Quantum cohomology, shift operators, and Coulomb branches

Abstract

Given a complex reductive group $G$ and a $G$-representation $\mathbf{N}$, there is an associated Coulomb branch algebra $\mathcal{A}_{G,\mathbf{N}}^\hbar$ defined by Braverman, Finkelberg and Nakajima. In this paper, we provide a new interpretation of $\mathcal{A}_{G,\mathbf{N}}^\hbar$ as the largest subspace of the equivariant Borel--Moore homology of the affine Grassmannian on which shift operators (and their deformations induced by flavour symmetries) are defined without localizations. The proofs of the main theorems involve showing that the defining equations of the Coulomb branch algebras reflect the properness of moduli spaces required for defining shift operators. As a main application, we give a very general definition of shift operators, and show that if $X$ is a smooth semiprojective variety equipped with a $G$-action, and $f \colon X \to \mathbf{N}$ is a $G$-equivariant proper holomorphic map, then the equivariant big quantum cohomology $QH^\bullet_G(X)$ defines a family of closed Lagrangians in the Coulomb branch $\mathrm{Spec}\mathcal{A}_{G,\mathbf{N}}$, yielding a transformation of 3d branes in 3d mirror symmetry. We further apply our construction to recover Teleman's gluing formula for Coulomb branches and to derive new generalizations of the Peterson isomorphism.