  We show that $2$d 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 $2$d adjoint QCD
confines for $N >2$, except for test charges of $N$-ality $N/2$, which are
deconfined. In other words, $\mathbb Z_N$ center symmetry is unbroken for odd
$N$ and spontaneously broken to $\mathbb Z_{N/2}$ for even $N$. All of these
results are confirmed by explicit calculations on small $\mathbb{R}\times S^1$.
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 $N$, $2$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 $8$. As a result, it fits into the classification of $(1+1)$d SPT phases of
interacting Majorana fermions in an interesting way.

High Energy Physics - Theory;Strongly Correlated Electrons