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

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.