Boundaries, mirror symmetry, and symplectic duality in 3d N = 4 $$ \mathcal{N}=4 $$ gauge theory
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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.