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.
Highlights
In this paper, we introduce and study various families of half-BPS boundary conditions in three-dimensional N = 4 gauge theories that preserve a two-dimensional N = (2, 2) super-Poincare algebra
We introduce several families of N = (2, 2) UV boundary conditions in 3d N = 4 gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches
By studying two-dimensional compactifications of 3d 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
Summary
An Omega background quantizes the algebra of Coulomb-branch line operators, and boundary conditions produce. Aganagic and Okounkov [49] study holomorphic blocks (cf [50]) of 3d N = 4 theories These are partition functions on D2 × S1, defined using a topological twist that treats Higgs and Coulomb branches symmetrically (in contrast to our Omega backgrounds). A boundary condition in 3d leads to a second brane BH under this reduction, and the space of open string states Hom(Bcc, BH ) is exactly the module that we call BH [30, 53] This 2d setup was used by [53] to construct representations of simple Lie algebras, connecting to much of the same mathematics that we study in this paper. The operators that generate the modules BH (or BC) — i.e. partition functions are solutions for the difference/differential equations that we set up in the current paper
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.