Gauge Theory, Geometric Langlands and Vertex Operator Algebras

The mathematical definition of Coulomb branches of 3d N=4 gauge theories gives ring objects in the equivariant derived Satake category. We have another fundamental example of a ring object, namely the regular sheaf. It corresponds to the 3d N=4 QFT T[G], studied by Gaiotto-Witten. We also have operations on ring objects, corresponding to products, restrictions, Coulomb/Higgs gauging in the `category' of 3d N=4 QFT's. Thus we conjecture that arbitrary 3d N=4 QFT with G-symmetry gives rise a ring object in the derived Satake for G.

In this talk, I plan to review the global Langlands correspondence in the de Rham setting. The focus will be on the `big picture': the formulation of the correspondence, its expected properties, and possible approaches towards its proof.