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

TL;DR

The paper analyzes two-dimensional adjoint QCD with a single massless adjoint Majorana fermion, deriving mixed ’t Hooft anomalies through a mod 2 index theorem. Anomaly matching then constrains infrared dynamics, predicting confinement for most N and a partial center-symmetry breaking for even N, along with exact Bose–Fermi degeneracies when N is even. It further connects the finite-N physics to symmetry-protected topological phases in one spatial dimension, and examines dynamics on R×S^1 to provide semiclassical validation of the anomaly-based conclusions. At large N, the spectrum exhibits Hagedorn-like growth for both bosons and fermions, while cancellations in the graded density of states persist, signaling intricate nonperturbative structure in this non-supersymmetric theory.

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.

Figures (4)

• Figure 1: Sketch of the flow of eigenvalues of the massless Dirac operator as a function of the bosonic field $a_{\mu}$ on an orientable manifold. The non-zero eigenvalues come in quartets, with two modes with eigenvalue $+\mathrm{i} \lambda$ and another two with eigenvalue $-\mathrm{i} \lambda$. We also sketch a pair of Dirac zero modes with opposite chirality. Since quartets of non-zero modes must go through zero together, the number of e.g. right-handed zero modes mod $2$ is a topological invariant, which we call $\zeta$. For clarity of presentation, all degenerate eigenvalues are shown slightly offset from each other.
• Figure 2: A sketch of the string tension in 2d $SU(N)$ adjoint QCD for even $N$ as a function of $N$-ality $k$. We draw a smooth curve, but of course for finite $N$ the spectrum of string tensions is discrete. The string tension must vanish at $k = N/2$ to satisfy 't Hooft anomaly matching. The sketch also shows the (generically) four-fold degeneracy of the string tension predicted by our discussion. Note that this double bump structure is required only for even $N$. It is not required for odd $N$.
• Figure 3: Examples of tunneling events between center-breaking minima on $\mathbb R\times S^1$ with anti-periodic boundary conditions. The blue lines depict the minimal action tunneling events without fermion zero modes. The red lines (marked with crosses) represent tunneling events with robust fermion zero modes. When $N=5$, the $5-1 = 4$ zero modes found in the Abelian ansatz are all lifted in generic field configurations because $\zeta = 4\text{ mod }2 = 0$. As a result, the $\mathbb{Z}_5$ center symmetry is completely unbroken. When $N=6$, instantons associated with tunneling between neighboring minima carry $\zeta = 6-1 \text{ mod } 2 = 1$ robust fermion zero modes, and are forbidden. However, instantons associated with tunneling between next-to-nearest neighbors carry $\zeta = 2(6-1) \text{ mod }2 = 0$ robust zero modes, and are not forbidden. As a result, the $\mathbb Z_3$ subgroup of center symmetry remains unbroken.
• Figure 4: Sketch of the spectrum of 2d massless adjoint QCD with even $N$ on $\mathbb{R} \times S^1$. The theory has two vacua with opposite fermion parity. These vacua remain degenerate even at finite volume due to the 't Hooft anomaly involving $(\mathbb{Z}_2)_F \times (\mathbb{Z}_2)_{\chi}$. The Hilbert spaces built using either one of these vacua exhibit a very powerful conspiracy involving the masses of bosonic and fermionic states. They are not paired energy-level by energy-level, but their overall distribution is such that there is no Hagedorn scaling in the $(-1)^F$-graded density of states.

Theorems & Definitions (2)

• Theorem 1
• Theorem 2