Three lectures on classical integrable systems and gauge field theories

TL;DR

This paper explores the deep connections between classical integrable systems, gauge theories, and geometric correspondences, highlighting the Symplectic Hecke transformation and its applications to notable models like the Calogero-Moser system.

Contribution

It introduces the Symplectic Hecke correspondence linking different integrable systems and applies it to classical models, expanding the understanding of their geometric and physical relations.

Findings

Established the Symplectic Hecke correspondence for elliptic Calogero-Moser and Euler-Arnold systems.

Connected integrable systems with gauge field theories and self-duality equations.

Provided geometric interpretations of integrable models within gauge theory frameworks.

Abstract

In these lectures I consider the Hitchin integrable systems and their relations with the self-duality equations and the twisted super-symmetric Yang-Mills theory in four dimension follow Hitchin and Kapustin-Witten. I define the Symplectic Hecke correspondence between different integrable systems. As an example I consider Elliptic Calogero-Moser system and integrable Euler-Arnold top on coadjoint orbits of the group GL(N,C) and explain the Symplectic Hecke correspondence for these systems.