that connects Calogero-Ruijsenaars-type integrable systems with opers, DAHAs, and enumerative geometry through quantum/classical dualities and 3d mirror symmetry. It establishes and utilizes a network of correspondences: oper spaces encode classical phase spaces, QQ-systems map to XXZ Bethe roots and Miura opers, and vertex functions in quantum K-theory coincide with eigenfunctions of tRS-type Hamiltonians, tying spectral data to Laumon spaces and affine Laumon invariants. The review also outlines the DELL, eRS, and CM families and demonstrates predictions for dualities and spectra, while highlighting important open problems, such as oper models for elliptic affine systems and the construction of double-elliptic DAHA. Overall, the article builds a bridge between integrable systems, representation theory, and enumerative geometry, enabling the computation of spectra and geometric invariants from oper-theoretic and quantum K-theoretic data.

Abstract

This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of opers, representations of double affine Hecke algebras, enumerative counts to quiver varieties, to name just a few. We formulate several conjectures and open problems. This is a contribution to the proceedings of the conference on Elliptic Integrable Systems and Representation Theory, which was held in August 2023 at University of Tokyo.

Quantum Geometry, Integrability, and Opers

TL;DR

This work surveys a unifying framework that connects Calogero-Ruijsenaars-type integrable systems with opers, DAHAs, and enumerative geometry through quantum/classical dualities and 3d mirror symmetry. It establishes and utilizes a network of correspondences: oper spaces encode classical phase spaces, QQ-systems map to XXZ Bethe roots and Miura opers, and vertex functions in quantum K-theory coincide with eigenfunctions of tRS-type Hamiltonians, tying spectral data to Laumon spaces and affine Laumon invariants. The review also outlines the DELL, eRS, and CM families and demonstrates predictions for dualities and spectra, while highlighting important open problems, such as oper models for elliptic affine systems and the construction of double-elliptic DAHA. Overall, the article builds a bridge between integrable systems, representation theory, and enumerative geometry, enabling the computation of spectra and geometric invariants from oper-theoretic and quantum K-theoretic data.

Abstract

This review article discusses recent progress in understanding of various families of integrable models in terms of algebraic geometry, representation theory, and physics. In particular, we address the connections between soluble many-body systems of Calogero-Ruijsenaars type, quantum spin chains, spaces of opers, representations of double affine Hecke algebras, enumerative counts to quiver varieties, to name just a few. We formulate several conjectures and open problems. This is a contribution to the proceedings of the conference on Elliptic Integrable Systems and Representation Theory, which was held in August 2023 at University of Tokyo.
Paper Structure (15 sections, 17 theorems, 48 equations, 2 figures)

This paper contains 15 sections, 17 theorems, 48 equations, 2 figures.

Table of Contents

  1. Introduction
  2. $(G,q)$-Opers, tRS model, and XXZ Chain
  3. The tRS model and $(SL(N),q)$-opers
  4. Quantum/Classical Duality -- Magnetic Frame
  5. Quantum/Classical Duality -- Electric Frame
  6. 3d Mirror Symmetry
  7. Limits and Degenerations
  8. Elliptic Opers
  9. Enumerative Geometry and Quantum Many-Body Systems
  10. Truncations to Macdonald Polynomials
  11. The DELL System
  12. The Elliptic RS Model
  13. The Calogero-Moser Systems
  14. Spaces of Opers for Elliptic Many-Body Systems
  15. DAHAs

Key Result

Theorem 2.5

Polynomials $\{\mathcal{V}_k(z)\}_{k=1,\dots, r}$ provide the solution to the $QQ$-system so that $Q^+_j(z)=\mathcal{V}_j(z)$. The polynomials $Q^+_j, Q^-_j$ for $j=1,\dots,r$ can be presented using minors where $F_i(z)=W_{r-i}(q^{i-r}z)$, are the quantum Wronskian and the Vandermonde matrix respectively.

Figures (2)

  • Figure 1: Notations for the diamond in Fig. \ref{['fig:diamond2023']}
  • Figure 2: The Diamond

Theorems & Definitions (24)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: KSZ
  • Theorem 2.6
  • Theorem 2.7: KSZ
  • Corollary 2.8
  • Theorem 2.9: Koroteev:2023aa
  • Proposition 2.10: Koroteev:2022vg
  • ...and 14 more