Geometric Langlands Program

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This program is dedicated to the investigation of the geometric Langlands, its relationship to other areas of mathematics, and its relationship to physics.

The Langlands Program, launched by Robert Langlands in the late 60's, ties together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic functions.  The Langlands conjecture predicts that there is a correspondence between n-dimensional representations of the Galois group of a number field and automorphic representations of the group GL(n) over the ring of adeles of this field.  This conjecture has an analogue when the number field is replaced by the field of functions on a smooth projective curve defined over a finite field.  In this setting, this conjecture has a geometric version, called the geometric Langlands correspondence.  The role of the Galois group is now played by the fundamental group of the curve X, and the role of the automorphic representations is played by the so-called Hecke eigensheaves on the moduli spaces of G-bundles on X.  This reformulation allows one to consider curves defined over arbitrary fields, such as the field of complex numbers.

There is a further generalization of this conjecture where the group GL(n) is replaced by a reductive algebraic group G.  It relates Hecke eigensheaves on the moduli space of G-bundles on the curve X to homomorphisms from the fundamental group of X to the so-called Langlands dual group.  Recently, a lot of progress has been made in constructing the geometric Langlands correspondence, particularly, in the works of A. Beilinson and V. Drinfeld.  Some of the constructions use methods from physics, more specifically, conformal field theory.

It has long been suspected that the Langlands correspondence is somehow related to various dualities observed in quantum field theory and string theory. Both the Langlands correspondence and the dualities in physics have emerged as some sort of non-abelian Fourier transforms.  Moreover, the Langlands dual group that is essential in the formulation of the Langlands correspondence also plays a prominent role in the S-dualities that are ubiquitous in physics (and was in fact rediscovered by physicists P. Goddard, J. Nuyts and D. Olive).  One of the goals of this project is to search for the underlying reason for these duality patterns in mathematics and physics and to develop a suitable language of representation theory and geometry to understand these patterns from a unifying point of view.

This program is partially supported by DARPA and the NSF.