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A few recent developments in 2d (2,2) and (0,2) theories

In this note we summarize a few of the many recent developments in two-dimensional quantum field theories. We begin with a review of the current state of quantum sheaf cohomology, a heterotic analogue of quantum cohomology. We then turn to dualities: we outline the current status of (0,2) mirror symmetry, and then outline recent work on two-dimensional gauge dualities. In particular, we describe how many two-dimensional gauge dualities in both (2,2) and (0,2) supersymmetric gauge theories can be understood simply as different presentations of the same infrared (IR) geometry. We then discuss (not necessarily supersymmetric) two-dimensional nonabelian gauge theories in which a subgroup of the gauge group acts trivially on massless matter. We describe how these theories 'decompose' into disjoint unions of other theories indexed by discrete theta angles, a fact which in other contexts has proven to have implications for interpretations of certain gauged linear sigma models (GLSMs) and for Gromov-Witten invariants of stacks. We conclude with a discussion of recent developments in infinitesimal moduli of heterotic compactifications.