Global symmetries greatly enrich the phase diagram of quantum many-body systems. As well as symmetry-breaking phases, symmetry-protected topological (SPT) phases have symmetric ground states that cannot be connected to a trivial state without a phase transition. There can also be symmetry-enriched critical points between these phases of matter. I will demonstrate these phenomena in phase diagrams constructed using the N-state chiral clock family of spin chains.  [Based on joint work with Paul Fendley and Abhishodh Prakash.]

The chiralization in the title denotes a certain procedure which turns cluster X-varieties into q-W algebras. Many important notions from cluster and q-W worlds, such as mutations, global functions, screening operators, R-matrices, etc emerge naturally in this context. In particular, we discover new bosonizations of q-W algebras and establish connections between previously known bosonizations. If time permits, I will discuss potential applications of our approach to the study of 3d topological theories and local systems with affine gauge groups. This talk is based on a joint project with J. Shiraishi, J.E. Bourgine, B. Feigin, A. Shapiro, and G. Schrader.

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.