| Abstract: |
It has long been known that supersymmetric (SUSY) ground states of QFTs that flow to a sigma-model with target X are closely related to certain cohomology theories of X. A more recent observation is that 3d SUSY QFTs quantised on an elliptic curve can be endowed with Berry connections, whose structure is able to capture the equivariant elliptic cohomology of the Higgs branch. I will present the first results of an ongoing effort to relate more broadly the structure of Berry connections of SUSY QFTs to various kinds of (generalised) cohomology theories. Focusing on 2d (2,2) models, whose Berry connections are generalised periodic monopoles, I will sketch how one kind of spectral data (difference modules) is related to novel difference equations for brane amplitudes and hemisphere partition functions. These difference equations are quantisations of spectral varieties that are related to the equivariant quantum cohomology of the target. I will moreover briefly mention how another kind of spectral data (vector bundles with filtrations) is instead related to its equivariant quantum K-theory. These aspects fit into a rich web of relations between various topics in mathematics. |
