Quantum Langlands dualities of boundary conditions, $D$-modules, and conformal blocks

We review and extend the vertex algebra framework linking gauge theory constructions and a quantum deformation of the Geometric Langlands Program. The relevant vertex algebras are associated to junctions of two boundary conditions in a 4d gauge theory and can be constructed from the basic ones by following certain standard procedures. Conformal blocks of modules over these vertex algebras give rise to twisted D-modules on the moduli stacks of G-bundles on Riemann surfaces which have applications to the Langlands Program. In particular, we construct a series of vertex algebras for every simple Lie group G which we expect to yield D-module kernels of various quantum Geometric Langlands dualities. We pay particular attention to the full duality group of gauge theory, which enables us to extend the standard qGL duality to a larger duality groupoid. We also discuss various subtleties related to the spin and gerbe structures and present a detailed analysis for the U(1) and SU(2) gauge theories.