Berry Connections for 2d (2,2) Theories, Monopole Spectral Data (generalised) Cohomology Theories

We study Berry connections for supersymmetric ground states of 2d𝒩=(2,2) GLSMs quantised on a circle, which are generalised periodicmonopoles. Periodic monopole solutions may be encoded into difference modules,as shown by Mochizuki, or into an alternative algebraic construction given interms of vector bundles endowed with filtrations. By studying the ground statesin terms of a one-parameter family of supercharges, we relate these twodifferent kinds of spectral data to the physics of the GLSMs. From thedifference modules we derive novel difference equations for brane amplitudes,which in the conformal limit yield novel difference equations for hemisphere orvortex partition functions. When the GLSM flows to a nonlinear sigma model withKähler target X, we show that the two kinds of spectral data are related todifferent (generalised) cohomology theories: the difference modules are relatedto the equivariant quantum cohomology of X, whereas the vector bundles withfiltrations are related to its equivariant K-theory.