Given a representation of a reductive group, Braverman--Finkelberg--Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from supersymmetric gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a kind of cohomological Hall algebra.

We develop a theory of Springer fibres related to Braverman--Finkelberg--Nakajima's construction. We use these Springer fibres to construct modules for (quantized) Coulomb branch algebras. In doing so, we 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.