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Anomaly Matching in the Symmetry Broken Phase: Domain Walls, CPT, and the Smith Isomorphism

by Itamar Hason, Zohar Komargodski, Ryan Thorngren

Submission summary

As Contributors: Ryan Thorngren
Arxiv Link: (pdf)
Date accepted: 2020-03-31
Date submitted: 2020-02-03
Submitted by: Thorngren, Ryan
Submitted to: SciPost Physics
Discipline: Physics
Subject area: High-Energy Physics - Theory


Symmetries in Quantum Field Theory may have 't Hooft anomalies. If the symmetry is unbroken in the vacuum, the anomaly implies a nontrivial low-energy limit, such as gapless modes or a topological field theory. If the symmetry is spontaneously broken, for the continuous case, the anomaly implies low-energy theorems about certain couplings of the Goldstone modes. Here we study the case of spontaneously broken discrete symmetries, such as Z/2 and T. Symmetry breaking leads to domain walls, and the physics of the domain walls is constrained by the anomaly. We investigate how the physics of the domain walls leads to a matching of the original discrete anomaly. We analyze the symmetry structure on the domain wall, which requires a careful analysis of some properties of the unbreakable CPT symmetry. We demonstrate the general results on some examples and we explain in detail the mod 4 periodic structure that arises in the Z/2 and T case. This gives a physical interpretation for the Smith isomorphism, which we also extend to more general abelian groups. We show that via symmetry breaking and the analysis of the physics on the wall, the computations of certain discrete anomalies are greatly simplified. Using these results we perform new consistency checks on the infrared phases of 2+1 dimensional QCD.

Ontology / Topics

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Domain walls Goldstone modes Quantum chromodynamics (QCD) Quantum field theory (QFT)

Published as SciPost Phys. 8, 062 (2020)

Reports on this Submission

Anonymous Report 1 on 2020-2-21 Invited Report

  • Cite as: Anonymous, Report on arXiv:1910.14039v2, delivered 2020-02-21, doi: 10.21468/SciPost.Report.1525


1- This paper provides a useful way to evaluate ‘t Hooft anomalies of higher-dimensional theories from those of lower-dimensional theories.

2- For such a technique, a careful identification of the symmetry of domain walls in spontaneously broken phase is important. They point out the importance of the canonical CPT symmetry.

3- This logic can be reversed, and the knowledge of anomaly for higher-dimensional field theories is helpful to obtain light excitations on the domain wall, when the anomaly in higher-dimensional QFT is matched by SSB of discrete symmetry. This study has an interesting application for this purpose, too.


Nothing in particular.


This paper focuses on how anomaly matching can be satisfied when discrete symmetry is spontaneously broken. When discrete symmetry is spontaneously broken, one can consider the domain wall which connects different domains. When the anomaly is present, the domain-wall physics is constrained by that anomaly and should contain light excitations similar to that of Jackiw-Rebbi mechanism.

In order to get a rigorous understanding on domain-wall physics, the authors studied carefully about the symmetry on the wall. Especially, they consider the following question first: If $Z_2$ symmetry is spontaneously broken on the bulk, does the domain wall have unbroken $Z_2$ symmetry? The answer turns out to be slightly nontrivial: The combination of the broken $Z_2$ and $CPT$ gives a good symmetry on the wall.

Then, what happens if the bulk theory has an ‘t Hooft anomaly for such $Z_2$ symmetry? To satisfy the anomaly matching, the domain wall theory should support a nontrivial QFT with lower-dimensional ‘t Hooft anomaly. This is partly known in previous studies, but they make a more rigorous connection between the original anomaly and the anomaly on the wall. This allows us to reconstruct the anomaly of higher-dimensional QFT from anomalies of lower-dimensional QFTs, which are more tractable in many cases.

These are very interesting and useful results in the study of QFTs, so I strongly recommend its publication.

I only have one question. The authors obtained a lower-dimensional QFT by considering the domain wall with the frustrated boundary condition along a line or a large circle. Instead, we can obtain another lower-dimensional QFT with the same b.c. along a small circle and integrating out non-zero KK modes. Do we have a similar subtlety of the symmetry and anomaly related to the combination of CPT invariance also in such cases?

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Author Ryan Thorngren on 2020-04-07
(in reply to Report 1 on 2020-02-21)
answer to question

Dear Colleague,

Thank you for your report.

Regarding your question, we also expect that this story will be in some part repeated in the study of defects. For instance, if U is an ordinary Z/2 symmetry, that is---unitary and squaring to the identity, it is well-known that in general there is no induced Z/2 action on the defect Hilbert space, since one may have to resolve crossings between the U insertions. For instance, for bosons in 1+1D, it is known that the Z/2 anomaly is captured by the nontrivial crossing relations of the U lines.

However it appears that one can indeed devise an action of U*CPT on the defect Hilbert space, and the associated anomaly in 0+1D is the Kramers degeneracy protected by this anti-unitary symmetry. (This seems to reflect the negative Frobenius-Schur indicator of the Z/2 vortex in the 2+1D theory.)

From the perspective of the Smith isomorphism, something like this probably happens in general dimensions, at least in this symmetry class. Indeed, in the first two paragraphs of the proof of theorem 4.1 we show that compactification with a U-twist is inverse to the Smith homomorphism (an isomorphism in this case).

On the other hand, this is the direction of the proof which fails in the other symmetry classes. Indeed, for time reversal, we don't know a good way to define the defect Hilbert space, but even for the other unitary Z/2 class with U^2 = (-1)^F something seems amiss.

best wishes,
Itamar Hason, Zohar Komargodski, and Ryan Thorngren

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