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Fusion Surface Models: 2+1d Lattice Models from Fusion 2-Categories

by Kansei Inamura, Kantaro Ohmori

Submission summary

Authors (as registered SciPost users): Kansei Inamura · Kantaro Ohmori
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-06-30 07:44
Submitted by: Ohmori, Kantaro
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We construct (2+1)-dimensional lattice systems, which we call fusion surface models. These models have finite non-invertible symmetries described by general fusion 2-categories. Our method can be applied to build microscopic models with, for example, anomalous or non-anomalous one-form symmetries, 2-group symmetries, or non-invertible one-form symmetries that capture non-abelian 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 three-dimensional classical statistical models, which are 3d analogues of the 2d Aasen-Fendley-Mong height models. In the construction, the "symmetry TFTs" for fusion 2-category symmetries play an important role.

Current status:
Awaiting resubmission

Submission & Refereeing History

Reports on this Submission

Anonymous Report 2 on 2023-9-29 (Invited Report)


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. Well-referenced with extensive context


A few statements which are not directly obvious to the non-initiated reader or lack details (see below)


This is an excellent paper constructing in a general and conceptual way statistical 3d models and quantum 2+1d models with 2-category 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 non-expert 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. p36-37, 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 p29-30 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 Aasen-Fendley-Mong 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.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

Anonymous Report 1 on 2023-8-28 (Invited Report)


1. main topic of broad interest in hep-th, cond-mat, and math-phys
2. interesting new results: categorification of earlier work on lattice models and generalized symmetries
3. clearly written


1. few mathematical imprecisions (which can mostly be easily resolved, see below)


This excellent preprint constructs new lattice models in 2+1 dimensions from data in a given spherical fusion 2-category C. This is already of considerable interest, but in addition the authors also exhibit various invertible and non-invertible symmetries of their model, also constructed from C. This is extremely timely. The construction can be viewed as a non-trivial categorification of the work of Aasen-Fendley-Mong 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 3-dimensional TFTs of Turaev-Viro-Barrett-Westbury 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 Turaev-Viro-Barrett-Westbury type. Similar remarks are believed to be true in one dimension higher. In particular, the 4-manifold invariants of Douglas-Reutter 2018 are constructed from spherical fusion 2-categories (and they are believed to be the partition functions of 4d oriented TFTs). Since the authors make heavy use of the Douglas-Reutter 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 non-invertible symmetries described by 3d oriented TFTs are relevant, then the "orbifold data" of Carqueville-Runkel-Schaumann 2017 are the relevant non-invertible 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 Carqueville-Runkel-Schaumann 2017 (and Carqueville-Müller 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 Barrett-Meusburger-Schaumann 2012 seems to be relevant here. Also in (2.4) and similar black-and-white 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 Barrett-Meusburger-Schaumann 2012.

10. Page 18, first sentence of last full paragraph: It is a _spherical_ pivotal structure.

11. Page 22, (3.1): Douglas-Reutter 2018 construct 4-manifold 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 2-category.

18. Page 37, first sentence of Section 4.4.2: Usually, 1-form 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 non-trivial morphisms.

20. Page 43, last paragraph: Please explain in what sense the 1-morphisms 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 Carqueville-Müller 2023, which appeared after the preprint) rigorously describe a broad class of defects in TVBW models.

  • validity: high
  • significance: high
  • originality: high
  • clarity: high
  • formatting: excellent
  • grammar: good

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