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, \mathbb Z_N ℤ N center symmetry is unbroken for odd N N and spontaneously broken to \mathbb Z_{N/2} ℤ N / 2 for even N N . All of these results are confirmed by explicit calculations on small \mathbb{R}\times S^1 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 8 . As a result, it fits into the classification of (1+1) ( 1 + 1 ) d SPT phases of interacting Majorana fermions in an interesting way.