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$QQ$-systems and tropical geometry

Rahul Singh, Anton M. Zeitlin

TL;DR

This work develops tropical-geometry methods to study $QQ$-systems and their $qq$-system limits, linking Bethe Ansatz equations of the XXZ spin chain and Gaudin models to the geometry of Nakajima quiver varieties and their quantum K-theory. It proves that isolated solutions of the infinite systems can be locally deformed to nearby finite systems under small Cartan twists, with Puiseux-series dependence under generic conditions, and provides explicit algorithmic construction in the $\mathfrak{sl}_2$ case. The authors deploy tropical bases, initial forms, and the Fundamental Theorem of Tropical Algebraic Geometry to establish finiteness and analyticity of deformations, enabling practical computation of Bethe roots via $Q$-functions. The results create a bridge between representation theory, geometric representation theory, and integrable systems, offering concrete procedures for obtaining eigenvalues of quantum tautological operators and for solving related Bethe equations in core models like XXZ and Gaudin.

Abstract

We investigate the system of polynomial equations, known as $QQ$-systems, which are closely related to the so-called Bethe ansatz equations of the XXZ spin chain, using the methods of tropical geometry.

$QQ$-systems and tropical geometry

TL;DR

This work develops tropical-geometry methods to study -systems and their -system limits, linking Bethe Ansatz equations of the XXZ spin chain and Gaudin models to the geometry of Nakajima quiver varieties and their quantum K-theory. It proves that isolated solutions of the infinite systems can be locally deformed to nearby finite systems under small Cartan twists, with Puiseux-series dependence under generic conditions, and provides explicit algorithmic construction in the case. The authors deploy tropical bases, initial forms, and the Fundamental Theorem of Tropical Algebraic Geometry to establish finiteness and analyticity of deformations, enabling practical computation of Bethe roots via -functions. The results create a bridge between representation theory, geometric representation theory, and integrable systems, offering concrete procedures for obtaining eigenvalues of quantum tautological operators and for solving related Bethe equations in core models like XXZ and Gaudin.

Abstract

We investigate the system of polynomial equations, known as -systems, which are closely related to the so-called Bethe ansatz equations of the XXZ spin chain, using the methods of tropical geometry.

Paper Structure

This paper contains 19 sections, 15 theorems, 51 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Acknowledgements
  3. Preliminaries
  4. $qq$-systems
  5. $QQ$-systems
  6. Puiseux series
  7. Motivation: Quiver Varieties, QQ-Systems, and Integrable Models
  8. Quiver Varieties and Infinite $QQ$-Systems
  9. Motivation: $QQ$-Systems, Quantum K-Theory, and Integrable Models
  10. Main results
  11. Deformation of solutions of infinite $qq$-systems
  12. Deformation of solutions of infinite $QQ$-systems
  13. Tropical geometry
  14. qq-systems
  15. Proofs
  16. ...and 4 more sections

Key Result

Theorem 4.1

Let $\{q_{\pm}^{i}(z)\}_{i=1}^{r}$ be an isolated solution of an infinite $qq$-system infiniteqqsystem. Then there exist $\epsilon>0$ and polynomials $\{q_{\pm}^{i,Z^{H}}(z)\}_{i=1}^{r}$ that lift $\{q_{\pm}^{i}(z)\}_{i=1}^{r}$ and solve the system qqsystem for all $Z^{H}\in\mathfrak{h}$ satisfying

Theorems & Definitions (54)

  • Definition 1
  • Definition 2
  • Remark 2.1
  • Example 1
  • Definition 3
  • Definition 4
  • Remark 2.2
  • Example 2
  • Definition 5
  • Theorem 4.1
  • ...and 44 more