Anomalies, a mod 2 index, and dynamics of 2d adjoint QCD

We show that 2 2 d adjoint QCD, an SU (N) S U (N) gauge theory with one massless adjoint Majorana fermion, has a variety of mixed ’t Hooft anomalies. The anomalies are derived using a recent mod 2 2 index theorem and its generalization that incorporates ’t Hooft flux. Anomaly matching and dynamical considerations are used to determine the ground-state structure of the theory. The anomalies, which are present for most values of N N, are matched by spontaneous chiral symmetry breaking. We find that massless 2 2 d adjoint QCD confines for N >2 N > 2, except for test charges of N N -ality N/2 N / 2, which are deconfined. In other words, ZN ℤ N center symmetry is unbroken for odd N N and spontaneously broken to Z₍/₂ ℤ N / 2 for even N N. All of these results are confirmed by explicit calculations on small R S¹ ℝ × S 1. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when N N is even. Furthermore, for most values of N N, 2 2 d massive adjoint QCD describes a non-trivial symmetry-protected topological (SPT) phase of matter, including certain cases where the number of interacting Majorana fermions is a multiple of 8