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 ndimensional
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 socalled Hecke eigensheaves on the moduli spaces of Gbundles 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 Gbundles on the
curve X to homomorphisms from the fundamental group of X to the
socalled 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 nonabelian Fourier transforms.
Moreover, the Langlands dual group that is
essential in the formulation of the Langlands correspondence also plays
a prominent role in the Sdualities 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.
