Quantum cohomology, shift operators, and Coulomb branches

TL;DR

This work builds a unified framework linking Braverman–Finkelberg–Nakajima Coulomb branches with (equivariant) quantum cohomology through shift operators defined without localization. By introducing a refined description of the quantized Coulomb branch as a subalgebra cut out by a stratified quotient ${\mathcal S}$ of a universal bundle, the authors define generalized Seidel and shift operators via section-counting on moduli of maps into Seidel spaces, while proving independence from resolutions and compatibility with flavour deformations. The main results show that the images of the Seidel homomorphism and the shift operator action lie in $QH_G^ullet(X)$ without localization, yielding a closed Lagrangian family in the Coulomb branch and enabling a broad non-equivariant limit analysis. These constructions recover Teleman’s gluing formula for Coulomb branches and generalize the Peterson isomorphism to general reductive groups with equivariant Novikov data, with broad applications to 3d mirror symmetry and open–closed brane correspondences. Collectively, the paper provides a versatile toolkit for computing genus-zero Gromov–Witten data via Coulomb-branch geometry and for understanding 3d/2d mirror phenomena in a representation-theoretic setting.

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.

Theorems & Definitions (147)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Remark 1: Assumption on $X$
• Remark 2: Independence of the choice of representation
• Remark 3: Relation with 2d and 3d mirror symmetry
• Definition 4: 3dmirrorfunctoriality
• Theorem 5
• Definition 1.1: BFN