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

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.

Accept in alternative Journal (see Report)