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

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

This is not the current version.

Submission summary

As Contributors: Theo Jacobson
Arxiv Link: (pdf)
Date submitted: 2019-09-18 02:00
Submitted by: Jacobson, Theo
Submitted to: SciPost Physics
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.

Ontology / Topics

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Quantum chromodynamics (QCD) Topological materials
Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 1908.09858v3 on 13 March 2020

Reports on this Submission

Anonymous Report 3 on 2019-11-2 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1908.09858v2, delivered 2019-11-02, doi: 10.21468/SciPost.Report.1288


The authors have studied 2 dimensional adjoint QCD by using the modern technology of global anomaly matching involving various symmetries and semiclassical analysis.


Some point needs to be clarified.


This is a very interesting paper which studies the dynamics of 2 dimensional adjoint QCD which is strongly coupled and cannot be solved exactly. I have one major question and a few minor questions/comments listed below.

A major question:

1) I would like to ask whether the center symmetry is really broken to a subgroup. For example, in a semiclassical analysis with periodic boundary condition, there are two states given in (7.54). It is natural to organize the states to be eigenstates of the fermion parity $(-1)^F$ because it is unbroken in the large volume limit. The two states in (7.54) are such eigenstates. Each of them breaks the chiral symmetry, but preserves the center symmetry according to the analysis above (7.54).

In fact, the mixed anomaly between the chiral symmetry and the center symmetry only requires that one of the symmetries is broken. It is not necessary (although possible) that both of them are broken.

The argument the authors give for the breaking of the center symmetry is given in Sec 5.1. I was not fully convinced by the argument there. The argument is that the domain wall for the chiral symmetry breaking has $N$-ality $N/2$, and both sides of the domain walls have the same energy. However, the criterion for the center symmetry breaking is by Wilson loops which (at least naively) do not interpolate two vacua related by the chiral symmetry breaking. If we are just in one of the chiral symmetry breaking vacua, I think it is possible that the charge with $N$-ality $N/2$ is confined in that one vacuum. (I personally suspect that the center symmetry is not broken at all.)

At least the authors should comment on how to interpret the result of the semiclassical analysis, because (after taking the eigenstates of $(-1)^F$) the center symmetry is not broken in (7.54).

Minor questions/comments:

2) The authors seem to criticize the theory without the 4-fermi interactions. However, I think there is nothing wrong to set them to be zero. Without 4-fermi interactions, the theory is super-renormalizable. In super-renormalizable theory, a marginal term is not necessary for renormalization (in the same sense that irrelevant terms are not necessary for renormalization of renormalizable theories). So it is consistent to set them to be zero. It is just a matter of taste whether to call it "fine tuning". (In other words, if it is "fine tuning" to set them to zero, it is also "fine tuning" to set all irrelevant operators to zero.)

3) I think there is a possible reason that the chiral symmetry is broken in the case $N=4n+3$. If we require a smooth large $N$ limit, they may be nonzero for all sufficiently large values of $N$ regardless of the anomaly matching. Otherwise, the chiral symmetry breaking must occur at the subleading order of large $N$ expansion. That is logically possible, but a bit surprising if it is true.

4) The authors have discussed some SPT phase interpretations of the anomaly after adding a mass term to the fermions in a few paragraphs below (6.1). A small comment is that the SPT phase with the charge conjugation symmetry, which is $Z_2$, can be seen, for example in Table 1 of
Neglecting the parity (or time-reversal) symmetry, the relevant group is $\Omega^{spin}_2(BZ_2)=Z_2 \times Z_2$. It is indeed consistent with what the authors found in the paper by more explicit analysis (after a mass deformation).

Requested changes

I would like to request the authors to address (or at least comment on) the major point above, because it is related to the main result of the paper. The minor points are optional.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: perfect
  • grammar: perfect

Anonymous Report 2 on 2019-10-31 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1908.09858v2, delivered 2019-10-31, doi: 10.21468/SciPost.Report.1280


1) The paper is very clear, very well written.
2) The paper contains many interesting results that contradict old results. The authors explain where the previous works had made mistakes. Fixing incorrect results is of course an important part of scientific progress.
3) The authors attack the problem from several directions and apply a variety of new theoretical ideas to understand the fate of the theory. These include for example anomaly matching and semiclassical analysis on $\mathbb{R}\times S^1$. Deriving the same results using different methods makes the conclusions of the paper very convincing.
4) The results of the paper strengthen the relations between 2d and 4d gauge theories. In addition to confinement and chiral symmetry breaking, adjoint QCD in 2 and 4 dimensions also share the surprising feature of Bose-Fermi degeneracy without supersymmetry as pointed out by the authors. This relation can hopefully help in understanding the origin of this peculiar phenomenon.


1) While the authors invest a lot in settling the disagreements with refs [34,35], this is not the case with the numerical results of refs [5,36,46,49]. The authors are aware of this problem but didn't try to explain it.
2) The conclusion of the discussion around equations (8.2),(8.3) seems correct but the explanation is confusing. If the operators $a_B^\dagger$ and $a_F^\dagger$ are built out of both zero and non-zero modes, it might be that $a_B^\dagger|0>_B$ for example is identically zero while $a_B^\dagger|0>_F$ is not. I think that a better way to rephrase the argument is to start from the zero modes sector which is explicitly Bose-Fermi degenerate as seen in section 7.2.1 (assuming the result can be generalized to any even N), and then act on these states with the operators $a_B^\dagger$ and $a_F^\dagger$, built now only out of non-zero modes.
3) The construction of the spectrum in equation (7.58) is very beautiful. However, it is not clear from the presented analysis whether the vacuum states are twofold or fourfold degenerate. These two possibilities are of course physically distinct and lead to different patterns of symmetry breaking. This depends on the values and signs of the parameters $\epsilon_{1,2}$. Since the authors know in principle how to compute these parameters, it can be interesting to see a small discussion about the possible true vacua and whether the result is universal or depends on the microscopic parameters of the 2d theory (or at least a comment if the computation is too hard).


The authors revisit an old problem of 2d adjoint QCD. Using a wide set of modern tools, such as generalized symmetries and their associated anomalies, new index theorem, SPT classification and more, they arrive to conclusions which partially disagree with previous results. The authors are fully aware of these discrepancies and it is clear that they put a lot of thought and effort in order to explain the source for the contradictions. The paper is very well written, very clear and the arguments presented are convincing. On the way, the Authors also clarify several points that could have been a source for confusion. The first example for this is the discussion about the hidden assumptions in the statement that discrete 1-form symmetries cannot be broken in two dimensions. Another example is a comment about the importance of charge conjugation symmetry for whether 8 Majorana fermions are in a non-trivial SPT phase or not. To summarize, the paper is interesting and important for several reasons. It fixes incorrect statements about the vacuum of adjoint QCD, contains new and interesting results such as the analysis of anomalies and their realization in the vacuum, the Bose-Fermi degeneracy and more, it clarifies and explains in a clear way several subtle points, and sharpen the application of new (and old) theoretical ideas to strongly interacting field theories.

Requested changes

No changes are needed except for some small typos:
1) Page 2 below eq (1.1): "an Majorana"-> "a Majorana"
2) Page 12 two lines below eq (4.7): "the the"->"the"
3) Page 17, item (b): "are longer zero modes"-> "are no longer zero modes"
4) Page 26 line 9: "that someone in its parameter space"-> "that somewhere in its parameter space"
5) Page 28 line 4 from the end: "in the sense then"-> "in the sense that"
6) Page 34 before eq (7.26): "these quasi-ground state"->"these quasi-ground states"
7) Page 39 line 8: "each other the action of"-> "each other under the action of"
8) page 45 before eq (8.14): "is dominated the Casimir"-> "is dominated by the Casimir"

  • validity: top
  • significance: high
  • originality: high
  • clarity: top
  • formatting: excellent
  • grammar: excellent

Anonymous Report 1 on 2019-10-7 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:1908.09858v2, delivered 2019-10-07, doi: 10.21468/SciPost.Report.1213


This is a beautifully written paper, bringing some very modern tools in quantum field theory (discrete 't Hooft anomalies and generalized global symmetries) to bear on an old problem, the dynamics of adjoint QCD in 1+1 dimensions. The calculation of the discrete anomalies is very well documented and quite convincing. With some further assumptions on the dynamics of the theory (trying to rule out that the theory is topological or conformal) the authors obtain some remarkable results about the physical properties of the theory, including statements about when chiral symmetry is broken (as a function of N, the number of colors) and about an exact Bose/Fermi degeneracy in the spectrum.

I'd like the paper to be eventually published, but there are two issues I'd like to see discussed a little more clearly:

1) The authors disagree with much of the previous literature. As such their results should be evaluated critically and it is their obligation to try to do their best to identify shortcomings in the previous literature. They do a remarkable job in discussing how their analysis avoids the conclusion of [35]. [35] is a based on formal arguments that rely on certain assumptions and the authors make a good case that these assumptions don't hold. What is more troubling to me is that the new results presented here disagree with [5,36,46,49]. These references present explicit numerical calculations which, in a theory that is mostly of theoretical interest and hence does not allow experimental verification, should be the ultimate arbiter. I believe the authors should do more to explain why these numerical studies fail to find the physics they claim to be correct. This does not have to be as detailed and explicit as the nice comparison to [35], but a little more should be said.

2) The authors find a Bose/Fermi degeneracy without supersymmetry (SUSY). Their main argument for the lack of SUSY is that the UV theory is manifestly non-SUSY. But is there a chance that SUSY emerges at low energies? It would be nice to see a few more comments/details on how one can tell from the answer for the physical spectrum that SUSY is broken. Do fermions and bosons interact differently despite their degenerate masses?

  • validity: -
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  • originality: -
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