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

by Aleksey Cherman, Theodore Jacobson, Yuya Tanizaki, Mithat Ünsal

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Authors (as registered SciPost users): Theo Jacobson
Submission information
Preprint Link:  (pdf)
Date accepted: 2020-04-02
Date submitted: 2020-03-13 01:00
Submitted by: Jacobson, Theo
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • High-Energy Physics - Theory
Approach: Theoretical


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.

Author comments upon resubmission

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We are very grateful to the referees for their positive comments and helpful questions and suggestions. We have edited our paper to take them into account, and are providing a description of the changes and comments on the questions below.

1) Two of the referees, Referee 1 and 2, asked us to comment on the discrepancy between our results and the previous results of numerical studies, especially Refs. [5,36,46,49]. Indeed, some of them claim deconfinement for massless adjoint QCD, and none of them observe Bose-Fermi degeneracy. We want to discuss the problems about confinement/deconfinement and about Bose/Fermi degeneracy separately.

About confinement/deconfinement, our results indeed agree with [5], but disagree with the interpretation of the results in [46]. Ref.[5] shows that there are infinitely many Regge-like trajectories in the large-N limit, with a density of states that exhibits Hagedorn growth. This is consistent with the hypothesis of confinement, but not with the hypothesis of deconfinement. Ref.[46] gives a reinterpretion of the same computation, and argues that some states on higher Regge trajectories can be decomposed into combinations of states from lower Regge trajectories, and uses this to claim deconfinement. However, Ref.[46] and later papers are just giving interpretation by looking at the spectrum of energies of single-trace states, but do not show that those single-trace states are indeed unstable in the large-N limit. Therefore, we do not think that this counts as persuasive evidence for deconfinement. The numerical and analytic calculations of the string tensions in [36,49] involve some uncontrolled approximations, involving e.g. neglect of parton-number changing processes. Consequently, we do not find the results persuasive.

About Bose/Fermi degeneracy, our claim is that two vacua related by broken discrete chiral symmetry have opposite fermion parity. Since all the numerical studies are using light-cone quantization, they are studying the spectrum of states created by local operators acting on *one* of these vacua. The fact that the other vacuum exists and has opposite fermion parity was not known prior to our work. Therefore, it is natural that Bose/Fermi degeneracy has not been observed. We have clarified this issue in the paper in the beginning of Section 8.

2 ) About the second question by Referee 1, we do not think that there is low-energy SUSY, for a number of reasons. First, if SUSY was only emergent, Bose-Fermi degeneracies could not be exact for all states - yet they are. If the theory did have exact SUSY, but it were spontaneously broken, then there would be two vacua with opposite fermion parity, as we have seen in 2d adjoint QCD. But spontaneously broken SUSY requires a massless Goldstino fermion to be present in the spectrum. There is no evidence for such a massless excitation from any of the studies of 2d adjoint QCD, including our study. Indeed, there is a point in parameter space of the massless theory where the absence of a massless fermion can be shown analytically, as discussed at the beginning of our Section 5, where we review an argument by Kutasov from the 1990s.

To take into account these answers to the questions, we have reorganized and improved the exposition of Section 8, which focuses on Bose-Fermi degeneracies and their interperation. We have also added a discussion of the relation between our work and numerical studies of 2d adjoint QCD from the 1990s at the end of Sec. 9.


3) About the second comment by Referee 2, we agree on their suggestion. We now separate the role of fermionic zero mode in Sec. 7.2.1, and the creation/annihilation operators are created out of non-zero modes.

4) About the third comment by Referee 3, the sign of four-fermion couplings generically depends both on the original four-fermion couplings $c_1, c_2$, and the perturbative corrections out of the gauge coupling, and we do not have a universal answer. Nevertheless, the symmetries allow us to discuss the degeneracies of the states, and this is the primary purpose of that section. So, we added a comment that we do not compute the matching of those coefficients.


5) Referee 3 asked whether it is really true that center symmetry is broken to subgroup, and asked us to explain this in the context of the analysis of Eq. 7.54.

We have clarified the discussion in that section and in several other places. Center symmetry is indeed broken to a subgroup due to the speciality of $1$-form symmetry in $(1+1)$ spacetime dimensions. Since our space is just a line, putting a test particle divides our space into two disconnected pieces. Therefore, we should now look for the lowest-energy states on each side while satisfying the boundary condition.

The mixed anomaly between the center and chiral symmetries suggest that the chiral generator has the charge $N/2$. Therefore, when the test particle has the charge $N/2$, we can consistently choose the ground states on both sides. We clarify this point by adding the footnote 13.

About the non-contractible loops on the torus, there is another subtlety due to the mixed anomaly between fermion-parity and chiral symmetry. When we take the periodic boundary condition, this anomaly survives under compactification, and thus the deconfined noncontractible line should become a composite of the Polyakov loop and the fermionic KK zero mode, $C, C^\dagger\sim \tr(P\psi_{\pm})$. We point out the existence of this subtlety in the footnote 13, and gives more explicit comments in Sec. 7.2.1.

6) Referee 3 argued that it is consistent to set the four-Fermi terms to zero, and said that this is not a fine-tuning.

We agree that it is consistent to set the symmetry-allowed four-Fermi to zero in a superrenormalizable theory, in the sense that these classically marginal terms are not necessary as UV counter-terms.

Still, we believe that it is customary to consider all the renormalizable interactions that are consistent with symmetry. Therefore, in the spirit of effective field theory, it is useful to consider the implications of classically marginal four-fermion interactions which are consistent with chiral symmetry.

Taking into account Referee 3's suggestion, we changed our presentation about the renormalizability and fine-tuning in order to clarify these points.

7) About the third comment of Referee 3, we also think that the smooth large-N assumption is reasonable on $\mathbb{R}^2$, and the chiral symmetry breaking would occur for a reasonable range of the parameter space. Still, we wanted to point out the logical possibility that there can be a region with non-zero measure that the N=4n+1 theory shows the unique ground state. Indeed, the $S^1$ compactified setup is an example of such a unique ground state.

8) Taking into account Referee 3's fourth comment, we changed our wordings in Sec. 6.

We again thank the referees for their detailed and very helpful comments!

Best regards,
Aleksey Cherman, Theodore Jacobson, Yuya Tanizaki, Mithat Unsal

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.

Published as SciPost Phys. 8, 072 (2020)

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