This article is cited in 4 papers

The large $N$ limits of integrable models

M. A. Olshanetsky

Institute for Theoretical and Experimental Physics (Russian Federation State Scientific Center)

Abstract: 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.

Key words and phrases: Blanchfield form, Seifert form, algebraic transversality.

MSC: 19J25, 57C45

Received: March 4, 2002

Language: English

DOI: 10.17323/1609-4514-2003-3-4-1307-1331

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