Oleg Sheinman. “Separation of Variables for Hitchin systems“

Title: Separation of Variables for Hitchin systems 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 which is a great modern achievement. Both of them apply to Hitchin systems. 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 by means of SoV and prove their integrability. By means of the method of generating functions (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. Every Hitchin system by definition is related with a certain complex reductive group G referred to as the structure group. In the case G=GL(n) I’ll give an explicit theta function formula for solutions. Then I’ll explain that in the cases G=SO(n), G=Sp(2n) there emerges a certain obstruction for a similar theta function solution related to the peculiarities of the inversion problem for Prim varieties. If the time admits, I’ll argue that the above definition of Hitchin systems is equivalent to the conventional one.