Large N limit of integrable models
Abstract
We consider a large
N
limit of the Hitchin type integrable systems.
The first system is the elliptic rotator on
G
L
N
that corresponds to the Higgs bundle of degree one over an elliptic curve with a marked point. This system is gauge equivalent to the
N
-body elliptic Calogero-Moser system, that is derived 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 the 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.