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The Zoo of Opers and Dualities

Peter Koroteev, Anton M. Zeitlin

TL;DR

This work develops a unified oper-theoretic framework for spaces of $SL(r{+}1)$-opers and their deformations, revealing a rich network of quantum/classical dualities that connect quantum spin chains to classical many-body systems. It constructs and analyzes $Z$-twisted $(SL(r{+}1),q)$-opers and $(SL(r{+}1),\epsilon)$-opers with regular singularities, and shows how Miura reductions, the $QQ$/$qQ$-systems, and the Bethe Ansatz encode both quantum spectral data and classical Lax structures. The authors establish precise isomorphisms between Bethe/theory spaces and the phase spaces of $tRS$/$tCM$ models, via the intersection of Lagrangian varieties, and extend these correspondences through 3d mirror symmetry to a web including $rRS$/$rCM$ and elliptic extensions. The results unify aspects of geometric representation theory, quantum cohomology/K-theory, and classical integrable systems, providing concrete Lax matrices, Hamiltonians, and duality maps with potential applications to enumerative geometry and gauge theories.

Abstract

We investigate various spaces of $SL(r+1)$-opers and their deformations. For each type of such opers, we study the quantum/classical duality, which relates quantum integrable spin chains with classical solvable many body systems. In this context, quantum/classical dualities serve as an interplay between two different coordinate systems on the space of opers. We also establish correspondences between the underlying oper spaces, which recently had multiple incarnations in symplectic duality and bispectral duality.

The Zoo of Opers and Dualities

TL;DR

This work develops a unified oper-theoretic framework for spaces of -opers and their deformations, revealing a rich network of quantum/classical dualities that connect quantum spin chains to classical many-body systems. It constructs and analyzes -twisted -opers and -opers with regular singularities, and shows how Miura reductions, the /-systems, and the Bethe Ansatz encode both quantum spectral data and classical Lax structures. The authors establish precise isomorphisms between Bethe/theory spaces and the phase spaces of / models, via the intersection of Lagrangian varieties, and extend these correspondences through 3d mirror symmetry to a web including / and elliptic extensions. The results unify aspects of geometric representation theory, quantum cohomology/K-theory, and classical integrable systems, providing concrete Lax matrices, Hamiltonians, and duality maps with potential applications to enumerative geometry and gauge theories.

Abstract

We investigate various spaces of -opers and their deformations. For each type of such opers, we study the quantum/classical duality, which relates quantum integrable spin chains with classical solvable many body systems. In this context, quantum/classical dualities serve as an interplay between two different coordinate systems on the space of opers. We also establish correspondences between the underlying oper spaces, which recently had multiple incarnations in symplectic duality and bispectral duality.
Paper Structure (23 sections, 36 theorems, 129 equations, 1 figure)

This paper contains 23 sections, 36 theorems, 129 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Integrable systems and enumerative geometry
  3. Twisted $(^LG,q)$, $(^LG,\epsilon)$-opers and dualities
  4. Zoo of Opers
  5. Elliptic Integrable Systems
  6. Structure of the Paper
  7. Acknowledgements
  8. $Z$-twisted $(SL(r+1),q)$-opers and $(SL(r+1),\epsilon)$-opers with regular singularities
  9. Miura $(SL(r+1),q)$-opers and $(SL(r+1),\epsilon)$-opers
  10. $(SL(r+1),q)$-opers, the $QQ$-system and Bethe Ansatz
  11. The tRS model and $(SL(N),q)$-opers
  12. Quantum/Classical Duality
  13. $(SL(r+1),\epsilon)$-opers and tCM Model
  14. $SL(r+1)$-Opers with Regular Singularities: Rational and Trigonometric Z-Twists
  15. Rationally $Z$-twisted Miura $SL(r+1)$-opers and $qq$-systems.
  16. ...and 8 more sections

Key Result

Proposition 2.5

Let $S_{r+1}$ be the symmetric group of $r+1$ elements. There are exactly $(r+1)!$ Miura opers for a given $Z$-twisted $(SL(r+1),q)$-oper if $Z$ is regular semisimple.

Figures (1)

  • Figure 1: The network of dualities between various types of opers and related integrable systems. Short vertical lines are the quantum/classical dualities, diagonal arrows show the double scaling limits between the models, while dashed lines designate the action of symplectic/bispectral dualities. The momenta $p$ and coordinates $x$ of the many body systems may take values in $\mathbb{C}^\times$ or $\mathbb{C}$ which is displayed in the figure.

Theorems & Definitions (60)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6: KSZkoroteev_zeitlin_2023
  • Theorem 2.7: Frenkel:2020
  • Theorem 2.8
  • Definition 2.9
  • Proposition 2.10
  • ...and 50 more