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

We show that 22d adjoint QCD, an SU(N)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 22 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 NN, are matched by spontaneous chiral symmetry breaking. We find that massless 22d adjoint QCD confines for N >2N>2, except for test charges of NN-ality N/2N/2, which are deconfined. In other words, \mathbb Z_NℤN center symmetry is unbroken for odd NN and spontaneously broken to \mathbb Z_{N/2}ℤN/2 for even NN. All of these results are confirmed by explicit calculations on small \mathbb{R}\times S^1ℝ×S1. We also show that this non-supersymmetric theory exhibits exact Bose-Fermi degeneracies for all states, including the vacua, when NN is even. Furthermore, for most values of NN, 22d 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 88. As a result, it fits into the classification of (1+1)(1+1)d SPT phases of interacting Majorana fermions in an interesting way.