SciPost Phys. 8, 072 (2020) ·
published 5 May 2020

· pdf
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 groundstate 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 nonsupersymmetric theory exhibits exact BoseFermi
degeneracies for all states, including the vacua, when $N$ is even.
Furthermore, for most values of $N$, $2$d massive adjoint QCD describes a
nontrivial symmetryprotected 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.