We show that 2d 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 2d adjoint QCD confines for N > 2, except for test charges of N-ality N/2, which are deconfined. In other words, ZN center symmetry is unbroken for odd N and spontaneously broken to ZN/2 for even N. All of these results are confirmed by explicit calculations on small R × S1. 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, 2d 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.
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We are very grateful to A. Armoni, D. Dorigoni, S. Dubovsky, L. Fidkowski, I. Klebanov, D. Ku-tasov, E. Poppitz, S. Sen, M. Shifman, A. Smilga, and R. Thorngren for comments and discussions. We also thank an anonymous referee for some comments that significantly improved our exposition. Part of this work was done during the conference “Topological Solitons, Non-perturbative Gauge Dynamics and Confinement 2” at the University of Pisa, and A. C. and Y. T. are grateful for the kind hospitality of the organizers. A. C. is supported by startup funds from UMN, T. J. is supported by a UMN CSE Fellowship, Y. T. is supported by JSPS Overseas Research Fellowships, and M. Ü. is supported by the U.S. Department of Energy via the grant DE-FG02-03ER41260.