The large $N$ limits of integrable models

We consider the large $N$ limits of Hitchin-type integrable systems. The first system is the elliptic rotator on ${\rm GL}_N$ that corresponds to the Higgs bundle of degree 1 over an elliptic curve with a marked point. This system is gauge equivalent to the $N$-body elliptic Calogero–Moser system, which is obtained from the Higgs bundle of degree zero over the same curve. The large $N$ limit of the former system is the integrable rotator on the group of the non-commutative torus. Its classical limit leads to an integrable modification of 2D hydrodynamics on the two-dimensional torus. We also consider the elliptic Calogero–Moser system on the group of the non-commutative torus and consider the systems that arise after the reduction to the loop group.