BFN Springer Theory

Given a representation N of a reductive group G, Braverman–Finkelberg–Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from 3d gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a convolution algebra, using a vector bundle over the affine Grassmannian of G. This vector bundle over the affine Grassmannian maps to the space of loops in the representation N. We study the fibres of this maps, which live in the affine Grassmannian. We use these BFN Springer fibres to construct modules for (quantized) Coulomb branch algebras. These modules naturally correspond to boundary conditions for the corresponding gauge theory. We use our construction to partially prove a conjecture of Baumann–Kamnitzer–Knutson and give evidence for conjectures of Hikita, Nakajima, and Kamnitzer–McBreen–Proudfoot. We also prove a relation between BFN Springer fibres and quasimap spaces.