Instanton Density Operator in Lattice QCD from Higher Category Theory

1. Well-written motivation.
2. Detailed review of the status of the problem.
3. Novel math tools.

1. Lack of clear statement of results.
2. The validity of the proposal is not tested.

A well-known problem is that the most popular lattice discretization of gauge theories and sigma-models do not respect the topology of these models. In the continuum the space of classical configurations has several topologically distinct components, on the lattice the space of configurations is connected. This disagreement is supposed to be resolved only after the continuum limit is taken. It would be desirable to have a lattice discretization which correctly reflects the expected topological classification before taking such a limit. For some models such discretizations exist, but the most interesting case (non-abelian Yang-Mills in 4d) remains open. The paper under review proposes a lattice discretization of 4d Yang-Mills with a microscopically defined instanton number, as well as other related models (sigma-model with target S^3). The proposal is quite complicated and is not clear to the reviewer whether it "works". I am not even sure what this would mean. The only way to test the proposal, as far as I can see, is to perform numerical simulations of the model and verify that it behaves qualitatively the same as the usual discretization of 4d Yang-Mills. Section 5 provides a sort of mathematical motivation for the proposal, but it is really hard for me to see the connection between these abstract considerations and the concrete proposal in Section 4. I would also add that the proposal itself is spread over many pages. It would be a good idea to formulate it more concisely.

1. Formulate the proposed discretization of 4d Yang-Mills clearly and concisely.
2. If possible, provide some tests of the proposal.
3. The mathematical part of the paper does not really help to understand the proposal. While it informs the author's thinking, it does not make the paper more convincing and just adds to its length. I suggest dropping it altogether.

Ask for major revision

1- intriguing motivation
2- detailed motivation and review of background
3- potential high impact, potential high novelty
4- potentially relevant sophisticated mathematical tools identified correctly

1- key definitions remain hard to parse
2-consistency checks of key definitions seem lacking
3-relation between key sections 4 and 5 remains vague

+++ General statement +++

The author's motivation and broad proposal is in intriguing, and success of this ambitious approach would have high impact. From the introductory sections, I was highly motivated to see the proposed solution.

However, the main section 4 still is largely motivational in style and the key fine-print of the proposed definitions remains unclear to me after spending some time mulling over it.

I would urge the author to restate the key definitions (64) and (71) in something closer to math style, where every ingredient is explicitly declared. This concerns particularly the arguments of the key term $W_2(..)$ and $W_3(...)$, respectively. Define them, concretely.

Morever, consistency checks are missing: After introducing all the new fields, there should be an argument that the resulting lattice models are indeed locally still suitably equivalent to the ordinary ones which they are meant to enhance. This seems far from obvious and needs an argument. (Note that a broad appeal to Elitzur's theorem advertized in the introduction has little bearing on this, as it does not concern the higher gauge fields nor the various constraints introduced by the authors)

Section 5 is a review of sophisticated mathematics that the authors plausibly argues to be necessary for coming up with these definitions, and it may serve a purpose as exposing some of this math to a math-remote pure physics community -- but its actual relation to the proposal in section 4 is left quite unclear.

It may be worth going for a much shorter article which just defines the proposed lattice models concretely. If the category/homotopy theory helps with establishing their intended behaviour then add that proof explicitly, but if the category/homotopy theory is just a vague motivation for the author, then leave it away and instead focus on tangible results.

+++ List of comments +++

Here is a list of random comments going linearly through the document.

general comment:
The text speaks throughout about "higher category theory", but what it really uses is just "higher groupoid theory" also known as (simplicial-)homotopy theory. This conflation is common in the literature, but it may be worthwhile to beware of it.In simplicial homotopy theory, the notion of "anafunctors" is much further developed, known as maps out of cofibration resolutions.

p. 4 "theoretical appeal[s]"

p. 4: appeal to Elitzur's theorem:
However, Elitzur's theorem applies to ordinary gauge transformation, while the lattice models that the author is about to introduce have (also) higher gauge symmetries. It is maybe not a priori clear that the analog of Elitzur's theorem will still apply to these.

p. 9 "only [a] sketched"

p. 9 "that we care [about]"

p. 12 "in the below"
either: "below" or "in the following"

p. 12 "we do not say ... is constant"
the mathematical term is: it is *locally constant*

p. 13: 
Figure below (7): It is hard to see from the figure on the right what is being illustrated.

p. 21 equation (18):
probably "f_c" and "s_c" should be "f_p" and "s_p"

p. 39: "at the 2d boundary between two patches"
What must be meant instead is "the is 2d overlap of two patches"

p. 39, 40: "Stoke's " must be "Stokes' "

p. 41 Section 4.1

Just to note that only now, over 40 pages into the text, does the first promised definition begin to slowly emerge. This lack of conciseness not only puts a burden on the reader at this point but may also not be helpful for the author's own thought development.

At some point the motivational commentarey must be set aside and an actual, concrete detailed definition must be written down and checked to make sense.

p. 41 "To understand why..."
To understand this one may simply and immediately observe that the fiber over +1 or -1 is a singleton but the fiber over any other point is a set of two elements.

p. 42 "What this extra S^2 does... will be explained later."
Why not just say where it will be explained.

p. 42: "Note that while m_l is a two-valued label it by no means for a Z_2 group"
The technical term for this is that m_l is an element of a *torsor* over Z_2.
And the words "by no means" are misleading: A simple means makes any torsor a group: namely the choice of any one element.

p. 42: "each patch of Y now has an open boundary"
Strictly speaking, since the components of Y are open balls (as opposed to closed balls), they do not have any boundary in the technical sense, much less an open boundary.

p. 43 "the contributions from all other continuum paths"
This does not seem to be true: There are paths that stay neither entirely in the patch SU(2)\{+1} not entirely in the patch SU(2)\{-1}.

This is the beginning of me feeling increasingly unsure about the definition that is incrementally being sketched here.

p. 43 "pick a representative path"
It is unclear why to pick any representative paths at all.

p. 46 "it is easy to picture the following desired properties for mu"

I find the logic now hard to follow. Easier than incrementally motivating a definition would be to just state it and then discuss that it satisfies desired properties.

It looks like a definition of the argument of W_2 in (59) is being indicated. And this is indeed needed to make of the key claim (64) to follow. But what exacty the definition of (59) is I am not sure from reading the text. This ought to be clarified.

p. 50 equation (64)
This seems to be the statement of the first main new proposal of the article -- it might want to be highlighted as such.
I am left wondering to know that definition (64) is consistent. Apart from it reproducing the intended topological charges -- which the author has motivated but maybe not proven -- one also needs to check backwards-compatibility, namely that all the new degrees of freedom added (notably the hat-n_l) do not locally change the intended dynamics of the sigma model.
This may be clear to the author, but it is far from clear to this referee at this point.

p. 54 equation (71)
This seems to be the statement of the second main new proposal of the article -- it might want to be highlighted as such.
The same comments apply as to the analogous statement (64) on p. 50 above, only more urgently so, since the actual definition of the term W_3(...) now is even less clear to me than the previous W_2(term).
Also, reading ahead I get the impression that discussion around (126) on p. 106 is meant to be relevant here in giving this definition of (64). If so, this ought to be said. If not, it needs to be said what else (126) is about.

p. 52: "Recall in the case... patches were chosen to be invariant under conjugation..."
Better to give an explicit equation number from which the reader can specifically recall this, since it is not easy to find.

p. 53: 
At this point I'd like to see a concise definition of the construction. The discussion of the ingredients has been spread out over many pages -- which may be good for motivation --- but it makes it hard to know at a glance what all the symbols mean.
Maybe one could point ahead for such a definition, to around (126)

p. 53, equation (71): 
The first integral sign seems to be lacking its "dg"

p. 71, item 2:This is maybe the first point that the notation "BEG" appears, which is used again many pages later (pp. 106) without (re-definition)
Clarification is needed for what is meant, and also for this choice of notation.
Note that usually, "EG" denotes the universal G-bundle, or else the simplicial complex 
 (WG)_n = G_n x G_{n-1} x ... x G_0
This happens to be a simplicial group (arXiv:1204.4886) hence has a further delooping via a bi-simplicial construction, which would deserve to be called B E G.But this does not appear to be what the author is after here.
For one, EG has contractible homotopy type, and hence so does its delooping, which would make it unsuitable for the author's purpose.

p. 106, equation (126): 
Best to (re-)state the definition of "Y" 
an open cover of G?
a more general surjective submersion of G?
subject to which conditions?

This is crucially important now to make sense of the discussion, and leaving it unclear casts doubt on the whole edifice.

1- state precisely the actual definitions of the lattice models
2-add consistency checks that these definitions are indeed backwards compatible with the ordinary ones.
3-clarify the claimed relation of the lattice models to homotopy theory and gerbes

Ask for major revision