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

TL;DR

This work provides a physical framework for symplectic duality by linking 3d $ ext{N}=4$ boundary conditions to holomorphic Lagrangian branes on Higgs and Coulomb branches, further quantized by Omega-deformation. It introduces three boundary-condition families (Neumann, Generic Dirichlet, Exceptional Dirichlet) and analyzes their IR images and corresponding modules for the quantized Higgs/Coulomb algebras, with detailed abelian-theory realizations via hypertoric geometry. A 2d compactification perspective connects these boundary data to categories $oldsymbol{ ext{O}}_H$ and $oldsymbol{ ext{O}}_C$ and exhibits a physical path to Koszul duality through wall-crossing of a B-type twist of a 2d theory $ ext{T}_{2d}$, predicting precise correspondences between simple, Verma, tilting, and projective modules, including generalized Whittaker-type objects. The paper further develops thimbles, boundary monopole operators, and enriched boundary conditions, and applies these ideas to Abelian theories to spell out explicit mirror-symmetric correspondences via hyperplane arrangements and Gale duality. Overall, it provides concrete physical avatars for the algebraic structures arising in symplectic duality and frames a roadmap for deriving Koszul-type equivalences from 3d to 2d reductions and wall-crossing.

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.

Figures (36)

• Figure 1: Bulk operators in the Omega-background acting on boundary operators, which define a module for the bulk algebra $\hat{{\mathbb C}}[\mathcal{M}_H]$. Here we have $\mathcal{O}_2^{\rm bulk}\mathcal{O}_1^{\rm bulk}|\mathcal{O}^{\rm bdy}\rangle$.
• Figure 2: The flow of a UV boundary condition to holomorphic Lagrangians in Coulomb- and Higgs-branch sigma-models, and its quantizations in the presence of Omega backgrounds.
• Figure 3: Top: A picture of derived category $\mathcal{O}$, with six distinguished collections of modules, and various functors represented as isometries. Koszul duality is the vertical reflection '$!$'. Bottom: chambers and generalized exceptional collections in the category of boundary conditions for the B-type twist of $\mathcal{T}_{2d}$ for real $Z_\nu(m,t)$. Koszul duality is wall crossing from $\mathrm{Im}\, \widetilde{m}_\rho < 0$ to $\mathrm{Im}\, \widetilde{m}_\rho > 0$.
• Figure 4: Top: the Higgs-branch images of Neumann boundary conditions $\mathcal{N}_\varepsilon$ for SQED with $N=2$ hypermultiplets with $t_{\mathbb R}>0$. Bottom: the corresponding $\mathfrak{sl}_2$ modules that these boundary conditions define in the $\widetilde{\Omega}$-background, for $t_{\mathbb C}/\epsilon = k$ a positive integer (here $k=4$).
• Figure 5: Applying gradient flow to deform $\mathcal{N}_L^{(C)}$ into a holomorphic Lagrangian that is invariant under $U(1)_t$ and supported on the upward-flow cycles $\mathcal{M}_C^>[t_{\mathbb R}]$. The intersection with downward-flow cycles $\mathcal{M}_C^<[t_{\mathbb R}]$ is preserved, and slides toward the vacuum locus $\mathcal{M}_C^0[t_{\mathbb R}]$.