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

1. Methodology is a bit outdated. The mathmatical theory of fusion 2-categories has been intensely devoloped in recently years. However, this work did not embrace those developments and used simple-minded treatments, sometimes proceed by guessing or trial-and-error. The general framework presented in Section 2 is incomplete and inaccurate. A number of errors and inaccuracies can be found in the details of the examples.
2. Organization is poor. The majority of the manuscript is devoted to technical details of S3 and D8 examples. This is also a sign that the authors lack confidence in their general theory. If the general framework and algorithm could be laid down clearly, the reader should be able to reproduce examples, and then only essential details of (important) examples need to be listed.
3. Not clearly written. In particular, the design of conventions and notations is messy and difficult to follow. For example, in Table 2, the same object (BDir and 2VecS3) is given different names, which is technically unnecessary and only make things complicated. New symbols often appear without clear explanation.

The manuscript needs substantial polishment before it can meet the publication criteria of SciPost Physics. Here are some potential suggestion: 1. Improve the section on the general framework, making it repeatable. 2. On the exmaples, focus on the interesting "superstar" examples, and reduce the amount the details on the well-known or less-interesting examples. 3. See also the requested changes.

1. Under Table 1, the names all-boson, emergent-fermion, and the abbreviations AB and EF date back to the original papers [65] and its sequal Phys. Rev. X 9, 021005. [52] relies heavily on the idea of these two works but failed to address the credit. Appropriate citations should be added. Footnote 2 is questionable. EF-type fusion 2-categories are by definition fermionic fusion 2-categories. It is the Drinfeld centre (or SymTFT) of an EF-type fusion 2-category that is a bosonic 4d topological order. Compare, in 2d, sVec is a fermionic symmetry and Z(sVec) is the toric code.
2. Below (2.18), "The relation to H-double cosets is due to the fact that the number of H-conjugacy classes into which a G-conjugacy class splits is equal to the number of double cosets HgH that intersect the G-conjugacy class." The statement should be more precise. Only saying "the number is equal" is pointless.
3. Table 3, row 3: What is the two group $\mathbb G^{(2)}$ when $H$ is not normal and $G/H$ is not a group?
4. Equation (2.32): What is the precise meaning of vacua? (It seems to mean idempotent local operators by later parts) How is this formula derived?
5. Above equaiton (2.42), the notation ($G/H$-SSB)$\boxtimes$($H$-SPT) is misleading for a mixed SSB and SPT phase since it indicates a decoupled stacking.
6. The term "topological local operators" looks self-contradictary; by definition topological means non-local. Here the notion of locality secretly changes upon compatification of the SymTFT: a topological operator in the 4d SymTFT becomes a local operator in the 3d theory after compatification. It may be better to use just local operator.
7. Discussions around equation (2.46) are questionable. Where does the basis $|g\rangle$ come from? $D^R(g)$ is a matrix acting on $R$, shouldn't one choose basis from $R$? Otherwise how can (2.46) make sense?
8. Various terminology and notions are used without clear explanations on their first occurrence: (un)twisted sectors, Triv in (3.55), $1_{00}$ in (3.67).
9. Around (3.65) and (3.103): It is weird to "assume" how $L^{1,2}$ interacts with $v_0$. Shouldn't this be computable from their algebras? Can't the author compute $L^{1,2}v_0$ and $L^{1,2}v_1$? How can these be "indistinguishable" and require some additional assumption to proceed?
10. (3.118), new symbol $\lambda$ does not come with an explanation.
11. (3.147), wrong label on the physical boundary.
12. Above (3.223), "Therefore each vacuum realizes a T topological order described by T = Z(C)." C needs explanation. Also, the authors keep distinguishing physical theories (boundary conditions) from their mathematical (categorical) descriptions, but here they ironically equate them.
13. (4.33), the same graph on both sides.

Ask for major revision

1- Interesting class of gapped phases, especially what the authors refer to as the "super-star" phase where a trivial and a topologically ordered phase coexist.

2-Several concrete and detailed examples.

1-There are several typos throughout the paper, both grammatical and mathematical. I urge the authors to proofread it thoroughly.

2-Perhaps this is my ignorance, but I am confused by the use of $\otimes$ vs $\boxtimes$ and $\oplus$ vs $\boxplus$. Could the authors explain the difference, at least in a footnote, when these notations are introduced for the first time? This could be, for example, around Table 2.

Publish (easily meets expectations and criteria for this Journal; among top 50%)