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We study Berry connections for supersymmetric ground states of 2d$\mathcal{N}=(2,2)$ GLSMs quantised on a circle, which are generalised periodicmonopoles, with the aim to provide a fruitful physical arena for mathematicalconstructions related to the latter. These are difference modules encodingmonopole solutions due to Mochizuki, as well as an alternative algebraicdescription of solutions in terms of vector bundles endowed with filtrations.The simultaneous existence of these descriptions is an example of aRiemann-Hilbert correspondence. We demonstrate how these constructions arisenaturally by studying the ground states as the cohomology of a one-parameterfamily of supercharges. Through this, we show that the two sides of thiscorrespondence are related to two types of monopole spectral data that have adirect interpretation in terms of the physics of the GLSM: the Cherkis-Kapustinspectral variety (difference modules) as well as twistorial spectral data(vector bundles with filtrations). By considering states generated by D-branesand leveraging the difference modules, we derive novel difference equations forbrane amplitudes. We then show that in the conformal limit, these degenerateinto novel difference equations for hemisphere or vortex partition functions,which are exactly calculable. Beautifully, when the GLSM flows to a nonlinearsigma model with K\"ahler target $X$, we show that the difference modules arerelated to deformations of the equivariant quantum cohomology of $X$, whereasthe vector bundles with filtrations are related to the equivariant K-theory.

Berry Connections for 2d $(2,2)$ Theories, Monopole Spectral Data & (Generalised) Cohomology Theories