Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory

TL;DR

This paper explores boundary conditions in 3d $ ext{N}=4$ gauge theories, revealing their connections to mirror symmetry, symplectic duality, and Koszul duality through algebraic and geometric analysis.

Contribution

It introduces new boundary conditions, constructs explicit models for abelian theories, and proposes a physical origin for symplectic duality linking physics and mathematics.

Findings

Explicit boundary conditions for 3d $ ext{N}=4$ theories

Construction of mirror symmetry interfaces

Proposal of symplectic duality as a physical phenomenon

Abstract

We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d $\mathcal{N}=4$ gauge theories and their boundary conditions, we propose a physical origin for symplectic duality - an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.