Gapped Phases in (2+1)d with Non-Invertible Symmetries: Part I

Many examples are examined in great detail.

1. Several statements contain mathematical inaccuracies that could mislead readers.

2. The treatment of gapped boundaries in the 4d TQFT is largely physical in tone. Some notations clash with standard notations in math and mathematical physics literature and might cause confusion.

4.Several arguments and examples recur in slightly different guises. Consolidating overlapping material will sharpen the narrative and reduce the overall length without sacrificing clarity.

The manuscript presents a study of 2+1D gapped phases via the categorical Landau paradigm. The authors discuss gapped boundaries of various SymTFTs(4d TQFTs) and apply the sandwich construction to obtain 2+1D gapped phases with generalized symmetries. Numerous examples are examined with great details. Although the current focus is on abelian, untwisted Dijkgraaf–Witten theories as bulk SymTFTs, the methods carry over straightforwardly to arbitrary fusion 2‑categorical symmetries, as the authors note they will pursue in future work.

Overall the manuscript is clearly written and meets SciPost's publication criteria. However there are several mathematical inaccuracies and other issues that should be addressed before publication, as detailed in the requested changes below.

1. On page 5, there is a typo “the existence of s that”.

2. On page 6, the authors discuss the classification of fusion 2-categories up to Morita equivalence. There are two locations where the statements are incorrect. 2.1 “​​In short the main result is that all fusion 2-categories are gauge related to $2\text{Vec}_G^\tau$”. The authors might mean “bosonic fusion 2-categories” or “fusion 2-categories of bosonic type”.

2.2 “A similar statement for the fusion 2-categories of fermionic type also holds by working with $\text{SVec}$ and allowing $M$ to have $\mathcal{Z}_2=\text{SVec}$”. The phrase “working with $\text{SVec}$” is ambiguous. It is also not true that all fusion 2-categories of fermionic type is Morita equivalent to $2\text{Vec}_G^\tau \boxtimes \Sigma M$ for a slightly degenerate $M$, as that would imply any fermionic fusion 2-category $\mathcal{C}$ satisfies $\mathcal{Z}_2\Omega \mathcal{C}=\text{SVec}$. The general case is that$\mathcal{Z}_2\Omega C$ is super-Tannakian. The correct variant of (2.1) for the fermionic case is that any fermionic fusion 2-category is Morita equivalent to $\mathcal{C}\boxtimes \Sigma M$ where $\mathcal{C}$ is a graded extension of $2\text{SVec}$, and $M$ is an MTC. This is 2211.04917 theorem 4.1.6.

1. On page 7. The authors describe the gapped symmetry boundary as being specified by bulk operators that end on the boundary, and mention that in 1+1D the analog are Lagrangian algebras. Notably condensable and Lagrangian algebras in higher dimensional TFTs had been defined and studied in 2105.15167, 2307.02843, 2403.07813, and provide a unified mathematical description for gapped boundaries of TFTs in any dimensions. It seems odd to say the 1+1d analog are Lagrangian algebras, as it suggests the 2+1d case is described by something else.

2. On page 46. The authors give a classification of gapped boundaries of 4d DW theories. This classification is incorrect. If the 4d DW theory has nontrivial twist, then a gapped boundary is given by a subgroup H<G, together with a $\textit{twisted}$ H-crossed extension of an MTC. When the $\mathcal{H}^4$ obstruction is nontrivial, there is no (untwisted)H-crossed extension.

3. On page 70. Eq.(6.87), the LHS should be $\mathcal{Z}2(\widetilde{M)$}}.

4. Section 6.2 seems redundant given the more general discussion in section 5.1. E.g. (6.87)(when corrected) is simply (5.10) when H=A and Eq. (6.83) is simply (5.9) when H=A, etc.

Ask for minor revision

1-The analysis of gapped phases in (2+1)d is both new and interesting.
2-The treatment is very explicit.
3-Many concrete examples are provided.

1-The authors only consider the case of abelian groups with trivial twist.
2-The content of section 3 is by now quite standard, both in the Math and Physics literatures.
3-The preprint is fairly long.

The preprint that is being submitted certainly meets SciPost's publication criteria. However, as it does contain a number of inacurracies detailed below, I am of the opinion that they should be adressed before this preprint can be accepted.

0-There is a typo in the first line of the second paragraph of page 5.
1-Footnote 5 is incorrect.
2-The question of which boundary conditions admit a topological interface admit a well-known mathematical description in terms of so-called Witt equivalence. This ought to be mentioned. Please see the article ''On the structure of the Witt group of braided fusion categories'' by Davydov-Nikshych-Ostrik. In particular, the question of which boundary conditions are not gauge related to the Dirichlet boundary is intimately related to fact that the Witt group associated to $\mathrm{Rep}(\mathbb{Z}_2)$ is non-trivial.
3-The notation $\boxtimes_{\mathbb{Z}_2}$ is potentially confusing: I find that it clashes somewhat with the relative tensor product. I would recommend modifying it, or at least introducing it more carefully.
4-The theta construction on page 45 is missing some data. Namely, in the presence of an anomaly $\tau$, there is no \textit{canonical} choice of diagonal action. (Unlike the case when the anomaly is trivial.)
4.5-Relatedly, for ease of comparison with the other treatments of similar questions that have appeared, it may be worth emphasizing that even when the anomaly $\tau$ is trivial, there is still a choice of anomaly $\omega$ for the $G$-symmetric 3d TFT. This is brought up later, but I think the general reader would appreciate this point being discussed explicitly near the beginning of section 5.
5-I believe that the categories on page 53 have also been extensively analyzed by Delcamp-Tiwari and Decoppet-Yu.
6-I fail to understand why Tables 1 and 2 are not symmetric along the diagonal. This should be explained.
7-On page 70, the class $\omega$ is already determined by $\widetilde{\mathcal{M}}$. It can therefore not be chosen.
8-Equation (6.96) is not quite correct. I suppose that the authors are trying to say that the left hand-side is some kind of extension?

Ask for minor revision

no example with non-invertible symmetry.

The manuscript studied 2+1d gapped phases with symmetries, using the trending framework of SymTFT. This is indeed an important and timely question in the field. Overall, the work is clearly written and well-organized.

The title claims that the symmetries can be non-invertible, however, in the details and examples only invertible symmetries are studied. In the referee's opinion, the semi-categorical, semi-field-theoretical approach taken in this work is far from sufficient for dealing with genuine non-invertible higher symmetries. Neither the examples with invertible higher symmetries nor the use of SymTFT is novel to the community. Therefore, this submission hardly meets the criteria of SciPost Physics but does meet those of SciPost Physics Core, where it could be published.

1. Footnote 5 on page 6 is incorrect. It is $2\mathrm{Vec}_G^\tau$ which should be replaced by a fermionic strongly fusion 2-category (a group extension of 2$\mathrm{SVec}$) while $\mathcal M$ remains to be non-degenerate (See Theorem 4.1.6. in arXiv:2211.04917).
2. Section 3.5 the "only if" direction under eq. (3.43) is not logically sound. Why it is not possible to gauge something both in $\mathfrak{B}_\mathrm{Dir}$ and $\mathfrak{T}$ to obtain $\mathfrak{B}_\mathrm{Dir}$? It may be proved based on a more general result (see Theorem 3.29. and Corollary 3.30. in arXiv:2312.15958) but for now the argument is not strong enough.

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