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Fusion Surface Models: 2+1d Lattice Models from Fusion 2Categories
by Kansei Inamura, Kantaro Ohmori
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Authors (as registered SciPost users):  Kansei Inamura · Kantaro Ohmori 
Submission information  

Preprint Link:  https://arxiv.org/abs/2305.05774v2 (pdf) 
Date submitted:  20230630 07:44 
Submitted by:  Ohmori, Kantaro 
Submitted to:  SciPost Physics Core 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We construct (2+1)dimensional lattice systems, which we call fusion surface models. These models have finite noninvertible symmetries described by general fusion 2categories. Our method can be applied to build microscopic models with, for example, anomalous or nonanomalous oneform symmetries, 2group symmetries, or noninvertible oneform symmetries that capture nonabelian anyon statistics. The construction of these models generalizes the construction of the 1+1d anyon chains formalized by Aasen, Fendley, and Mong. Along with the fusion surface models, we also obtain the corresponding threedimensional classical statistical models, which are 3d analogues of the 2d AasenFendleyMong height models. In the construction, the "symmetry TFTs" for fusion 2category symmetries play an important role.
Current status:
Reports on this Submission
Strengths
1) The paper is of foundational importance as it provides explicit 3d (resp. 2+1d) lattice models in which arbitrary fusion2category symmetries are implemented explicitly and exactly.
2) Several wellknown models arise as special cases of the construction presented in this paper.
Report
The authors give an explict construction of a 2d quantum spin model and a related 3d statistical model in terms of a 3d state sum topological quantum field theory, called DouglasReutter theory. The input is a spherical fusion2category, and the remarkable aspect of the construction in the present paper is that the resulting models automatically realise the input fusion2category as topological symmetry. That is, in the 3d statistical model, the objects describe topological surface defects, the 1morphisms topological line defects, and the 2morphisms topological point defects. The implementation in the 2d quantum spin model is analogous. The paper generalises a construction by Aasen et al. which happens in one dimension less.
In my view, the paper is of foundational importance as it provides explicit 3d (resp. 2+1d) lattice models in which arbitrary higher symmetries (as described by an arbitrary fusion2category) are implemented explicitly and exactly, and the authors demonstrate that several wellknown models arise as special cases of their construction. The paper should certainly be published.
The presentation is necessarily somewhat technical, as this is the case already for the description of fusion2categories. The authors write a wellreadable extended introduction which explains their results. In particular the review of the model by Aasen et al. in terms of fusion1categories is very helpful to set the stage.
Requested changes
The changes below are just suggested, not requested.
1) p.2: "In 1+1 dimensions, finite noninvertible symmetries are generally described by fusion categories [6, 7, 9]."  Here I find the word "generally" misleading. Indeed, in general one does not expect a fusion category (which is both finite and semisimple), and instead one expects symmetry categories which are neither finite nor semisimple. Also, the first reference I am aware of which states how topological defects in a 2d quantum field theory give rise to a (pivotal) 2category (of which a fusion category is a special case) is the paper by Davydov et al. 1107.0495 (your reference [26])  indeed, that paper does not exclusively deal with topological field theories.
2) p.2: "In higher dimensions, finite noninvertible symmetries are expected to be described by fusion higher categories"  As in the 2d comment above, I do not share this expectation, and instead expect that also in higher symmetries the categories describing topological defects are neither finite nor semisimple. The term "fusion higher categories" appears in several papers, but as far as I am aware, there is no mathematical definition of what this means beyond fusion2categories. The present formulation suggests that "fusion higher categories" are already an existing concept.
3) p.2: footnote 3 "The statisticalmechanical model presented by them had also appeared in [124]."  I was under the impression that Aasen et al [33] is more general than [124] as [33] allows general fusion categories, while [124] require the existence of a fibre functor.
4) p.3: section 1.2  Here fusion categories are described, but then on p.5 also quantum dimensions are presented as part of the data. These are not part of a fusion category, but require more structure, namely that of a spherical fusion category. It might be worth to point this out to avoid confusion. The same comment applies to p.7 where "a fusion category C" is stated as the input for the AFM model, but the definition of the model refers to quantum dimensions, and so one is using a spherical structure on C.
5) p.6: "Ising category. A basic example of a noninvertible fusion category arises in the critical Ising model [7, 130]."  Your reference [22] (Fröhlich et al, condmat/0404051) would also be appropriate here.
6) p.7: footnote 9 "The idea of using the TuraevViro model to construct and study 2d statisticalmechanical systems had already appeared in [21]"  Is that the reference you wanted to give? That paper is concerned with the relation of 3d topological quantum field theory and 2d conformal field theory.
7) p.7: footnote 10: Technically, boundary conditions are given by module categories together with a module trace. Just giving the module category does not determine the value of the TQFT on a 3ball.
8) p.9 "The Hamiltonian H of the model is derived by expanding the transfer matrix of the AFM height model as T = id_H − ϵ H + O(ϵ^2)..."  What is epsilon here? The AFM model as it is reviewed does not contain an anisotropy parameter, so what limiting process is being referred to here when writing O(eps^2)?
9) p.11 "In particular, we can naturally expect that finite generalized symmetries in 2+1 dimensions are generally described by fusion 2categories."  Provided these symmetries are also semisimple, which I would not expect to be the case in general.
10) p.17 "Nonchiral topological phases with fusion 2category symmetries."  Is this related to the commuting projector model by Huston et al. in 2208.14018?
11) p.18 "In particular, the unit object I of a fusion 2category is simple."  That makes it sound like a consequence of the previous sentence. But is it not an independent requirement on a fusion2category?
12) p.23 "In this case, eq. (2.7) implies that we can shrink the left boundary N × {0} to a point"  Could you expand this a little bit? E.g. why is the topology of the boundary you are shrinking not important for the state that is being generated. And does this procedure not destroy how the fusion2category symmetry is implemented?
13) p.26 "We call the 3d classical statistical model defined by the above partition function a 3d height model."  Could you elaborate more on this? In the 2d situation, choosing the input fusion category and the object rho appropriately (e.g. the fundamental representation in affine su(2), or anything with an A_r fusion graph), the dynamical label can only change by one unit per lattice spacing, which in my understanding is the reason for the name height model. Does a similar example exist in the 3d case, i.e. that the model describes a height variable in a 3d volume?
14) p.29 "The appropriate quantum counterpart of the 3d height model is obtained by restricting the state space of the above 2+1d model to a specific subspace of H."  Here the the point is made that the relation of the transfer matrix and Hamiltonian description requires the transfer matrix to have no kernel. In the present case, this leads to the requirement to pass to a smaller Hilbert space H_0, which is formulated as the image of a projector T_0. The reasoning is that T can be replaced by T_0 T T_0. But this only shows that the complement of H_0 lies in the kernel of T, not that T has no kernel on H_0. Question: Why is the reduction form H to H_0 sufficient, i.e. why is T injective on H_0?
15) p.29 "the Hamiltonian (4.5) does not mix states in H0 and those in H \ H0"  The formulation H \ H0 appears in a number of places in the paper. To me this is confusing as H \ H0 is no longer a vector space. Technically, this formulation is also wrong: Take v in H0 and w in the kernel of T_0, then v+w is in H\H0 but it gets mapped to H0. It may be better to write H as a direct sum of H0 and H' or so, and say H0 and H' do not get mixed. Similar comments apply to all other occurences of H \ H0 in the paper.
16) p.30 "gives the partition function of the 3d height model when ϵ ≪ 1"  Where does the parameter epsilon appear in the 3d height model you define? And where do you show this limiting statement in the paper? I agree that the models are similar, but the concrete limit statement seems stronger.
17) p.37 "whose representative is chosen to be a unit object I"  Is that B as a regular module over itself in this case? If so, one could point it out for concreteness.
18) p.37 "The fusion 2category Mod(B)"  The construction presented in the paper requires a spherical fusion2category, at least the DouglasReutter theory does. Could you explain and/or provide a reference how the spherical structure is defined in this case?
19) p.41 "Anomalies of an invertible 1form symmetry A are characterized by the Fsymbols and Rsymbols"  This defines the braided monoidal structure. Do I understand the setting correctly in that you actually need a ribbon category here? If so, maybe add a comment on the ribbon structure. Or, equivalently, say what your spherical structure is. Do you assume dim(a) = 1 for all ain A? E.g. in the 2nd point in (5.5) it is natural to also consider dim(eta)=1, which then describes the braided category of super vector spaces with supertrace as spherical structure.
20) p.42 Eqn (5.6)  What happened to the sum over A which is still present in (5.4)? I am guessing that one can always drop the unit e of A from the sum. If so, it might be clearer to point this out.
21) p.43 section 5.3  A general question: Do you happen to know what the braided category of anyons is in your 2d topological phases? E.g. does one get B if you choose C = Mod(B) for a ribbon category B?
Strengths
1. Broad and relevant results for 3d/2+1d models
2. Clearly written
3. Detailed figures explaining the relevant geometric intuition behind the formalism
4. Concrete examples with explicit expressions
5. Wellreferenced with extensive context
Weaknesses
A few statements which are not directly obvious to the noninitiated reader or lack details (see below)
Report
This is an excellent paper constructing in a general and conceptual way statistical 3d models and quantum 2+1d models with 2category symmetry. Such constructions provide valuable insights into the often challenging physics of d>2 systems. The main technical steps are exposed in a clear way, with many detailed figures making the paper accessible even to the nonexpert reader. The unitarity of the models is also carefully studied. The authors display an extensive knowledge of the relevant literature and the broader mathematical and physical context. The acceptance criteria are clearly met. Some possible corrections/improvements are suggested below.
Requested changes
Changes/Comments/Suggestions/Questions :
1. p3, third paragraph "an ’t Hooft anomaly" (typo)
2. p10, below eq. (1.6) "Here, Fint denotes the set of a simple object..." > the set of all simple objects... ?
3. p22, DR subscript in eqs. (2.6)(2.7)
4. p3637, eq. (4.30) and around. It is not entirely clear from the discussion what is the exact status of eq. (4.30). Is it proven or do we simply expect it to hold in all physically sensible cases ? If it is not proven, is it imposed or conjectured under some assumptions ?
5. The discussion p2930 motivating the introduction of the restricted space $\mathcal{H}_0$ (which is a crucial piece of the construction) could be more detailed. In particular:
a. Recalling the definition of the anisotropic limit for example as in eq. (3.44) in AasenFendleyMong could be useful.
b. The transfer matrix cannot be written as $\hat{T}=\exp(\epsilon H)$ because it has a large kernel which is exactly projected out by $\hat{T}_0$. It seems that $\hat{T}$ and $\hat{T}_0$ should commute. Is it the case ? If not why $\hat{T}=\hat{T}_0\epsilon H \hat{T}_0+O(\epsilon^2)$ and not $\hat{T}=\hat{T}_0\epsilon\hat{T}_0 H \hat{T}_0+O(\epsilon^2)$ ? It should be correct if $H$ is hermitian but is it true in general ?
c. Is there some mathematical and/or physical intuition as to why the transfer matrix only propagates states of $\mathcal{H}_0$ ? Some discussion would be welcome as it is a distinguishing feature of the 2+1d construction and does not seem to happen in 1+1d.
Author: Kansei Inamura on 20240222 [id 4318]
(in reply to Report 2 on 20230929)
We would like to thank the referee for the constructive comments and helpful suggestions to improve the manuscript. We implemented the requested changes in the updated version of our paper. Please see below for our responses to the comments in the report.

To be precise, the prefix 't seems to be pronounced as [ət], and hence the indefinite article "an" in front of 't Hooft would be correct, please see websites such as https://webspace.science.uu.nl/~hooft101/ap.html and https://en.wikipedia.org/wiki/%27t.

We clarified the definition of $\mathcal{F}_{\mathrm{int}}$ below eq. (1.6).

Fixed.

Equation (4.30) holds in general. We added the derivation of eq. (4.30) at the end of Section 4.3.
5.a. We clarified the meaning of the anisotropic limit in the last three sentences of the paragraph that contains eq. (4.6).
5.b. The operators $\hat{T}$ and $\hat{T}_0$ commute with each other because we have $\hat{T} = \hat{T}_0 \hat{T} \hat{T}_0$ and $\hat{T}_0^2 = \hat{T}_0$. We explained this in footnote 36.
5.c. We added a way to intuitively understand the projector $\hat{T}_0$ and state spaces $\mathcal{H}$ and $\mathcal{H}_0$ on page 30. There, we also added sentences explaining why this new feature arises in this dimension.
Strengths
1. main topic of broad interest in hepth, condmat, and mathphys
2. interesting new results: categorification of earlier work on lattice models and generalized symmetries
3. clearly written
Weaknesses
1. few mathematical imprecisions (which can mostly be easily resolved, see below)
Report
This excellent preprint constructs new lattice models in 2+1 dimensions from data in a given spherical fusion 2category C. This is already of considerable interest, but in addition the authors also exhibit various invertible and noninvertible symmetries of their model, also constructed from C. This is extremely timely. The construction can be viewed as a nontrivial categorification of the work of AasenFendleyMong 2020 (here and below all years refer to the first arxiv version of the given paper) on lattice models in 1+1 dimensions and their symmetries described by 3dimensional TFTs of TuraevViroBarrettWestbury type.
The acceptance criteria are clearly met as soon as the authors will have addressed the comments listed under "Requested changes" below.
Requested changes
1. Page 2, first full paragraph: Please clarify what models precisely you have in mind here. For example, ordinary fusion categories correspond to 3d _framed_ TFTs of state sum type, while _spherical_ fusion categories produce 3d oriented TFTs of TuraevViroBarrettWestbury type. Similar remarks are believed to be true in one dimension higher. In particular, the 4manifold invariants of DouglasReutter 2018 are constructed from spherical fusion 2categories (and they are believed to be the partition functions of 4d oriented TFTs). Since the authors make heavy use of the DouglasReutter construction, one could suspect that orientations play a bigger role than framings. Please clarify whether this is the case.
2. Page 2, first sentence of second full paragraph: If noninvertible symmetries described by 3d oriented TFTs are relevant, then the "orbifold data" of CarquevilleRunkelSchaumann 2017 are the relevant noninvertible symmetries.
3. Page 5, itemization: Note that in a general fusion category, left and right quantum dimensions can be different. It seems that "fusion category" should be "spherical fusion category".
4. Page 10, text before (1.7): Please explain why the action of a on an element in the Hilbert space spanned by elements in Figure 9 again yields a linear combination of such vectors. Naively, one could expect a condition on how a fuses with rho etc.
5. Page 16, last sentence in paragraph on "Symmetry": Is this statement proven somewhere in the paper?
6. Page 16, first sentence of last item: Since everything comes with orientations, maybe the "orbifold completion" of CarquevilleRunkelSchaumann 2017 (and CarquevilleMüller 2023, which appeared after the preprint) is more to the point here. Similarly in the next item, in Footnote 26, and in Section 5.3.
7. Page 17, second sentence in Section 2.1: This is not expected to be true in general, e.g. not for twisted sigma models.
8. Page 18, second full paragraph: The work of BarrettMeusburgerSchaumann 2012 seems to be relevant here. Also in (2.4) and similar blackandwhite 3d diagrams.
9. Page 18, text before (2.1): For this to be a definition, it would be necessary to make sense of spheres, including caps and cups. This can be done with the results of BarrettMeusburgerSchaumann 2012.
10. Page 18, first sentence of last full paragraph: It is a _spherical_ pivotal structure.
11. Page 22, (3.1): DouglasReutter 2018 construct 4manifold invariants, but not quite a full TFT, and also no boundary conditions for such a TFT. It is expected that such a TFT and boundary conditions can be constructed. Please clarify what exactly Z_DR and boundary condition mean here.
12. Page 25, first line: What is the relative height of the new vertex pt to make pt*[ijkl] oriented?
13. Page 29, first full sentence after (4.3): Why do these matrices commute?
14. Page 29, (4.4): Instead of using the index p three times, one could use three indices y, g, b for the three colors.
15. Page 29, last paragraph: Why is $\hat T_0$ a projector?
16. Page 30, first paragraph in Section 4.2: Please explain the origin of the name "reflection positivity" here.
17. Page 30, Footnote 34: Please clarify that stacking with an invertible 2d TFT can change the (pivotal) structure of the spherical fusion 2category.
18. Page 37, first sentence of Section 4.4.2: Usually, 1form symmetries come from representations of delooped groups. Please explain that here it is meant in a more general sense, and why it makes sense to use the same name in the more general sense.
19. Page 39, first paragraph of Section 4.4.3: Please explain in what sense 2Vec$_G^\omega$ does not have nontrivial morphisms.
20. Page 43, last paragraph: Please explain in what sense the 1morphisms f and g (which are by definition module functors in this example) are given by the object $\sigma$.
21. Page 46, Footnote 47: Meusburger 2022 (and CarquevilleMüller 2023, which appeared after the preprint) rigorously describe a broad class of defects in TVBW models.
Author: Kansei Inamura on 20240222 [id 4317]
(in reply to Report 1 on 20230828)
We would like to thank the referee for the careful reading and constructive feedback. We revised the manuscript to address the referee's comments. Please see below for our responses to the requested changes listed in the report.

All fusion categories and fusion 2categories in the paper are supposed to be spherical. We clarified this point in the last two sentences of footnote 2.

We believe that the "orbifold datum" appearing in [CarquevilleRunkelSchaumann 2017] represents the data of condensation, or gauging. While we find it not particularly relevant to this paragraph, it becomes pertinent when discussing the condensation/gauging/orbifolding of the DR theory on page 48. Therefore, we cited it in conjunction with [GaiottoJohnsonFreyd 2019] in the updated manuscript, although we had already cited this reference as [172] in the same section of our original manuscript. We appreciate the referee for pointing out this reference.

We commented on the sphericity in the last item below eq. (1.1).

We elaborated on the action of $a$ in the caption of Figure 10.

We explained why the symmetry operators commute with the Hamiltonians in the last sentence of the paragraph on "Symmetry." Although we did not explicitly demonstrate the commutativity of these operators in the paper, we hope it should be clear from the construction that our models have the said fusion 2category symmetries.

We clarified what we mean by the condensation completion in footnote 22. We expect that the condensation completion is equivalent to the orbifold completion in the case of our interest because a $\Delta$separable Frobenius algebra in a pivotal fusion category is automatically symmetric due to [FuchsRunkelSchweigert hepth/0204148, Corollary 3.10] and [Schaumann 1206.5716, Lemma 2.9], see also [CarquevilleMüller 2307.06485, Remark 2.6].

We believe that our expectation should hold in unitary theories, which exclude the example that the referee mentioned. We added "In unitary theories" to the second sentence in Section 2.1 to clarify this point.

We included the reference [BarrettMeusburgerSchaumann 2012] in the first paragraph of Section 2.1.

We clarified below eq. (2.1) that the caps and cups are implicitly used in the definition of the quantum dimension.

A pivotal structure, even if not spherical, guarantees that the left and right dimensions of a 1morphism agree with each other. The pivotal structure is said to be spherical if the quantum dimension of every object is equal to the quantum dimension of its dual. Our terminology follows that of [DouglasReutter 2018, Sections 2.2 and 2.3].

We added footnote 29 to clarify the meaning of $Z_{\mathrm{DR}}$ in the presence of boundaries. We would also like to point out that the state sum invariants of oriented 4manifolds with boundaries are defined in [Walker 2021], which we briefly review in Section 2.2 of our paper.

The relative height of $\mathrm{pt}$ is lower than any other vertices of a 4simplex $\mathrm{pt}*[ijkl]$. We articulated this in a sentence above eq. (3.4).

The local transfer matrices on plaquettes of the same color commute with each other because they do not have overlaps. We clarified this point in the first full sentence below eq. (4.3).

Fixed.

The reason why $\hat{T}_0$ is a projector is that the Dirichlet boundary decorated by the trivial coloring is topological in the imaginary time direction. We elaborated on this point in the fourth full sentence below eq. (4.6).

We explained the origin of the name "reflection positivity" in footnote 40.

We explained that stacking with an invertible 2d TFT multiplies the quantum dimension by its partition function on a sphere, see footnote 39 (labeled as footnote 34 in the original manuscript). The stacking of an invertible 2d TFT and an object of a fusion 2category is another object of the same fusion 2category equipped with the same pivotal structure.

We clarified what we mean by 1form symmetries in footnote 48.

The fusion 2category $2 \mathrm{Vec}_G^{\omega}$ has no nontrivial 1 and 2morphisms in the sense that the Hom category $\mathrm{Hom}(g, h)$ is $\mathrm{Vec}$ when $g = h$ and empty otherwise. We clarified this point in the first paragraph of Section 4.4.3.

The category of module endofunctors of the regular module over a ribbon 1category $\mathcal{B}$ is braided equivalent to $\mathcal{B}$, and hence 1morphisms $f$ and $g$ are given by objects of $\mathcal{B}$. We explained this below eq. (5.10).

We included the reference [Meusburger 2022] in footnote 54, which was originally footnote 47.
Author: Kansei Inamura on 20240222 [id 4319]
(in reply to Report 3 on 20240102)We would like to thank the referee for the insightful comments and questions, which helped us improve the presentation of our results. The following is our response to each of the comments in the report.
We clarified in footnote 2 that we focus on finite noninvertible symmetries of unitary bosonic theories, which we expect to be described by fusion (higher) categories. We also cited [Davydov et al. 1107.0495] in the sentence where we first mentioned fusion category symmetries in 1+1 dimensions.
As in the response to the comment above, we clarified that we consider finite noninvertible symmetry in unitary theories. Also, we clarified where the definition of fusion higher categories is given or proposed in footnote 3.
As the referee pointed out, while the construction in [124] is based on the TuraevViro TQFT as in [33], the one in [124, Section 7.2] depends on a fiber functor. We modified footnote 4 accordingly.
We articulated in footnote 2 that we exclusively work on spherical fusion categories in our paper. We also commented on sphericity when we introduced the quantum dimension. We also clarified at the bottom of page 7 that the input fusion category of the AFM height model is spherical.
We added the reference [23] (originally referenced as [22]) to [7, 130] on page 6.
We replaced the reference [21] in footnote 10 with the paper [124] by Freed and Teleman.
We incorporated the comment.
We elaborated on the anisotropic limit of the AFM height model in footnote 15.
We clarified that we restrict our attention to symmetries of unitary theories, which we expect to be semisimple.
To our understanding, the commuting projector model in 2208.14018 is the 3+1d WalkerWang model with boundary, which realizes a surface topological order that is Witt equivalent to the input modular category $\mathcal{A}$ of the bulk WalkerWang model. The boundary condition of their model is specified by the choice of an $\mathcal{A}$enriched fusion category $\mathcal{X}$, which is indeed an algebra object of the fusion 2category $\mathrm{Mod}(\mathcal{A})$. Their model put on a slab as in the symmetry TFT construction realizes a purely 2+1d nonchiral topological order, which is also realized in our model if we choose $\mathcal{C} = \mathrm{Mod}(\mathcal{A})$ and $A = \mathcal{X}$.
We clarified that the simplicity of the unit object of a fusion 2category is an independent requirement.
We explained why we can shrink the Dirichlet boundary to a point when we compute the partition function of the DouglasReutter TFT. We note that we do not shrink the boundary to a point when the boundary is decorated by a topological defect network that implements the fusion 2category symmetry.
We named our model 3d height model simply because it is a natural threedimensional generalization of the 2d height model. We explicitly mentioned this below eq. (3.9). We are not aware of an example of a fusion 2category that is similar to the representation category of the affine Lie algebra $\mathfrak{su}(2)_k$.
The transfer matrix $\hat{T}$ is not necessarily injective on $\mathcal{H}_0$. However, this is not an issue. The point of restricting the state space to $\mathcal{H}_0$ is that once we pass to the smaller space $\mathcal{H}_0$, we can find the completely anisotropic limit where the transfer matrix becomes the identity and we can thus derive the quantum Hamiltonian by expanding the transfer matrix $\hat{T}$ around this limit. We elaborated on this point at the end of the paragraph of eq. (4.6).
Thank you for pointing out this mistake. We replaced $\mathcal{H} \setminus \mathcal{H}_0$ by $\mathrm{ker}(\hat{T}_0)$.
We clarified how we define the anisotropic limit of the 3d height model below eq. (4.6).
We clarified that the unit object of $\mathrm{Mod}(\mathcal{B})$ is the regular $\mathcal{B}$module in the first paragraph of Section 4.4.2.
We added a reference to the spherical structure on $\mathrm{Mod}(\mathcal{B})$ in footnote 49.
We suppose that $\mathrm{dim}(a) = 1$ for all elements $a$ of a finite abelian group $A$. We clarified this point below eq. (5.1).
The unit element of $A$ gives rise to a constant term in the Hamiltonian and thus can be removed from the summation without loss of generality. This was explained in footnote 45 of the original manuscript. We moved this explanation to the main text below eq. (5.6) in the revised manuscript.
As we briefly mentioned in the reply to comment 10, if we choose $\mathcal{C} = \mathrm{Mod}(\mathcal{B})$ and $A = \mathcal{X}$ for a ribbon category $\mathcal{B}$ and a $\mathcal{B}$enriched fusion category $\mathcal{X}$, our commuting projector model in Section 5.3 realizes a nonchiral topological order described by the Drinfeld center $Z(\mathcal{X})$ of $\mathcal{X}$. In particular, since the Hamiltonian of the model in Section 5.3 is the sum of commuting projectors, we do not believe that this specific model can realize a chiral topological phase for any choice of $\mathcal{C}$ and $A$. However, we do expect that our general model constructed in Section 4 can realize chiral topological phases in some parameter regions.