Supersymmetric gauge theories, quantisation of moduli spaces of flat connections, and Liouville theory

This is the 11th article in the collection of reviews “Exact results in N=2 supersymmetric gauge theories”, ed. J. Teschner. It describes an approach to understanding the 4d/2d relations discovered by Alday, Gaiotto and Tachikawa by establishing a triangle of relations between the zero mode quantum mechanics obtained by localisation of class $\mathcal{S}$ theories, the quantum theory obtained by quantisation of Hitchin moduli spaces, and conformal field theory.