Abstract

Hitchin Systems, Mirror Symmetry, and the Geometric Langlands Conjecture

Ron Donagi
University of Pennsylvania

In this talk we will state and outline a proof of the classical limit of the Geometric Langlands Conjecture, and discuss its relation to the full "quantum" conjecture. Concretely, we show that the Hitchin integrable system for a simple complex Lie group $G$ is dual to the Hitchin system for the Langlands dual group ${^L}G$. In particular, the general fiber of the connected component $\Higgs_0$ of the Hitchin system for $G$ is an abelian variety which is dual to the corresponding fiber of the connected component of the Hitchin system for ${^L}G$. The relation of this Hitchin duality to the full GLC can be interpreted as a "classical limit" of a quantum phenomenon; but there is also the tantalizing possibility, closely related to recent ideas from physics, that the Hitchin duality, appropriately interpreted, may actually give a solution of the full GLC.
This is based on the non-abelian Hodge theory of Simpson, Mochizuki and Sabbah, along with the calculation of Koszul cohomologies - a subject I learned from Mark, oh so many years ago.