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

We study Berry connections for supersymmetric ground states of 2d $\mathcal{N}=(2,2)$ GLSMs quantised on a circle, which are generalised periodic monopoles, with the aim to provide a fruitful physical arena for mathematical constructions related to the latter. These are difference modules encoding monopole solutions due to Mochizuki, as well as an alternative algebraic description of solutions in terms of vector bundles endowed with filtrations. The simultaneous existence of these descriptions is an example of a Riemann-Hilbert correspondence. We demonstrate how these constructions arise naturally by studying the ground states as the cohomology of a one-parameter family of supercharges. Through this, we show that the two sides of this correspondence are related to two types of monopole spectral data that have a direct interpretation in terms of the physics of the GLSM: the Cherkis-Kapustin spectral variety (difference modules) as well as twistorial spectral data (vector bundles with filtrations). 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 or vortex partition functions, which are exactly calculable. Beautifully, when the GLSM flows to a nonlinear sigma model with K\"ahler target $X$, we show that the difference modules are related to deformations of the equivariant quantum cohomology of $X$, whereas the vector bundles with filtrations are related to the equivariant K-theory.