Difference Equations: from Berry Connections to the Coulomb Branch

TL;DR

This paper explores the connection between Berry connections, difference equations, and Coulomb branch actions in supersymmetric gauge theories, providing a 3d perspective and explicit computations for abelian models.

Contribution

It introduces a 3d boundary perspective to recover spectral and difference equation results, linking Coulomb branch algebra actions to spectral data and monopoles.

Findings

Spectral variety supports sheaves from equivariant quantum cohomology

Difference equations govern brane amplitudes and vortex partition functions

Explicit computations for abelian GLSMs demonstrate the theoretical framework

Abstract

In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with K\"ahler vacuum moduli space $X$ and abelian flavour symmetry is the support of a sheaf induced by a certain action on the equivariant quantum cohomology of $X$. This action could be quantised to first-order matrix difference equations obeyed by brane amplitudes, and by taking the conformal limit, vortex partition functions. In this article, we elucidate how some of these results may be recovered from a 3d perspective, by placing the 2d theory at a boundary and gauging the flavour symmetry via a bulk A-twisted 3d $\mathcal{N}=4$ gauge theory (a sandwich construction). We interpret the above action as that of the bulk Coulomb branch algebra on boundary twisted chiral operators. This relates our work to recent constructions of actions of Coulomb branch algebras on quantum equivariant cohomology, providing a novel correspondence between these actions and spectral data of generalised periodic monopoles. The effective IR description of the 2d theory in terms of a twisted superpotential allows for explicit computations of these actions, which we demonstrate for abelian GLSMs.