Quantisation of Hitchin's moduli spaces - relations to conformal and topological field theory

The aim of this talk will be to explain two points of view on the quantisation of Hitchin's moduli spaces, and how they are related. There is one point of view directly relating the quantisation of Hitchin's moduli spaces to conformal field theory. Another point of view was proposed in work of Kapustin/Witten, Gukov/Witten, and Nekrasov/Witten. It describes the quantisation of Hitchin's moduli spaces using a two-dimensional topological sigma model which has Hitchin's moduli space as target space. Recent work of Frenkel, Gukov and myself offers hints on how these two points of view are related.