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Instanton Density Operator in Lattice QCD from Higher Category Theory
by JingYuan Chen
Submission summary
Authors (as registered SciPost users):  JingYuan Chen 
Submission information  

Preprint Link:  scipost_202407_00023v1 (pdf) 
Date submitted:  20240714 16:02 
Submitted by:  Chen, JingYuan 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
A natural definition for instanton density operator in lattice QCD has been long desired. We show this problem is, and has to be, resolved by higher category theory. The problem is resolved by refining at a conceptual level the YangMills theory on lattice, in order to recover the homotopy information in the continuum, which would have been lost if we put the theory on lattice in the traditional way. The refinement needed is a generalizationthrough the lens of higher category theoryof the familiar process of Villainization that captures winding in lattice XY model and Dirac quantization in lattice Maxwell theory. The apparent difference is that Villainization is in the end described by principal bundles, hence familiar, but more general topological operators can only be captured on the lattice by more flexible structures beyond the usual group theory and fibre bundles, hence the language of categories becomes natural and necessary. The key structure we need for our particular problem is called multiplicative bundle gerbe, based upon which we can construct suitable structures to naturally define the 2d WessZuminoWitten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) nonlinear sigma model, and the 3d ChernSimons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ YangMills theory. In a broader perspective, higher category theory enables us to rethink more systematically the relation between continuum quantum field theory and lattice quantum field theory. We sketch a proposal towards a general machinery that constructs the suitably refined lattice degrees of freedom for a given nonlinear sigma model or gauge theory in the continuum, realizing the desired topological operators on the lattice.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
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
Weaknesses
1 key definitions remain hard to parse
2consistency checks of key definitions seem lacking
3relation between key sections 4 and 5 remains vague
Report
+++ 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 fineprint 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 mathremote 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 twovalued 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 backwardscompatibility, namely that all the new degrees of freedom added (notably the hatn_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 (redefinition)
Clarification is needed for what is meant, and also for this choice of notation.
Note that usually, "EG" denotes the universal Gbundle, or else the simplicial complex
(WG)_n = G_n x G_{n1} x ... x G_0
This happens to be a simplicial group (arXiv:1204.4886) hence has a further delooping via a bisimplicial 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.
Requested changes
1 state precisely the actual definitions of the lattice models
2add consistency checks that these definitions are indeed backwards compatible with the ordinary ones.
3clarify the claimed relation of the lattice models to homotopy theory and gerbes
Recommendation
Ask for major revision