Abstract

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 \( \mathrm{Higgs}_0 \) 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.