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

Understand

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.