SciPost Submission Page
Non-invertible Symmetries and Higher Representation Theory II
by Thomas Bartsch, Mathew Bullimore, Andrea E. V. Ferrari, Jamie Pearson
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Thomas Bartsch · Andrea Ferrari |
| Submission information | |
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| Preprint Link: | scipost_202402_00004v1 (pdf) |
| Date submitted: | 2024-02-03 11:31 |
| Submitted by: | Bartsch, Thomas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
In this paper we continue our investigation of the global categorical symmetries that arise when gauging finite higher groups and their higher subgroups with discrete torsion. The motivation is to provide a common perspective on the construction of non-invertible global symmetries in higher dimensions and a precise description of the associated symmetry categories. We propose that the symmetry categories obtained by gauging higher subgroups may be defined as higher group-theoretical fusion categories, which are built from the projective higher representations of higher groups. As concrete applications we provide a unified description of the symmetry categories of gauge theories in three and four dimensions based on the Lie algebra $\mathfrak{so}(N)$, and a fully categorical description of non-invertible symmetries obtained by gauging a 1-form symmetry with a mixed 't Hooft anomaly. We also discuss the effect of discrete torsion on symmetry categories, based a series of obstructions determined by spectral sequence arguments.
Author comments upon resubmission
• Four dimensions: As the mathematical literature on fusion 3-categories is less developed, our discussion in four dimensions is intentionally less conclusive and systematic as in the previous sections. In particular, we do not wish to include Reshetikin-Turaev type theories in our discussion of 3d TQFTs, as was clarified between versions 1 and 2 of the paper.
• Simple objects in 3Vec: We are not aware of a sensible equivalence relation that could be imposed on objects in the 3-category 3Vec to render the number of simple objects finite. Similarly for 3Rep(G), for which 3Vec acts as a “coefficient system”.
List of changes
Below, we summarise the relevant changes that we made in order to address the main points raised in the report:
• Summary of results: Added eq. (1.2) to the introduction to summarise and highlight the main findings of the paper (the classification of symmetry defects after gauging).
• Euler terms: Clarified our treatment of Euler terms in a footnote on page 23. These fix the partition function of a decoupled 2d TQFT on a 2-sphere. If the Euler terms are non-trivial, the associated topological surface defect acts universally by a non-trivial scalar multiplication on all local operators, including the identity operator. Since topological defects of this type are not particularly interesting from a symmetry perspective, we only consider canonically normalised (stable) 2d TQFTs with trivial Euler terms in what follows.
• Projective 2-representations: Irreducible projective 2-representations of a group G with 3-cocycle u can be labelled by a choice of subgroup H ⊂ G together with a trivialisation dc = u|_H. For generic H, such a trivialisation may not exist, in which case there exists no corresponding projective 2-irrep labelled by H. We clarified this point in a footnote on page 25.
• Typos: fixed the following typos:
- Psi ∈ C^{D-1} --> Psi ∈ C^D on page 3 above eq. (1.1)
- Fixed undefined references on page 24
• Notation: Gave further motivation for the notation H_g := H ∩ gHg^{-1} for the subgroup of H that leaves a given graded line defect V_g invariant when intersecting it, see page 9.
Current status:
Reports on this Submission
Strengths
1) The paper offers a clear mathematical perspective on several previously known physical constructions of non-invertible symmetries in D>2.
2) It clarifies what is the categorical characterization of several known group-theoretical non-invertible symmetries.
3) It discusses in detail many examples, so it is useful for the reader who wishes to reproduces the results.
Weaknesses
1) The paper is very long, extremely detailed, and many things are repeated several times without making a clear point.
2) The authors do not attempt to convince the reader that the explanation of what are the mathematical structures underling the known non-invertible symmetries has some relevance for physics. Hence the most physics oriented reader may have the feeling that the analysis is a purely academic exercise.
3) While the paper presents a general mathematical structure that seems to explain in a unified languale a varieties of phenomena known from a physics perspective, like the appearance of TQFT coefficients, and dressing of the defects with TQFT, the analysis shows that there is a mismatch (see report) between the mathematical formalism and the expectation from physics which the authors didn't attempt to fill or at least comment on.
Report
The paper analyzes systematically various group-theoretical higher-categories in D=2,3,4, giving a unified mathematical perspective on several previously known facts. It starts reviewing the D=2 case, where many things are well known, with a perspective which explains interesting phenomena discovered in the physics literature in higher D.
Several known non-invertible symmetries in D=3,4 are characterized as higher categories of higher-representations, and the authors discuss several examples which can help the reader to acquire familiarity with the type of computations involved.
One of the main interesting finding is the fact that, the dressing of topological defects with TQFT while gauging some (higher)subgroup which is known to be necessary in D>2, is already present in D=2: the anomaly of the bulk implies that the topological lines support a quantum mechanics with an anomaly, namely a projective representation, and one must stack a topological QM with an opposite anomaly while gauging. The necessity of stacking TQFT in D>2 is then understood similarly as higher-projective representations on the defect which must be cancelled with anomalous TQFTs.
However, while this clarifies previous findings in D=3, the situation in D=4 shows a clear mismatch between the mathematical formalism developed and previous finings in the physics literature. In particular the formalism developed in this paper seems to predict that the 3d TQFTs necessary for dressing topological defects while gauging are always of the Turaev-Viro type, but in many papers, starting with 2111.01141, it has been shown that there are simple examples (which should definitely be not beyond the formalism developed in the present paper) where one needs chiral TQFTs. While the authors makes some quick comment about this (e.g. at the end of section 4.5), I think this is a quite serious issue which would require ideally some extra analysis, or at least some more extended comment, explaining why and how the formalism developed here does not cover these cases.
Overall the paper is interesting and well written, and if the authors can fulfill the above main issue together with some other minor improvement I would advice the paper for publication.
I'll report here a list of comments, suggestions and questions, which the authors may want to consider in order to improve the presentation. In the "requested changes" I'll report instead things which I believe should be address before publication.
1) A comment on the second bullet point at pag. 6, which also applies to analogue discussions in D=3,4. The functors described are "fiber functors" and their existence is the categorical statement of the fact that the symmetry can be realized in a trivially gapped phase, i.e. it is anomaly free. Maybe it is worth explaining the relation with the first bullet point, to give a "physical" flavor of why anomaly free is equivalent to the existence of fiber functors.
2) The presentation in sec. 2.3.1 appears involved. Why do authors start with a definition of \Phi_g in eq. 2.11 and then successively switch to an alternative definition to get 2.14, instead of directly starting with the alternative? Same comment in D=3.
3) Most of the material in sec 2.4 is well known (in particular sec 2.4.1) and I would suggest to reduce it to improve readability.
4) Eq. (3.1) is confusing. I guess the authors have in mind placing the defects on S^2, otherwise the result of stacking depends on topology. I think that's the reason why after 3.1 they conclude the coefficients are integers instead of TQFTs. In section 4 they also comment that the appearance of TQFTs is peculiar of 3d defects in D=4 theories, but this is in contrast e.g. with the results of 2204.02407, I think precisely because the authors do not consider 2-manifolds other than spheres.
5) Beginning of sec. 3.4: H^3(K,A) should be twisted by \phi. Is this considered here?
6) Beginning of sec. 3.4: I'm not aware of a group-cohomology formulation of the cohomology of classifying spaces of 2-groups. For instance already for K trivial, the anomaly is classified by H^4(B^2A,U(1)). This has an algebraic formulation as the group of quadratic forms over A, but it is not of the group-cohomology type. The situation is even worst in higher D, and when the higher-group structure is non trivial, and as far as I know the anomaly is classified in general by something calculated by a spectral sequence. If the authors are aware of a simpler group-cohomology-like formulation it is worth reviewing it here.
7) Eq. 3.45 and below: the group cohomology should be twisted. It is not clear if the authors are using this or not.
Requested changes
1) Address somehow (either resolving it, or commenting about the mismatch) the above mention issue that the formalism contains a mismatch with previous physics analysis in D=4.
2) The notation in eq. (1.2) is not explained there, but only much below. I'd suggest to explain the notation every time a symbol appears for the first time.
3) Why is there a (-1) in eq. (2.3)? And in many other similar places. It is fine but doesn't match with the intro...
4) I guess there is a typo in the the first sentence of the second paragraph of section 2.6
5) An other mismatch (already present in the part I paper of the same authors) between the present mathematical formalism and the expectation from physics: physics intuition would suggest that condensates are labelled by subgroups instead of equivalence classes of them. It seems however that the math theory on 2reps produces the second option.
6) Typo first line second paragraph sec. 3.4: \hat{\mathcal{G}} ---> \hat{G}
7) Typo first line sec. 4.5: "now" repeated