We study supersymmetric Berry connections of 2d 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 arises 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 deformation 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 Kähler target X, we show that the difference modules are related to deformations of the equivariant quantum cohomology of X.

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The simplicial volume of a simplicial complex is a topological invariant
related to the growth of the fundamental group, which gives rise to a
semi-norm in homology. In this talk, we introduce the volume entropy
semi-norm, which is also related to the growth of the fundamental group
of simplicial complexes and shares functorial properties with the
simplicial volume. Answering a question of Gromov, we prove that the
volume entropy semi-norm is equivalent to the simplicial volume
semi-norm in every dimension. Joint work with I. Babenko.
 

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applied mathematics.