Global symmetries of quantum lattice models under non-invertible dualities

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.

The paper investigates how global symmetries transform under dualities associated with the gauging of a discrete group symmetry. The authors focus on cases where the full symmetry group is larger than the gauged subgroup and may include continuous symmetries. When gauging a non-normal subgroup, they show that the resulting dual symmetry is described by a ring of double cosets.

A particularly interesting example is the so-called "cosine symmetry" which the authors demonstrate corresponds to the double coset $\mathbb{Z}_2 \ O(2) / \mathbb{Z}_2$, rather than just a coset as previously suggested.

The authors also propose conjectures regarding the fusion algebra of duality operators and the relation between sectors of the original and dual Hilbert spaces. These conjectures imply a detailed decomposition of the dual model’s energy spectrum into contributions from symmetry-twisted sectors of the original model.

These general results are illustrated through four explicit examples of 1+1d lattice models with $S_3$ and Rep($S_3$) symmetries, including the spin-1/2 XXZ chain. While symmetries and dualities in these models have been studied before, previous works did not systematically track how the full continuous symmetries transform under duality. This systematic analysis allows the authors to uncover, for example, a previously unnoticed U(1) symmetry in one sector of the integrable Rydberg ladder.

Overall, the paper presents a clear advance in understanding the interplay between global symmetries and dualities, especially in the presence of continuous symmetries. I recommend publication in SciPost Physics after minor revisions, if any.

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.

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