SciPost Submission Page
Instanton Density Operator in Lattice QCD from Higher Category Theory
by Jing-Yuan Chen
Submission summary
| Authors (as registered SciPost users): | Jing-Yuan Chen |
| Submission information | |
|---|---|
| Preprint Link: | scipost_202407_00023v2 (pdf) |
| Date submitted: | 2025-02-19 23:47 |
| Submitted by: | Chen, Jing-Yuan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| 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 Yang-Mills 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 generalization---through the lens of higher category theory---of 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 Wess-Zumino-Witten term, 3d skyrmion density operator and 4d hedgehog defect for lattice $S^3$ (pion vacua) non-linear sigma model, and the 3d Chern-Simons term, 4d instanton density operator and 5d Yang monopole defect for lattice $SU(N)$ Yang-Mills 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 non-linear 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 multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
List of changes
Main:
1) The complimentary follow-up paper arXiv:2411.07195, which contains a detailed technical construction for the refined Yang-Mills theory, is cited wherever needed, especially in Section 4.2.
2) Splitted the original Section 5.4 into 5.4 and 5.5. Moreover, the contents being moved to Section 5.5 are significantly improved.
3) Discussions about the triviality of ET versus the non-triviality of the lattice theory is moved from Section 5.2 to Section 5.1.
4) In Section 2.1, after introducing the vortex fugacity, we added a paragraph emphasizing its role in improving the renormalization behavior (so that the "back-reaction" from the perspective of the "original model" is in fact not only not a problem, but an advantage). The idea has been reiterated in more general models later in the paper.
Besides these main changes, minor changes are made, fixing the typos and/or inaccuracies brought up by the 2nd referee.
Relevant references missed in the original version are added. They are:
[2], [7], [10] (the main follow-up), [15], [47], [66, 67] (both appeared after this paper was first posted), [76], [78], [79], [96], [97], [105],
Current status:
Reports on this Submission
Strengths
The advertised idea or research program sounds compelling.
Weaknesses
The promised substance of the proposal is still hard to discern and the advertised higher categorical construction is nowhere to be seen.
Report
In reaction to the previous reports on the article under consideration [Chen-2406.06673], both asking for substantially more details, the author has written another article [ZhangChen-2411.07195] and restricted the requested revision of the first article mainly to referencing this second article.
This is unusual procedure. Since I am asked to review the first article [Chen-2406.06673] and not the second [ZhangChen-2411.07195], and since the requested information is claimed to be relegated to the second which I am not asked to review, I think that already on formal grounds I have no choice but to reject the submission at this point, for not providing the requested major revision.
Still, I did read the new article [ZhangChen-2411.07195], which I am not asked to review but which I will comment on now, nevertheless.
On its p. 3, the new article [ZhangChen-2411.07195] claims that
"In this paper we will only describe the intuitive explicit construction, while directing any mathematical formality to [1]"
where [1] of course is the article actually under review here.
But this claim contradicts the claim in revision made by the author, that: "The complimentary follow-up paper arXiv:2411.07195, [...] contains a detailed technical construction."
In conclusion, both articles point to each other for more details.
It seems the only "technical details" which the first article (the one under review) provides on top of the second is its section on higher category theory. However, my complaint from the first report still stands, that this section had (and has) no tangible relation to the actual construction presented. Therefore there is no sense in which the provided discussion of higher category theory (which is largely sketchy anyways) provides details for either of the pair of articles we are looking at.
In fact, the new, second, articles claims on its p. 2 that
"The degrees of freedom altogether form a higher category structure (a suitable weak 4-group) [...] the language of higher category theory really is necessary here"
This sounds superficially like a plausible research plan, and it would be interesting to see this carried out, but the claim is at odds with the material presented, since it is nothing like this.
A potentially acceptable form of the author's claim based on the above quoted advertisement would look like this:
An actual definition of the 4-lattice as a 4-category, then an actual definition of a 4-groupoid coefficient resolving the Yang-Mills monodromies and an actual 4-functor from there to the 4-fold delooping 4-groupoid of Z projecting out the instanton number. Finally an actual definition of 4-functors from the lattice to these coefficients with an actual definition of the path integral over such 4-functors.
This is the kind of construction that the introduction of both articles lead the reader to expect to be presented. But the actual construction offered is not remotely close to this.
Instead, the actual definition, now (16) in the new article, is a decidedly non-categorical expression using a lengthy list of explanations of its symbols which I cannot vouch to be a precise definition, and which in any case seems too roundabout to base any further deductions on.
I am not saying that it might not work. Maybe the authors figured it out and the problem is just in the presentation. But since it is only the presentation that I can base judgement on, and since this presentation is far from living up to its claims, I do not recommend publication (neither of the article that I am asked to review nor, for what it's worth, of the one that was offered in place of its requested major revision).
In closing, I'll to point out references on actual higher-categorical/homotopical constructions of characteristic 4-classes from gauge data:
J.-L. Brylinski and D. A. McLaughlin:
"The geometry of degree-four characteristic classes and of line bundles on loop spaces I."
Duke Math. J., 75(3) (1994) 603–638
J.-L. Brylinski and D. A. McLaughlin:T
"The geometry of degree-4 characteristic classes and of line bundles on loop spaces. II"
Duke Math. J., vol 83 no 1 (1996) 105–139
J.-L. Brylinski and D. A. McLaughlin:
"Cech cocycles for characteristic classes",
Comm. Math. Phys., vol 178 no 1 (1996) 225–236
D. Fiorenza et al:
"Cech Cocycles for Differential Characteristic Classes"
Adv. Theor. Math. Physics vol 16 no 1 (2012) 149-250
with exposition in
D. Fiorenza et al:
"A higher stacky perspective on Chern-Simons theory"
in: "Mathematical Aspects of Quantum Field Theories"
Springer (2014) 153-211
Recommendation
Reject