Four-dimensional avatars of two-dimensional RCFT

We investigate a 4D analog of 2D WZW theory. The theory turns out to have surprising finiteness properties and an infinite-dimensional current algebra symmetry. Some correlation functions are determined by this symmetry. One way to define the theory systematically proceeds by the quantization of moduli spaces of holomorphic vector bundles over algebraic surfaces. We outline how one can define vertex operators in the theories. Finally, we define four-dimensional "conformal blocks" and present an analog of the Verlinde formula.