Vertex Operator Algebras and 3d N=4 gauge theories

TL;DR

This work introduces two mirror constructions of VOAs associated to special boundary conditions in 3d ${ m N}=4$ gauge theories, and posits precise links between these boundary algebras and bulk topologically twisted observables. It develops the framework of deformable $(0,4)$ boundaries, analyzes both H- and C-twists through explicit examples (including free hypermultiplets and vector multiplets, as well as $U(1)$, T[SU(2)], and T[SU(N)] theories), and tests the conjectured bulk-boundary correspondence via index calculations and BRST reductions. A central proposal is that the bulk operator algebra is recovered from the self-${ m Ext}$’s of the boundary VOA’s vacuum module, with detailed checks in several H-twist setups and a parallel program for C-twist boundaries that connects to four-dimensional GL-twisted constructions. The paper also outlines a broad landscape of future tests, including symmetry enhancements, Koszul dualities for line operators, and implications for Geometric Langlands and Symplectic Duality, establishing a versatile computational bridge between 3d gauge theories and VOA techniques.

Abstract

We introduce two mirror constructions of Vertex Operator Algebras associated to special boundary conditions in 3d N=4 gauge theories. We conjecture various relations between these boundary VOA's and properties of the (topologically twisted) bulk theories. We discuss applications to the Symplectic Duality and Geometric Langlands programs.