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Large N limit of spectral duality between the classical XXX spin chain and the rational reduced Gaudin model

R. Potapov

TL;DR

This work addresses the problem of formulating a large $N$ (infinite-dimensional) spectral duality between the classical $\mathfrak{gl}_M$ XXX spin chain and the $\mathfrak{gl}_N$ trigonometric Gaudin model by embedding the latter into the $A_{\hbar}$-algebra framework on the noncommutative torus. It develops the $A_{\hbar}$ rational reduced Gaudin model for rank-1 variables as a field theory on $\mathbb{T}^2$ with a Moyal star product, and constructs its dual infinite-dimensional XXX spin-chain description; the duality is encoded in a determinant-type identity for the spectral power series $\Gamma_{\infty}(v,z)$ and the monodromy $\tilde{T}(v)$. The core result is a direct infinite-dimensional analogue of the finite-dimensional spectral duality: the spectral curves of the Lax and monodromy data coincide, and the brackets preserve the $r$-matrix structure, ensuring commuting families of Hamiltonians. This work extends previous rational Gaudin dualities to the XXX/trigonometric setting, providing a field-theoretic realization of spectral duality at large $N$ via the noncommutative torus, with potential connections to 1+1 integrable dynamics and gauge-theoretic frameworks.

Abstract

We study the large $N$ limit of the spectral duality between the classical $\mathfrak{gl}_M$ XXX spin chain and the $\mathfrak{gl}_N$ trigonometric Gaudin model in its rational reduced form. The infinite-dimensional limit of the Gaudin model is constructed within the framework of the noncommutative torus algebra, following the approach of Hoppe, Olshanetsky and Theisen. In this formulation, the matrix dynamical variables are promoted to fields on the torus, and the corresponding Lax equations acquire the Moyal star-product structure. The dual model is obtained as the large $N$ limit of the $\mathfrak{gl}_M$ classical XXX spin chain with $N$ sites, represented by Laurent series satisfying a quadratic $r$-matrix relation associated with the rational solution of the classical Yang--Baxter equation.

Large N limit of spectral duality between the classical XXX spin chain and the rational reduced Gaudin model

TL;DR

This work addresses the problem of formulating a large (infinite-dimensional) spectral duality between the classical XXX spin chain and the trigonometric Gaudin model by embedding the latter into the -algebra framework on the noncommutative torus. It develops the rational reduced Gaudin model for rank-1 variables as a field theory on with a Moyal star product, and constructs its dual infinite-dimensional XXX spin-chain description; the duality is encoded in a determinant-type identity for the spectral power series and the monodromy . The core result is a direct infinite-dimensional analogue of the finite-dimensional spectral duality: the spectral curves of the Lax and monodromy data coincide, and the brackets preserve the -matrix structure, ensuring commuting families of Hamiltonians. This work extends previous rational Gaudin dualities to the XXX/trigonometric setting, providing a field-theoretic realization of spectral duality at large via the noncommutative torus, with potential connections to 1+1 integrable dynamics and gauge-theoretic frameworks.

Abstract

We study the large limit of the spectral duality between the classical XXX spin chain and the trigonometric Gaudin model in its rational reduced form. The infinite-dimensional limit of the Gaudin model is constructed within the framework of the noncommutative torus algebra, following the approach of Hoppe, Olshanetsky and Theisen. In this formulation, the matrix dynamical variables are promoted to fields on the torus, and the corresponding Lax equations acquire the Moyal star-product structure. The dual model is obtained as the large limit of the classical XXX spin chain with sites, represented by Laurent series satisfying a quadratic -matrix relation associated with the rational solution of the classical Yang--Baxter equation.

Paper Structure

This paper contains 14 sections, 3 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Purpose of the paper.
  3. $A_{\hbar}$ algebra and infinite-dimensional limit of Gaudin model
  4. Algebra of noncommutative torus and its representations
  5. The Poisson-Lie structure
  6. Infinite-dimensional Gaudin model
  7. Spectral duality of finite-dimensional models
  8. $\mathfrak{gl}_{N}$ rational reduced Gaudin model
  9. $\mathfrak{gl}_{M}$ classical XXX spin chain
  10. Statement of duality
  11. Large $N$ limit of spectral duality
  12. $A_{\hbar}$ rational reduced Gaudin model
  13. Dual model
  14. Statement of duality

Key Result

Theorem 1

Theorems & Definitions (3)

  • Theorem 1
  • Lemma 4.1
  • Theorem 2