Large N limit of spectral duality in classical 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 ${\rm gl}_N$ model is performed by means of the 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 a large $N$ limit of the ${\rm gl}_M$ Gaudin model with $N$ marked points written in the form of Gaudin model with irregular singularities.