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 ${\mathcal{A}}_{G,\mathbf{N}}^{\hslash }$ defined by Braverman, Finkelberg and Nakajima. In this paper, we prove a new characterization of ${\mathcal{A}}_{G,\mathbf{N}}^{\hslash }$ 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 regular, meaning that they are defined without localizations. The proofs involve showing that the defining equations of the Coulomb branch algebras precisely reflect properness of the moduli spaces used to define shift operators. As a main application, we show that if $X$ is a smooth semiprojective variety equipped with a $G$-action, and $f:X\to \mathbf{N}$ is a $G$-equivariant proper holomorphic map, then the equivariant big quantum cohomology $Q{H}_G^{\bullet }(X)$ 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. Regularity of shift operators also gives way to highly efficient computations in equivariant Gromov--Witten theory; in particular, we obtain a very short proof of Peterson isomorphism.

54 pages. Version 4: Rewrote the introduction to better emphasize the characterization of Coulomb branches in terms of properties of shift operators. Added section 6.5. We also gave a simpler proof of Peterson isomorphism and removed appendix C