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

TL;DR

This paper explores the mathematical structures underlying 2d supersymmetric theories, connecting Berry connections, monopole spectral data, and generalized cohomology theories, revealing new difference equations and relations to quantum cohomology and K-theory.

Contribution

It introduces a novel link between spectral data of supersymmetric ground states and generalized cohomology theories, providing new difference equations for brane amplitudes and partition functions.

Findings

Derived new difference equations for brane amplitudes.

Connected spectral data to equivariant quantum cohomology and K-theory.

Revealed relations between monopole solutions and algebraic vector bundles.

Abstract

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