Anomalies of Coset Non-Invertible Symmetries

1. The manuscript’s literature review is weak

2. The technical sections are not well organized and are difficult to follow

1. The manuscript suggests that coset symmetry arises from starting with a global symmetry group $G$ possessing a ’t Hooft anomaly classified by $[\omega_{D+1}] \in H^{D+1}(G, U(1))$, and subsequently gauging an anomaly-free subgroup $K \subset G$ with a discrete torsion inside $H^D(K, U(1))$. The authors need to clearly and explicitly state—preferably at the very beginning of the manuscript—the precise extent to which this interpretation holds.

2. Related to the point above, referring to the quadruple $(G, K, \omega_{D+1}, \alpha_D)$ as a “symmetry” can be somewhat confusing. The “fractional topological response” discussed in the manuscript appears to characterize the phase rather than the symmetry itself. Specifically, in the section “Description using bulk TQFT”, it is unclear how the fractional topological response—associated with $\alpha_D \in C^D(K,U(1))$—is encoded within the bulk TQFT data. The information contained in the higher fusion category, or equivalently the bulk TQFT together with a choice of gapped boundary condition, should only be sensitive to the cohomology class $[\omega_{D+1}]$ of the cocycle $\omega_{D+1}$ and an element of $H^{D+1}(K,U(1))$, rather than the specific cocycle $\alpha_D \in C^D(K,U(1))$. For example, in the $D=2$ case studied in Ref. [31], the data of the fusion category is specified by $(G, K, [\omega_3], [\alpha_2])$, depending only on the cohomology classes of the cocycles.

Additional Minor Comments:

1. It is important throughout the manuscript to maintain a clear distinction between cocycles and their cohomology classes. For clarity and precision, consistent notation should be adopted—for example, using $[\omega_{D+1}] \in H^{D+1}(G, U(1))$ for a cohomology class and $\omega_{D+1} \in Z^{D+1}(G, U(1))$ for a representative cocycle. The current usage is inconsistent and potentially misleading.

2. The term “generalized Frobenius-Schur indicator” is used ambiguously. The Frobenius-Schur indicator is an invariant associated with self-dual objects in fusion categories and should not be conflated with the full associator.

3. Equation (2.5) is difficult to parse. Each symbol and component should be properly defined before use. In particular, the notation $i^A$ lacks a clear definition—only some of its properties are mentioned.

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This paper studies twisted coset symmetries, i.e. symmetries obtained by gauging a (non-normal) subgroup $K$ of a group $G$, possibly in presence of a non-trivial twist $\omega$, often resulting in a non-invertible symmetry. In particular, the authors study anomalies of coset symmetries, where the anomaly is intended as the obstruction to realizing such symmetry in an SPT phase. The authors explore also anomalous scenarios that lead to non-trivial phases, such as gapless phases and phases that spontaneously break the symmetry. Moreover, several concrete realizations are discussed both in the continuum and in lattice models.

Let me list some comments and questions:

1. Second bullet point on pg. 4: reading later on it seems $\alpha_D$ can modify also the fusion rules of the symmetry;
2. When the sandwich construction for twisted coset symmetry defects is explained on pg. 7, it is stated that this works for a large class of symmetry defects. It would be interesting to mention some instances where this construction is not applicable;
3. Middle of page 12, before subsection 2.2.2.: why is it 2Rep$(\mathbb{Z}_2)$ when the boundary is 2D?;
4. Bottom line of pg. 12: should it be the magnetic defect for $a_2$ instead of the magnetic defect for $a_1$?;
5. Eq 2.29 has $\omega_4 = d \eta_3/2$, while in the introduction at the bottom of pg. 5 there is $\omega_4 = d \eta_3'/2$;
6. Point (1) on pg. 19: why the subgroups need to satisfy $[\omega_{D+1}|_{K,K'}]=1$? The above condition for a gapped boundary is $[\omega_{D+1}|_{K,K'}]=0$. Moreover the bullet point (1) seems redundant since a gapped boundary is already labelled by a subgroup $K$ such that $[\omega_{D+1}|_{K}]=0$;
7. In the previous paper by the authors [10], it is stated that the untwisted $S_3 / \mathbb{Z}_2$ coset symmetry cannot be realized in a trivially gapped theory (pg. 31). However $S_3 = \mathbb{Z}_3 \rtimes \mathbb{Z}_2$ and $\mathbb{Z}_3 \cap \mathbb{Z}_2 = { \text{id}}$, so this seems in contradiction with the conditions in section 3.2;
8. Bottom of pg. 20: instead of "$G_1$ needs to break to subgroup of $K_1$", "$G_1$ needs to break to a subgroup $K_1$";
9. In section 3.5, I find the term symmetric gapped phase slightly confusing, as this sometimes includes gapped phases that spontaneously break the symmetry, where this is realized as some permutation of the vacua. Also why $[\omega_{D+1}|_{H}]$ can still be non-trivial in general if a gapped boundary is labelled by a $H$ such that $[\omega_{D+1}|_{H}]=0$?;
10. In middle of pg. 31, the sentence "consistency condition on fusion can from the cocycle conditions" is probably missing a "come" before "from";
11. There are some grammar typos around the text that should be fixed, e.g. in the introduction "$K \subset G$ is a not a normal subgroup", etc.

Overall, I believe this paper is well-written and contains original and important results. Moreover the discussion is for arbitrary $D$ space-time dimensions and therefore very general and widely applicable, even though this may come sometimes at the cost of less concrete details. Therefore I recommend the paper for publication provided the above comments are taken into account.

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