Quantum cohomology, shift operators, and Coulomb branches

Abstract: Given a complex reductive group $G$ and a $G$-representation $\mathbf{N}$, there is an associated quantized Coulomb branch algebra $\mathcal{A}{G,\mathbf{N}}^\hbar$ defined by Braverman, Finkelberg and Nakajima. In this paper, we give a new interpretation of $\mathcal{A}{G,\mathbf{N}}^\hbar$ as the largest subalgebra of the equivariant Borel--Moore homology of the affine Grassmannian on which shift operators can naturally be defined. As a main application, we show that if $X$ is a smooth semiprojective variety equipped with a $G$-action, and $X \to \mathbf{N}$ is a $G$-equivariant proper holomorphic map, then the equivariant big quantum cohomology $QH_G^\bullet(X)$ defines a quasi-coherent sheaf of algebras on the Coulomb branch with coisotropic support. Upon specializing the Novikov and bulk parameters, this sheaf becomes coherent with Lagrangian support. We also apply our construction to recover Teleman's gluing construction for Coulomb branches and derive different generalizations of the Peterson isomorphism.