Fourier-transformed gauge theory models of three-dimensional topological orders with gapped boundaries

1. The paper presents all of the technical details of the calculations
2. The calculations appear to be technically correct
3. The results are potentially useful for advancing knowledge in the field

1. The authors need to give a clearer explanation of what can be learned from the correspondence that they establish
2. There are a number of points (see below) that are poorly explained or confusing.
3. The paper presents no concrete examples. These could be useful first to clarify certain points in an accessible way (see 2), but also to illustrate more concretely what can be learned from this approach that has not been described elsewhere in the literature. For example, perhaps the authors can use their approach to identify a class of gapped boundaries for non-abelian gauge models that were not previously known.
4. The paper has no discussion or conclusion.

The authors use the Fourier transform of general groups to re-write a conventional 3D gauge theory model with gauge group G in a basis where edges are instead labeled by group representations. They discuss how to do this both in the bulk and at the boundary, as well as how to make an explicit connection between the model in the representation basis and a 3D topological lattice model due to Walker and Wang. They use this to describe in general terms some gapped boundaries of these models that have not been described previously to the best of my knowledge.

The results that this paper obtains are certainly not surprising, and its main contributions to the literature are highly technical in nature, at the level of how in detail to carry out a mapping from one basis to the other. That said, this is a case where there is merit to writing out the technical details, and I could be persuaded that this work makes a sufficiently substantive contribution to the literature on 3D topological lattice models to warrant publication in Scipost. However, in its current form I find the paper to be very heavy on technical details and light on discussion of their physical interpretation and importance. I think that the authors could do a better job of:
(1) articulating what can be learned from this formulation, other than showing that two previously known models can be mapped one onto the other
(2) clarifying what their general framework says about simple examples, such as simple abelian theories, and then working out some specific examples of e.g. interesting gapped boundaries in non-abelian models.
(3) Making the connection to known models, and especially the Walker-Wang model, clearer (I have a number of specific comments about this below.).

ntroduction:
— I’m confused about the comparison to lattice gauge theory. Specifically: the topological models contain a different Hilbert space, in which the Gauss’ law constraint is satisfied only energetically (i.e. in the ground state). Thus the models discussed are very similar to gauge theory models with non-dynamical matter (in which charges can appear as static excitations). However, modulo this difference, the topological models are essentially a particular limit of lattice gauge theories, in which the fluctuations of the electric field are set to 0. (This is discussed, for example, in the PhD thesis of Mark De Wilde Propitious). I am similarly confused about what the authors mean when they say that lattice gauge theory has only local excitations: because of the constraint, charge excitations can live only at the ends of Wilson lines, and the constraint that the net magnetic flux passing through a closed surface must be an integer multiple of 2 pi is present in both theories. I suggest that the authors clarify this discussion.

Section 3:
— It would be helpful if the authors commented on the connections between their Fourier transform and a change of basis in a quantum gauge theory from the group basis to a basis of eigenstates of the electric field. Intuitively, I expected them to be related, but the need to introduce a braiding suggests otherwise.
— Similarly, the authors should clarify the following. In the original model, violations of the vertex term are clearly given by representations of G, while violations of the plaquette term are given by group elements. I don’t see a discussion of the excitations in the Fourier transformed model (when tails are included). This is implicit in the construction but some readers may benefit from a more detailed discussion.
— I am not sure I understand Eq. 39. In Walker and Wang’s formulation, the R factors appear in two ways. First, in the Hamiltonian, from “twisting” an edge relative to another edge with which it shares a trivalent vertex. Second, they dictate relative phases between different edge configurations, in which the configuration effectively has a twisted loop. The picture in Eq. (58) is the one relevant to evaluating the plaquette operator. Fig. 5 appears to be illustrating a situation with a crossing of the first type, in which the R matrix should be telling us about a relative phase between two configurations in a ground state. Could the authors clarify the connection between this and Eq. 38? It is not obvious to me.
— I am also confused about how we choose R. The authors seem to be implying in the discussion below (41) that there is *a* choice of R. However, for general G, I expected there to be multiple choices. First, naively I expected that taking R to be trivial (always +1) is always allowed, and that this is the choice that would have the same spectrum as the original G theory (i.e. bulk point-like charges and line-like vortex loops). Second, I would also have expected that there are other choices of R that satisfy the hexagon equations, and that these correspond to physically different theories in which the number of point particles in the bulk is reduced. For example, when the Hilbert space on each edge is just Rep(Z_2 ) = Vec(Z_2), there are two choices for R, and these yield the 3D Toric code (with point particles and vortices) and the “3D semion model” which does not have point-like bulk excitations. The authors should address this point more thoroughly in Section 3, so that readers can understand how much freedom there is in choosing R, and how the choice of R affects the final theory. In my view this is a very central question, since it determines whether we are discussing “the” Fourier transform of the model in question, or simply “a” Fourier transform…
— Please discuss the role of R at the boundary in more detail. Again, comparing the 3D Toric code to 3D semion model, we see that the former admits two distinct gapped boundaries (which are usually thought of as flux condensing and charge condensing), while the latter apparently only admits one. Hence I would have thought that there are some stipulations on what boundary condition(s) can be chosen, and that these depend on R.


Section 4:
— In the third paragraph of Section 4, the authors should specify whether they are discussing 2+1 D or 3+1 D topological orders. They should also clarify the relationship between this paragraph and the discussion of coupled layer constructions in the following paragraph.
— Loop like excitations in 3D abelian topological orders are, however, well understood. The authors should note this and include Ref. https://journals.aps.org/prb/abstract/10.1103/PhysRevB.91.165119 .
— The authors should add a discussion of the physics of these gapped boundaries. For example, given a subset of charges that can condense, what happens to loops that are brought to the boundary?


Section 5:
— I don’t understand what the authors mean when they say: “no loss of information is
“ the WW model does not contain charge and dyon excitations, just like the LW model.” Dyons in 3D theories would require point-like monopoles carrying charge — but the discrete gauge theories discussed here don’t have magnetic monopoles. So why are we talking about dyons at all?

Ask for major revision

1. The paper is well-organized and clearly explains the correspondence between lattice quantum field theory and the state-sum construction of TQFT.
2. The Frobenius algebra-based boundary theory is extended to the 3d model.
3. An explicit Fourier transformation between the G-lattice gauge theory and the Walker-Wang model is provided.

There are many typos that need to be revised.

In this work, the author investigates the gapped boundary theory of a 3d lattice model of topological order. Similar to the 2d case [JHEP01(2018)134], the construction is based on the Frobenius algebra within the input unitary fusion category $\mathrm{Rep}(G)$. Through the Fourier transform, which reformulates the original model into the fusion basis, they argue that the 3d lattice gauge theory is equivalent to the Walker-Wang model. This generalizes the well-established result in 2d, where lattice gauge theory is equivalent to the Levin-Wen string-net model.

The results are interesting and the paper is well-written. I believe this work meets the publication criteria for Sci-Post.

Before publication, I believe the paper requires thorough proofreading, as there are some typos that need to be addressed, and there may potentially be more.

P 5, Line -2 of the first paragraph, "oriented cubic lattice $\Gamma$ which boundaries", "which" should be "with".

P 6, Line 2 below Eq.(7), "operators are all commute with each other".

P 7, Caption of Fig 2, "that terminat on the boundary".

P 7, Line above Eq. (11), "the duality map and its inverse has presentations".

P 10, Line -4, "Additionaly" should be "Additionally".

P 13, Caption of Fig 4, "by contracting there indices", "with an free end".

P 15, Line 2 of the second paragraph, "larger Hilber space".

P 15, Caption of Fig 5, "as are edges labeleds by".

P 27, Above Fig 9, "comparing with the LW model."

Ask for minor revision