Speaker: Oleg K. Sheinman (Steklov Mathematical Institute, Moscow)
Date and time: 4.10.2023, 17:00 (GMT +03:00)
Title: Obstructions for Inverse Spectral Method and for Separation of Variables for Hitchin systems
Abstract: Inverse Spectral Method is a fundamental method with which the main progress of XX-XXIth centuries in the soliton theory is related. For the finite gap solutions, its algebraic-geometric version enables one to explicitly express such solutions via theta functions. It is also applicable to integrable systems with finitely many degrees of freedom, admitting a Lax representation. In particular, Krichever (2002) proposed a way to explicitly integrate Hitchin systems on the moduli space of all (semi-stable) holomorphic bundles. However, further investigations showed that there is a certain obstruction for Krichevers approach for the Hitchin systems on moduli spaces of G-bundles where G is a complex simple Lie group (which we refer to as the structure group of the system). This obstruction is related to the fact that dynamical poles of Baker-Akhieser functions are anavoidable in this case.
A similar phenomena can be observed for another fundamental method of the theory of integrable systems, namely for the Method of Separation of Variables. It gives theta function formulae for solutions of the GL(n) Hitchin systems but there appears an obstruction for the systems with simple structure groups. The last observation is related to the peculiarities of the inversion problem for Prim varieties.
In the talk, I shall define Hitchin systems via their Lax representation (due to I. Krichever), explain the Inverse Spectral Method for them, and give a quite simple calculation demonstrating the essence of the obstruction. Then I'll do the same for Separation of Variables.