Global symmetries of quantum lattice models under non-invertible dualities

1- The paper does a thorough job of describing the map of (twisted) Hilbert space subsectors of 1+1d spin/qubit models under (strong) duality/finite gauging operations. A "sandwiched construction" is presented for a subset of the symmetries of the dual Hamiltonian.

2- A number of concrete conjectures are made about the maps between symmetries and twisted sectors. These conjectures are tested in a well-understood family of duality-related models.

3- The authors emphasize the importance of a ring of double cosets that appears as part of the dual symmetry when gauging a finite group symmetry. This has not been emphasized in the literature.

1- The importance of the ring of double cosets is not really apparent from the paper. It is certainly natural to consider mathematically, but symmetries of quantum lattice models that flow to locally acting symmetries of quantum field theories in the IR limit usually have additional constraints. In particular, they must act in a reasonably defined "local" manner. One way to achieve this is to express symmetry operators as MPOs (matrix product operators). Could the authors clarify whether all elements of the above-mentioned ring have an MPO representation?

2- We understand that the authors are contextualizing their conjectures in known examples (e.g. their section 4 includes examples that have all been studied in Ref 13 and/or 22). However, to highlight the novelty of their conjectures, it would be great if they could provide some new examples.

3- The models referred to here are all integrable. While integrability doesn't play a direct role in the discussion, a (numerical) test of the conjectures in non-integrable models would make the evidence stronger.

4- There are numerous typos (some noted in the "Requested changes" section). The manuscript would greatly benefit from a careful proofreading.

1- "We take a down-up-earth approach..." in the 3rd paragraph of Page 2: is this a typo for "down-to-earth"?

2- In equation (2), $\phi\in U(1)$ is confusing. Should we understand the parameter $\phi$ to have circle compactification such that $\phi \sim \phi+\pi$ or $\phi \sim \phi+2\pi$?

3- In equation (18), should the second part read $\mathcal{D}^{(b)}{\text Izz}\in \text{ Hom}\left(\mathcal{H}},(-1,X)}^{\mathbb{Z2;1},\mathcal{H}\right)$}}^{[b], for consistency of notation?

4- Should one understand the right hand side of (21) to be an element of the ring of double cosets? If so, how should one understand the factor of $\frac 12$ when the ring is over $\mathbb Z$?

5- The discussion of symmetries in section 2.1 would benefit from a summary of the various symmetries in various Hilbert space subsectors, perhaps in the form of a table (similar to the one in Sec 4 but with more detail, such as labeling the twist sectors explicitly).

6- Just to clarify, is the O(2) in the parenthetical comment appearing at the end of the paragraph above eq (23) referring to the O(2) group generated by (24) and $\prod_{m=1}^{L/2} X_{2m-1}$?

7- Are factors of $\frac12$ missing on the r.h.s. of (26) and (27)? Otherwise these equations are inconsistent with (21).

8- Should the - sign in the middle expression of (34) be a + sign instead?

9-The authors sometimes refer to "non-Abelian" objects in fusion categories (e.g. below eqs. (35) and (98)) . This is different from the usual adjective of "non-invertible" used for them in the literature. Is this difference in terminology intentional? The word "non-Abelian" is confusing since the fusion of these objects is commutative, even though not invertible.

10- The notation in (42) is quite suggestive. Should one think of the superscripts as "charges" of the Rep(G) symmetry?

11- Below (64), the authors claim that the operators $\tilde O_s$ form a representation of a ring of double cosets, pointing to a demonstration in Appendix B. However, in Appendix B, this is only demonstrated in special examples. Could the authors provide a general demonstration of this in the Appendix?

12- A notation question: does $\text{dim}(G_g)$ in (67) mean the same thing as $|G_g|$?

13- In the last sentence of section 4.1, what does the notation $\mathbb{Z}[\mathbb{Z}_3 \backslash SU(2)/ \mathbb{Z}_3]/O(2)$ mean? Is this a quotient in the sense of rings?

14- There is a typo in (90): $s_{j+1}$ should be $h_{j+1}$.

15- The comment about the U(1) symmetry in the IRL model, below (117), is intriguing. Does the lack of a local charge density mean there is no MPO form for the symmetry operator?

Ask for major revision

1- The authors' goal is to understand how ordinary global symmetries transform under non-invertible mappings between models. This is a topic of great current interest. 2- They formulate a general conjecture on how the global symmetries behave, writing it in terms of a double coset. The conjecture is very plausible. 3- They demonstrate that their conjecture applies in terms of a set of examples related to the XXZ chain via non-invertible mappings.

1- The authors follow the unfortunate but currently popular practice of putting important details in appendices.

2- In particular, while the conjecture is stated in the paper, the definition of a double coset is put in an appendix, and the explanation of of the algebraic ring of double cosets is as well. Since this conjecture is their central result, it seems rather off-putting to not explain it properly in the body of the paper.

3- They also relegate the support of the conjecture into Appendix A, leaving little motivation in the main text. I'm guessing they did so in order to avoid mentioning fusion categories. This seems strange -- in the non-invertible symmetry world, fusion categories are frequently used, and likely are familiar to anybody interested in reading this paper.

4- Even stranger is the fact that the bulk of the paper is devoted to a single set of examples that have been already done to death in the literature. In particular, the work of Lootens et al in Refs 11,12, along with Eck and Fendley in Ref 13, discusses this set of examples (summarized in Figure 2) in detail. While I'm sure the authors provide some details not in the earlier work, that's not nearly as interesting as their conjecture.

5- As far as I can tell, no other examples are discussed in any meaningful detail. Thus their conjecture really has not been tested.

The authors need to make their conjecture and the general support for it stronger, and reduce the amount of discussion of known work. It's not simply a matter of moving appendices up to the text, but will require considerably more rewriting. It also will require giving some other examples in support. Not every detail needs to be worked out, but more general evidence in support of the conjecture must be provided.

Ask for major revision

1. The paper introduces and clarifies the notion of double coset symmetries and presents general results and conjectures on how Hilbert space sectors and global symmetries transform under the gauging of a discrete symmetry.

2. It revisits four well-known models related to $S_3$/Rep($S_3$) and systematically analyzes their full symmetry structure, including continuous and non-invertible symmetries, providing a more complete understanding than previous studies.

3. The paper is well-structured, pedagogical, and largely self-contained, with clear explanations and helpful figures that illustrate the duality mappings.

Questions / suggestions:

1. Does the general theory of symmetries under dualities developed in Section 3 apply to 0-form symmetries in arbitrary spacetime dimensions, or is it restricted to 1+1 dimensions?
2. After Equation (36), it is stated that “the symmetry S is described mathematically by a fusion category.” Since fusion categories only have a finite number of simple objects, and the paper explicitly treats continuous symmetries, I believe this statement is not correct.
3. In Subsection 3.1, the paper distinguishes between "strongly symmetric" and "weakly symmetric" dualities, but provides only a brief definition. Later, in the Conclusion, it is noted that strongly symmetric dualities correspond to gauging discrete groups. It would improve clarity to move this statement (and perhaps add a few sentences of physical explanation or examples) into Section 3.1, when the distinction is first introduced.
4. After Equation (51), it is stated that the Hilbert space of the original model with twisted boundary conditions decomposes with respect to Rep($G_g$) “as a consequence of gauging $G_g$ symmetry.” Since the equation concerns the original model before gauging, I found this wording confusing.
5. After Equation (98), it is inferred from the properties of the duality operators that a certain Frobenius algebra $1+s$ is gauged. This conclusion is not immediately obvious to me; a bit more explanation might be helpful here.
6. In Equation (116), the authors discover a previously unnoticed U(1) symmetry in a sector of the integrable Rydberg ladder, which is very interesting. While they note that this symmetry cannot be expressed in terms of local operators and do not write it out explicitly, I would still appreciate seeing an explicit (possibly nonlocal) expression for it. This could help to understand potential physical consequences of this symmetry.
7. In Equation (133), the solution $\alpha_b = \mathrm{dim}(b)$ to an ansatz for the fusion of duality operators is derived, and it is stated that Equation (47) should follow from this. However, Equation (47) appears to correspond to $\alpha_b = 1$, rather than $\alpha_b = \mathrm{dim}(b)$. Additionally, Equation (134) is unclear to me — it seems that the dagger may need to be placed on the second duality operator rather than the first?

Math typos:

1. In Equation (18), the superscript of the second duality operator should be b instead of -1, and the subscript of the twisted XXZ Hilbert space should be -1 instead of $\pi$, to match the notation used earlier in the paper.
2. In Equation (94), it seems that the terms $E_2^{s,s}$, $E_2^{s-1,s-1}$, and $E_2^{s+1,s+1}$ in the third and fourth row are missing a bar over their superscripts, given the twisted boundary conditions.
3. In Equation (144), it should be $g_2$ instead of $h_2$ on the right-hand side of the equality sign.

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