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

We show that 2d adjoint QCD, an SU(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 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, are matched by spontaneous chiral symmetry breaking. We find that massless 2d adjoint QCD confines for N > 2, except for test charges of N-ality N/2, which are deconfined. In other words, Z_N center symmetry is unbroken for odd N and spontaneously broken to Z_N/₂ for even N. All of these results are confirmed by explicit calculations on small R × S¹. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when N is even. Furthermore, for most values of N, 2d 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. As a result, it fits into the classification of (1 + 1)d SPT phases of interacting Majorana fermions in an interesting way.