Large N limit of spectral duality in classical integrable systems
R. Potapov, A. Zotov
TL;DR
The paper addresses the large N limit of spectral duality for classical Gaudin models by embedding Mat_N data into the noncommutative torus A_hbar and interpreting the Lax operators as A_hbar-valued, leading to a field-theoretic description on the torus. It develops two representations (infinite matrices and deformation quantization on T^2) that are connected by a basis change, yielding a 2+1D integrable field theory in the large N limit. A central result is the extension of Adams-Harnad-Hurtubise duality to this field-theoretic, large N setting with irregular singularities, culminating in a precise identity that equates the spectral data of the dual models. This work links 2D/3D integrable hydrodynamics on T^2 with large N Gaudin limits and irregular singularities, with potential implications for gauge theory, Seiberg-Witten geometry, and related dualities.
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.