Notes on nonabelian (0,2) theories and dualities

In this paper we explore basic aspects of nonabelian (0,2) GLSM's in two dimensions for unitary gauge groups, an arena that until recently has largely been unexplored. We begin by discussing general aspects of (0,2) theories, including checks of dynamical supersymmetry breaking, spectators and weak coupling limits, and also build some toy models of (0,2) theories for bundles on Grassmannians, which gives us an opportunity to relate physical anomalies and trace conditions to mathematical properties. We apply these ideas to study (0,2) theories on Pfaffians, applying recent perturbative constructions of Pfaffians of Jockers et al. We discuss how existing dualities in (2,2) nonabelian gauge theories have a simple mathematical understanding, and make predictions for additional dualities in (2,2) and (0,2) gauge theories. Finally, we outline how duality works in open strings in unitary gauge theories, and also describe why, in general terms, we expect analogous dualities in (0,2) theories to be comparatively rare.