On Spectral Data for $(2,2)$ Berry Connections, Difference Equations & Equivariant Quantum Cohomology

TL;DR

This paper explores the connection between supersymmetric Berry connections in 2d GLSMs, difference modules, and quantum cohomology, revealing new difference equations for brane amplitudes and partition functions.

Contribution

It introduces a novel link between difference modules, monopole spectral data, and GLSM physics, providing new difference equations for brane amplitudes and hemisphere partition functions.

Findings

Derived new difference equations for brane amplitudes.

Connected difference modules to equivariant quantum cohomology.

Showed how these equations simplify in the conformal limit.

Abstract

We study supersymmetric Berry connections of 2d $\mathcal{N}=(2,2)$ gauged linear sigma models (GLSMs) quantized on a circle, which are periodic monopoles, with the aim to provide a fruitful physical arena for recent mathematical constructions related to the latter. These are difference modules encoding monopole solutions via a Hitchin-Kobayashi correspondence established by Mochizuki. We demonstrate how the difference modules arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. In particular, we show how they are related to one kind of monopole spectral data, a quantization of the Cherkis-Kapustin spectral curve, and relate them to the physics of the GLSM. By considering states generated by D-branes and leveraging the difference modules, we derive novel difference equations for brane amplitudes. We then show that in the conformal limit, these degenerate into novel difference equations for hemisphere partition functions, which are exactly calculable. When the GLSM flows to a nonlinear sigma model with K\"ahler target $X$, we show that the difference modules are related to the equivariant quantum cohomology of $X$.