2d Mirrors in nonabelian 3d Mirror Symmetry

Abstract

We establish a connection between (nonabelian) equivariant 2d mirror symmetry and the geometry of Coulomb branches. In the context of 3d mirror symmetry, a Hamiltonian $G$-manifold $Y$ is expected to determine a complex Lagrangian subvariety ${\mathbb{L}}_Y^G$ of the Coulomb branch. Using transverse Hilbert schemes and nil-Hecke algebras, we develop an algebro-geometric framework for studying Coulomb branches and their Lagrangian subvarieties and formulate criteria for the existence of ${\mathbb{L}}_Y^G$ in terms of equivariant 2d mirror symmetry. We then reinterpret these criteria in terms of Lagrangian displaceability and prove the resulting statements using Lagrangian Floer theory.

27 pages. Substantially revised and expanded version: added new results and theorems, expanded the exposition and proofs