Geometric Langlands: Comparing the views from CFT and TQFT

The goal of my talk will be to discuss the relation between two approaches to the geometric Langlands program. The first has been proposed by Beilinson and Drinfeld, using ideas and methods from conformal field theory (CFT). The second was initiated by Kapustin and Witten based on a topological version of four-dimensional maximally supersymmetric Yang--Mills theory and its reduction to a two-dimensional topological sigma model. After discussing some issues complicating a direct comparison we will formulate a proposal for a precise relation between two main ingredients in the two approaches.