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

TL;DR

This paper explores the connections between gauge theory, vertex algebras, and the quantum Geometric Langlands Program, extending dualities and constructing new algebraic structures relevant to the Langlands correspondence.

Contribution

It introduces a framework linking boundary conditions in gauge theories to vertex algebras and D-modules, extending quantum Langlands dualities with new constructions and duality group analysis.

Findings

Constructed vertex algebras for all simple Lie groups.

Extended the duality group to include larger gauge theory symmetries.

Analyzed subtleties in spin and gerbe structures for specific gauge theories.

Abstract

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.