Large N limit of spectral duality in classical integrable systems

TL;DR

This paper explores the large N limit of spectral duality in classical integrable systems, connecting rational Gaudin models to 2D hydrodynamics on a torus through noncommutative geometry.

Contribution

It introduces a novel approach to analyze the large N limit of spectral duality using noncommutative torus algebra and describes the resulting integrable field theory.

Findings

Large N limit leads to a 2D hydrodynamics model on a torus.

Duality relates the large N limit of gl_N to a model with irregular singularities.

Application of noncommutative geometry techniques to integrable systems.

Abstract

We describe the large $N$ limit of spectral duality between rational Gaudin models introduced by Adams, Harnad and Hurtubise. The limit of the ${\rm gl}_N$ model is performed by means of a noncommutative torus algebra represented by the fields on a torus with the Moyal-Weyl star product. We apply the approach developed by Hoppe, Olshanetsky and Theisen to the Gaudin-type models and describe the corresponding integrable field theory (2d hydrodynamics) on a torus. The dual model is the large $N$ limit of the ${\rm gl}_M$ Gaudin model with $N$ marked points written in the form of the Gaudin model with irregular singularities.