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

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 Dmodules 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.

$$ \mathcal{W} $$ -algebra modules, free fields, and Gukov-Witten defects

“…Gukov-Witten defects were originaly introduced in the context of the ramification (inclusion of punctured Riemann surfaces) of the geometric Langlands program in [33]. In the recenly introduced corner approach to the geometric Langlands program from [4,84], Gukov-Witten defects play the role of generic modules for kernel VOAs. This work sets foundations for the inclusion of the ramification in this context.…”

Mirror symmetry of 3D N=4 gauge theories and supersymmetric indices

“…Type IIB brane construction is extended to further study of half-BPS boundary conditions and interfaces for four-dimensional N = 4 Super Yang-Mills (SYM) theory [8,9,10], quarter-BPS corner configuration for four-dimensional N = 4 SYM theory [11,12], half-BPS boundary conditions for three-dimensional N = 4 gauge theory [13] and two-dimensional N = (0, 4) supersymmetric gauge theories [14]. S-duality turns out to give a physical underpinning to Geometric Langlands Program [15,11,16,17] and Symplectic Duality [18,19].…”

W-algebras as coset vertex algebras

“…Recently there has been considerable interest in connecting four-dimensional supersymmetric gauge theories, the quantum geometric Langlands program and vertex algebras. On the vertex algebra side, one is interested in certain master chiral algebras that serve as a kernel for the quantum geometric Langlands correspondence [Gai18] and at the same time as a corner vertex algebra for the junction of topological Dirichlet boundary conditions in gauge theory [CG17,FG18]. Roughly speaking, physics predicts the existence of vertex algebra extensions of tensor products of vertex algebras associated to g, the Langlands dual L g of g, and the coupling Ψ.…”

“…These vertex algebra extensions are then expected to imply equivalences of involved vertex tensor categories and also spaces of conformal blocks (twisted D-modules). We refer to [FG18,Gai18] for recent progress in this direction.…”