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

### Submission summary

 As Contributors: Theo Jacobson Arxiv Link: https://arxiv.org/abs/1908.09858v3 (pdf) Date accepted: 2020-04-02 Date submitted: 2020-03-13 01:00 Submitted by: Jacobson, Theo Submitted to: SciPost Physics Academic field: Physics Specialties: Condensed Matter Physics - Theory High-Energy Physics - Theory Approach: Theoretical

### 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.

### Ontology / Topics

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Published as SciPost Phys. 8, 072 (2020)

### List of changes

-Altered equation 1.1 to include perturbatively marginal four-Fermi terms allowed by symmetries, and revised footnote 1 to reflect this change in presentation.

-Corrected the proof given in footnote 3 that we have found all internal symmetries of the theory.

-Revised the paragraphs at the end of page 6 regarding the four-Fermi terms to address comments made by Referee 3.

-Added footnote 13 to address questions raised by Referee 3 about the breaking of center symmetry to a subgroup.

-Changed the wording in the second to last paragraph of section 6, taking into account the fourth comment by Referee 3.

-Added a paragraph on page 39 clarifying which non-contractible line operator is deconfined on the cylinder with periodic boundary conditions.

-Added two new paragraphs just before section 8.1 to emphasize why numerical light-cone studies in the literature did not observe Bose-Fermi degeneracy.

-Changed the wording leading up to equation 8.2 to incorporate the suggestions of Referee 2.

-Added figure 4 to illustrate the Bose-Fermi degeneracy of the massless theory on the cylinder.

-Added a discussion at the top of page 46 about the vanishing of the charge-conjugation-twisted partition function at large N.

-Reorganized section 8 to include a subsection focusing on Bose-Fermi degeneracy without supersymmetry. We removed the misleading example of a free Majorana fermion in two spacetime dimensions, and added arguments ruling out spontaneously broken SUSY or emergent SUSY at long distances.

-Added a discussion in section 9 further commenting on the discrepancy between our results and numerical studies in the literature.

-Changed the wording in the third bullet point in section 10.

-Corrected typos generously pointed out by Referee 2.

### Submission & Refereeing History

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Resubmission 1908.09858v3 on 13 March 2020
Submission 1908.09858v2 on 18 September 2019