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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.
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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.
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.