Phases of adjoint QCD₃ and dualities

We study 2+1 dimensional gauge theories with a Chern-Simons term and a fermion in the adjoint representation. We apply general considerations of symmetries, anomalies, and renormalization group flows to determine the possible phases of the theory as a function of the gauge group, the Chern-Simons level k, and the fermion mass. We propose an inherently quantum mechanical phase of adjoint QCD with small enough k, where the infrared is described by a certain Topological Quantum Field Theory (TQFT). For a special choice of the mass, the theory has N = 1 supersymmetry. There this TQFT is accompanied by a massless Majorana fermion – a Goldstino signaling spontaneous supersymmetry breaking. Our analysis leads us to conjecture a number of new infrared fermion-fermion dualities involving SU, SO, and Sp gauge theories. It also leads us to suggest a phase diagram of SO/Sp gauge theories with a fermion in the traceless symmetric/antisymmetric tensor representation of the gauge group.