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

TL;DR

This paper investigates the anomalies, ground-state structure, and confinement properties of 2d adjoint QCD with an SU(N) gauge group, revealing spontaneous symmetry breaking, deconfinement for certain charges, and connections to topological phases.

Contribution

It introduces a mod 2 index theorem approach to analyze anomalies and provides new insights into the phase structure and topological classification of 2d adjoint QCD.

Findings

Confines for N>2 except for N-ality N/2 charges

Unbroken center symmetry for odd N, broken to Z_{N/2} for even N

Exact Bose-Fermi degeneracies for even N

Abstract

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.