Dualities between 2+1d fusion surface models from braided fusion categories

I thank the reviewer for their thoughtful questions.

1. As in 1+1d, the 2+1d dualities considered here change the quantum phases; for example, in the anisotropic limit the XXZ model with a unique gapped ground state is dual to a $\text{Rep}(S3)$ model with $D(S3)$ topological order in the same limit. Therefore, I generally expect duality implementations to require linear-depth unitary circuits. Some 1+1d dualities can be realized via constant-depth circuits supplemented with measurements (https://arxiv.org/abs/2311.01439), and extending this to 2+1d is an interesting direction. I’ve expanded the discussion of circuit realizations in the Conclusions. Regarding the second point, I agree that duality mappings in 2+1d should correspond to functors between module 2-categories.
2. Indeed, should be able to build 2+1d fixed-point fusion surface models from the data of a fusion 2-category and the 2+1d analogue of a special symmetric Frobenius algebra. As discussed in Section 5.3 in Inamura & Ohmori, taking the fusion 2-category 2Vec$_G$ and a separable algebra therein, i.e. a $G$-graded multifusion 1-category, produces a $G$ symmetry-enriched Levin-Wen model. In the present setting, which starts from a braided fusion category, the same mechanism should yield fixed-point Hamiltonians from commutative special symmetric Frobenius algebras in the braided category (braiding is the minimal extra structure needed in 2+1d). As you note, interpolating between different algebras would offer a natural route to critical transitions between different topological phases.

I thank the reviewer for the constructive feedback and insightful questions. Regarding the requested changes:

1. Thank you for the suggestion. I have expanded the two brief examples at the end of Section 6 into a dedicated subsection with two more examples, one based on $\mathcal{B}=\text{Rep}(S_3)$ and $M=\text{Vec}(\mathbb{Z}_3)$ and another one with $\mathcal{B}=su(3)_1$ and $\mathcal{M}=TY(\mathbb{Z}_3)$. In general, determining the symmetry fusion 2-category from $\mathcal{M}$ and $\mathcal{B}$ is a challenging mathematical task and outside the scope of this paper; the aim here is to present concrete 2+1D models that fit into this extended fusion surface model framework and are related by dualities.
2. The condensation defects correspond to hexagonal nets whose edges are labeled by the algebra object; this is now stated explicitly in Eq. (31) and the preceding paragraph. Their action is computed by fusing the net into the 2+1D fusion tree and evaluating with the F- and R-symbols.
3. Taking the braided fusion category $\mathcal{B}$ and the regular module category as input to the fusion surface model construction does not guarantee global 0-form symmetries (beyond those realized as condensation defects of 1-form symmetries), though they can exist accidentally or emerge at low energies. To build 0-form symmetries in from the start, one could instead choose a different $\mathcal{B}$-module category or begin with a fusion 2-category that is not connected. In particular, $G$ symmetry-enriched string-nets can be regarded as fusion surface models constructed from a $G$-graded fusion category regarded as a separable algebra over the fusion 2-category 2Vec$_G$, as explained in Section 5.3 in Inamura & Ohmori. The symmetry fractionalization class will be determined by the input data in this case. Gauging these models amounts to specifying a module 2-category of the input fusion 2-category, which has been explored in more detail in https://arxiv.org/pdf/2506.09177. I have clarified the connection to symmetry-enriched topological order in the Conclusions.
4. In Section 4.2 I take $\mathcal{B}=\text{Rep}(S_3)$ with the symmetric braiding to realize the duality to the XXZ model. Since all braiding phases are real, the Hamiltonian and any perturbations compatible with the fusion surface construction are real and time-reversal invariant, so I do not expect chiral topological order. I have added this clarification to Section 4.2. In contrast, if one uses $\mathcal{B}=\text{Rep}(S_3)$ with a non-degenerate braiding, the complex R-symbols generally lead to a fusion surface Hamiltonian that breaks time reversal (analogous to the $\mathbb{Z}_3$ Kitaev honeycomb model), making it a candidate for chiral topological order. Because the underlying $\text{Rep}(S_3)$ anyon chain is described by the c=1 free boson orbifold CFT at criticality, I would then expect chiral edge modes with $c=1$; a definite identification would require numerical simulations.