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

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.

222 pages, 36 figures; v2: reference added; v3: JHEP version

References in corpus (14)

• Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory
• Towards a mathematical definition of Coulomb branches of $3$-dimensional $\mathcal{N}=4$ gauge theories, II
• Phases Of N=2 Theories In 1+1 Dimensions With Boundary
• Supersymmetric gauge theory and the Yangian
• Deformation quantization and superconformal symmetry in three dimensions
• Ω-deformation and quantization
• AGT on the S-duality Wall
• A-branes and Noncommutative Geometry
• Duality walls and defects in 5d N=1 theories
• The Coulomb Branch of 3d $\mathcal{N}=4$ Theories
• A survey of hypertoric geometry and topology
• Algebra of the Infrared: String Field Theoretic Structures in Massive $\mathcal{N}=(2,2)$ Field Theory In Two Dimensions
• Sequencing BPS Spectra
• Gauge theory and mirror symmetry

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