Hitchin systems: how to solve them

Hitchin systems are remarkable finite-dimensional integrable systems intrinsically related to the moduli space of holomorphic $G$-bundles on a Riemann surface. There is a vast literature on algebraic, Lagrangian and differential geometry of Hitchin systems, and their generalizations like parabolic Hitchin systems, Sympsom systems. However, only a few works are devoted to a natural question: how to solve them. These are the works by J.C. Hurtubise, A. Gorski-N.Nekrasov-V. Rubtsov, K. Gawedzki, I. Krichever (all of them about 1990-2000), and several recent papers of the author. Also there is only a short list of explicitly resolved Hitchin systems, which includes systems with $G=GL(n)$ on a curve of arbitrary genus, and with $G=SL(2), SO(4)$ for genus 2. There exist basically two methods of exact solution of finite dimensional integrable systems. These are the classical method of Separation of Variables (SoV), and Inverse Spectral Method (ISM) which is a great modern achievement. Both of them apply to Hitchin systems, and both finally result in theta function solutions. However, so far the ISM is applicable only for $G=GL(n)$ while SoV is working also for simple groups. In this talk we focus on the method of Separation of Variables. It goes back to Hamilton and Jacobi, its modern form is due to Arnold and Sklyanin. Majority of classical (finite-dimensional) integrable systems had been resolved by means of SoV. As for Hitchin systems, Separation of Variables gives also a simplest way to define them. In the talk, I shall define Hitchin systems both following Hitchin'87, and by means of SoV, and prove their integrability. For the systems on hyperelliptic curves by means of methods of symplectic geometry I'll derive a fundamental fact that Hitchin trajectories are straight lines (windings) on certain Abelian varieties replacing Liouville tori in this context (namely, on Jacobians/Prymians of the corresponding spectral curves). In the case $G=GL(n)$ I'll give an explicit theta function formula for solutions, and explain how it can be generalized onto the case of a simple group $G$.