Difference Equations: from Berry Connections to the Coulomb Branch

TL;DR

This work develops a cohesive link between Berry connection spectral data for 2d $(2,2)$ GLSMs and the action of bulk 3d $ abla$-type Coulomb-branch algebras on boundary twisted-chiral operators, via a 3d sandwich construction that gauges the 2d flavor symmetry. By introducing an Omega-deformed framework, the authors promote the Coulomb-branch action to a noncommutative algebra $igl[oldsymbol ext{C}_oldsymbol{ ext{e}}[ ext{M}_C]igr]$ and extract first-order matrix difference equations for brane amplitudes and for vortex/hemisphere partition functions, with SQED$[N]$ serving as a concrete check. The boundary module structure, arising from coupling a 2d GLSM to a bulk 3d theory, yields a natural realization of the spectral variety as the image of the boundary condition in the bulk Coulomb-branch space $ ext{M}_C=(C^ imes imesC)^n$, and recovers quantum equivariant cohomology $QH_T^ullet(X)$ as a boundary module. These results illuminate a novel correspondence between spectral data of generalized periodic monopoles and Coulomb-branch actions on equivariant quantum cohomology, and provide computational leverage through twisted superpotentials and localization techniques. The framework generalizes to matter-coupled bulk theories and hints at connections to qKZ-type equations governing hemisphere/vertex functions, offering a versatile platform for exploring 2d-3d interplays in supersymmetric geometry.

Abstract

In recent work, we demonstrated that a spectral variety for the Berry connection of a 2d $\mathcal{N}=(2,2)$ GLSM with Kähler 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.

Figures (2)

• Figure 1: The sandwich construction realising the A-twisted $\mathcal{T}_{\text{2d}}$ theory as a pure 3d $\mathcal{N}=4$ abelian gauge theory with enriched boundary condition.
• Figure 2: The sandwich construction for difference modules