  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 $X$.
