On the Geometric Langlands Program
The classical Langlands correspondence manifests a deep connection between number theory and representation theory. In particular, it relates subtle number theoretic data (such as the numbers of points of a mod $p$ reduction of an elliptic curve) to more easily discernable data related to automorphic forms (such as the coefficients in the Fourier series expansion of a modular form on the upper half-plane). We will consider explicit examples of this relationship.
The geometric Langlands correspondence predicts that to each rank $n$ holomorphic vector bundle $E$ with a holomorphic connection on a complex algebraic curve $X$ one can attach an object called Hecke eigensheaf on the moduli space $Bun_n$ of rank $n$ holomorphic vector bundles on $X$.
A Hecke eigensheaf is a $D$-module on $Bun_n$ satisfying a certain property that is determined by $E$. More generally, if $G$ is a complex reductive Lie group, and $^L G$ is the Langlands dual group, then to a holomorphic $^L G$-bundle with a holomorphic connection on $X$ we should attach a Hecke eigensheaf on the moduli space $Bun_G$ of holomorphic $G$-bundles on $X$
It is possible to describe the connections between the Langlands Program and two-dimensional conformal field theory that have been found in the last few years. These connections give us important insights into the physical implications of the Langlands duality. We intend to follow the links between the classical class field theory and the Geometric Langlands correspondence.