4d gauge theories and classical integrable systems

Abstract:
A wide class of integrable classical systems (the so-called Hitchin systems) can be derived from a topological field theory in 2+1 dim. They include, in particular, the Calogero-Moser systems, the Toda models, integrable tops, the KdV equation, the Landau-Lifshitz equation. This approach allows to obtain immediately the Lax equations with a spectral parameter and the conservation laws. I plan to explain the general theory and then to illustrate it by two examples:
1. Elliptic Calogero-Moser systems
2. Elliptic tops
Then I establish an equivalence of these systems and explain, follow Kapustin-Witten, how this equivalence can be explain in the framework of 4-d topological theory by a presence of a monopole.