Instanton Density Operator in Lattice QCD from Higher Category Theory

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I sincerely thank the referee for seriously reading and thinking about the contents of both papers, and returning their opinions sharply. I do hope the referee can however re-evaluate the manuscript after reading this reply.

I have to first clarify the misinterpretation that the follow-up paper 2411.07195 was written “in reaction to” the previous reports by both referees. It was NOT. It was planned to be written as a complimentary paper, independent of the submission process of this first paper. (One can check that in the original version of this first paper, it was already stated that this follow-up paper was in preparation.) Papers appearing in sequential orders and serving complimentary roles, together tackling one problem, is completely normal; perhaps my coordination of timing was not that optimal here, making it as if the follow-up was written as an “unusual reaction" rather than what it really is---a completely normal, planned complimentary paper.

I feel I particularly need to clarify this, because in the second report the referee seemed to have developed a particularly negative tone/attitude, in relation to us writing a second paper. Let me state clearly: I whole-heartedly respect the referees' first reports and took them seriously; I implemented many suggested revisions thanks to the second referee, and particularly important revision/improvement was made in Section 5.5, changing the over-category perspective to the more generally applicable relative cohomology perspective, thanks to the referee's question on $BEG$; in my response I was bringing up the follow-up paper because I do think it contains (as planned) some of the key technical information that both referees were asking for---i.e. the expression of the partition function in traditional notations.

It seems the referee does consider this overall program to be compelling, and I am sincerely glad about it, despite the referee has judged the current paper's proposal as "hard to discern" and recommended a rejection. To be honest, regardless of the judgements of my current paper(s), my overall goal is that in the long run this program will work out throughout the relevant communities. If other future works can flesh out this program much better than these current papers do, I will only be more than happy.

That said, I will firmly defend my current papers and I of course want the one under review to be accepted. My response to the referee's objections are:

1. I totally disagree with the referee's judgement that the presentation of the complimentary paper 2411.07195 is "too roundabout to base any further deductions on".

2. I totally disagree with---and actually find it hard to understand---the referee's assertions that "the advertised higher categorical construction is nowhere to be seen", "there is nothing like this", and that the Section 5 on categories "is not remotely close to"/"has no tangible relation to" the actual construction presented.
I am willing to further improve my wordings in Section 5.5 and Section 4, so to further improve their connection, but that is only going to be incremental. It really is hard to understand the assertion that this connection is in the current version "nowhere to be seen".

3. The referee said "both articles point to each other for more details". This is of course true because they are supposed to serve complimentary roles, and the "more details" that they point to each other for are clearly DIFFERENT aspects of the same research. I don't see how this can in any way be problematic, and how this "contradicted the author's claim in revision".

I will elaborate on points 1 and 2. I believe 3 is self-evident.

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1.

It is the best to compare with Seiberg's Phys. Lett. B 148 (1984) 456, in order to respond to the referee's judgement on the presentation of our complimentary paper 2411.07195.

Seiberg's 1984 paper partly refined Luscher's 1982 geometrical construction by introducing a dynamical lattice 3-form U(1) gauge field $\tilde{k}$, which is nothing but the dynamical CS 3-form in our paper. Seiberg indeed also introduced the CS saddle function, $k(U)$, which is also a "symbol defined using a lengthy list of explanations" (in the referee's words).

What our paper 2411.07195 crucially achieved compared to Seiberg's 1984 paper is, we introduced different possible plaquette interpolations with suitable weights (as opposed to fixed plaquette interpolations), and introduced a CS sensitivity function $|\nu|$ with suitable properties (as opposed to the fixed constant $\alpha$ in Seiberg's paper), thereby explicitly resolving the singularities in Seiberg's definition (or more precisely, removing the unphysical and discontinuous "admissibility" condition that used to be imposed in order to artificially avoid those singularities). This comparison is mentioned at footnote 10 in our 2411.07195.

Regarding the presentation, one can see that both Luscher's and Seiberg's work contained construction of functions via lengthy technical steps. Our 2411.07195 is technical along the same line and to the same level. In fact, we have elucidated and removed some unnecessary technicality already (by using Wilson loops instead of Wilson lines for interpolation). We also made additional efforts to improve the readability/transparency by including many pictorial illustrations and intuitive explanations of the technical steps that we are doing, instead of asking the readers to decipher those long formulae themselves.

And over the decades there had been many variants of Luscher's construction, and they are all technical and lengthy at a comparable level. As far as I can see, this technicality is hard to avoid. From Luscher, to Seiberg, through other works over the decades, to us, the technicality has stayed similar, and I believe all these authors, ourselves included, have been trying the best of what they (we) could to present explicit definitions. Nobody tried to "roundabout" stuff. Of course, if someday someone resolved the technicality in one shot by coming up with some simple formulae doing all the same job, I will be extremely happy. But there is no hint of this (we tried very hard and only obtained our limited elucidation mentioned above), so before this happens, I don't see how there can be any problem working with the state-of-art technicality which is lengthy but well-defined.

In summary: In the previous report, the referee asked to see a detailed description of the path integral, so I said such a description, as planned, was presented in our complimentary work 2411.07195; we constructed it using a technical but well-defined state-of-art method. Compared to the published and well-established works of Luscher's and Seiberg's (needless to say, both are physicists of top calibre, and in particular Luscher's work has been very influential and useful), our work made the crucial advance of explicitly resolving the singularities (or more precisely, removing the unphysical "admissibility" condition)---the key problem in these pioneering works---and our presentation is at a level of technicality comparable to theirs, but elucidated and removed some parts of the technicality, and included many pictorial illustrations and intuitive explanations, which could only be helpful.

Given this comparison, I sincerely hope the referee can re-evaluate our complimentary paper 2411.07195, recognizing its presentation of the lengthy (but well-defined) construction as being what one can reasonably do at best rather than dismissing it as something "too roundabout to base any further deductions on". If, on the contrary, the referee still rather disregards our complimentary paper as being apparently inferior to the established papers of Luscher's and Seibergs, to the extent that it is an unacceptable "roundabout" not worthy for publication or even any further deductions/discussions, then I would have to solicit a concrete explanation from the referee of why.

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2.

The connection between higher category theory (if any) and our non-categorical description (as a usual lattice path integral) in Section 4 and in the complimentary paper 2411.07195 is, of course, a fair question to ask.

If there was a single inspiration that made this entire program possible, it was 1307.4793 by Gukov and Kapustin and the closely related 1308.2926 by Kapustin and Thorngren. This is acknowledged on page 29 of my manuscript under review. Section V.B of 1307.4793 and Section 5 of 1308.2926 brought to me the key insight that the d.o.f. in the Villain $PSU(N)$ gauge theory organize into a strict 2-group. And these two papers are considered important by the community as of today.

In these two seminal papers, when introducing the relation between the Villainized $PSU(N)$ gauge theory and the strict 2-group, there was no discussion to the level of mathematical rigor required by the referee (in order for my manuscript to be "potentially acceptable"). The discussions there were mostly in physicists' familiar terms. As physics papers, I don't think the connection thus made between Villainized $PSU(N)$ gauge theory and strict 2-group was therefore "sketchy" or "nothing tangible". To me, and perhaps to many other physicists as well, the connection thereby made was sufficiently evident, and truly inspiring.

And this is the kind of communication I want to provide to the physicist readers, albeit the problem now is intrinsically much more technically involved than the Villain case.

To better understand the referee's negative judgement, I would like to ask: For the physically well-established Villain $S^1$ NLsM, Villain $U(1)$ or $PSU(N)$ gauge theory, and spinon-decomposed $S^2$ NLsM, does the referee consider their relations to strict (higher) group(oid)s mathematically well-established in the mathematical physics community? If yes, does the referee think the presentation in Section 5 of the current manuscript helps a broader theoretical physics audience make that connection for these well-known models (given that categories are not something most theoretical physicists will think about at least as of now)? Does the referee agree that the discussion in Section 5 (especially the first few pages of Section 5.5) on these known models, though not pursuing mathematical rigor, is more comprehensive than the brief (but already inspiring enough) discussions in 1307.4793 Section V.B and 1308.2926 Section 5?

Now let's turn to the new models. Let me first summarize the mathematical ideas here, and then discuss my presentation.

If the referee wants, we can begin with the referee's suggested "actual higher-categorical/homotopical constructions" papers.
[While I appreciate suggestions, I don't understand why the referee put the word "actual" there. The mathematical literatures that I based my work on and properly cited---which will be mentioned in the below---are, by all measures, no less "actual".]
Among the referee's suggested papers, the one on $\mathbb{Z}$-valued Cech cocycles is not so relevant because it only captures the topological class but loses the continuous geometry of the original Lie group---the latter has been emphasized in my manuscript, because the refined d.o.f. must at least cover those in Wilson's traditional theory. The other four papers (among which, the last two by Fiorenza et al were indeed unknown to me before) are somewhat more relevant but still far from what we need.

Let's, say, look at the referee's suggested exposition 1301.2580 by Fiorenza et al, in particular its Section 3. This section is built around the stack map $BG\rightarrow B^3 U(1)$, applying it on different continuum manifolds. But does this directly achieve what we want? Clearly NOT. Stacks are defined on the Grothendieck topology of continuum manifolds; what we ultimately want is a lattice path integral with finite-dimensional d.o.f.. There is no further discussion in 1301.2580 that is directly helpful for moving from the stack description of $BG\rightarrow B^3 U(1)$ towards our actual goal.

If all that is needed is a precise categorical statement of the solution, then I have already stated it in Section 5.5: The proposed 4-group of degrees of freedom for the refined $G=SU(N)$ lattice Yang-Mills is the span of an anafunctor that realizes the generator of $\pi_0 \mathbf{H}(BEG, BG; B^5\mathbb{Z})$ (counting Yang monopole) in the ambient category of finite dimensional smooth Kan complexes (or its cubical counter-part instead of simplicial); and for pion NLsM it would be $\pi_0 \mathbf{H}(B|G|, |G|; B^4\mathbb{Z})$ instead (counting baryon non-conservation defect). (Except that in Section 5.5 $\pi_0 H(X; B^n\mathbb{Z})$ was expressed as the more familiar $H^n(X; \mathbb{Z})$.) And in fact the homotopy of the stack map $BG\rightarrow B^3 U(1)$ is naturally isomorphic to my claimed $\pi_0 \mathbf{H}(BEG, BG; B^5\mathbb{Z}) \cong H^5(BEG, BG; \mathbb{Z})$. So already here, I don't know what the referee might consider is the essential content in 1301.2580 that was missed in my manuscript. (Also, the Deligne-Cech evaluation of this stack map in Section 3.3 of 1301.2580 was already discussed in physicists' familiar notations at the beginning of my Section 4.) But my manuscript had to achieve more, i.e. an actual finite dimensional realization of this generator to be used as lattice d.o.f., which is why the "fancy" category language is at all useful to this traditional long standing problem. So we have to do the following unwinding, using the no-less-actual mathematical literature that I properly cited.

1) While stacks and stack maps are defined on the Grothendieck topology of continuum manifold, they have many other homotopically equivalent realizations, and I particularly looked into the anafunctor (bibundle) between (higher) groupoids, a perspective that I learned from 0911.2483 by Schommer-Pries. Why did I particularly focused on this realization? Because when applied to the simpler case of $BG\rightarrow B U(1) \rightarrow B^2\mathbb{Z}$ discussed in Section 2 of 1301.2580, this anafunctor realization indeed reduces to the strict 2-group Villain structure that I learned from 1307.4793 and 1308.2926. So this must be the right route to take.
There is another route to motivate the same idea (which was my actual original motivation when I started this research three years ago). If we want a direct generalization of Villainization, then, as said in my Section 3, the 3-connected cover on the Whitehead tower will be needed. But the 3-connected cover is an infinite dimensional space, so we need a categorical realization of it (idea learned from Baez and Huerta 1003.4485) that is finite dimensional. This again leads to the same higher groupoids and anafunctors, thanks to the same paper 0911.2483 by Schommer-Pries.

2) The realization of anafunctor between $BG\rightarrow B^3 U(1)$ (now seen as groupoids instead of stacks) has been studied as "multiplicative bundle gerbe" in math/0410013 by Carey et al (which was also cited in the referee's suggested 1301.2580). But they only gave an existential statement about finite dimensional realization, not an actual description. (Similarly in 0911.2483 by Schommer-Pries.)

3) To find an actual finite dimensional description of a multiplicative bundle gerbe, we can begin with that of a bundle gerbe, realizing $|G|\rightarrow B^2 U(1)$. For $G=SU(N)$ an explicitly conjugation-equivariant construction of bundle gerbe has fortunately been given in hep-th/0205233 by Gawedzki and Reis, and the description is easily understandable to physicists.

4) Now the key task is to deloop this known description of a bundle gerbe into that of a multiplicative bundle gerbe. Most mathematical literatures only gave an existential statement about this. The paper that aimed to constructively introduce a delooping protocal was Waldorf's 1201.5052, in particular Section 7, method 2. It makes use of path spaces, and a rigorous treatment requires the path space's diffeological structure. And a key map is need, denoted as $\mathcal{A}$ there, which was however not explicitly defined.
The actual study of this key map $\mathcal{A}$ was done in Waldorf's previous paper 1004.0031, which was, alas, not manifestly expressed either. It was described as a stable isomorphism between the bundle gerbe given in hep-th/0205233 by Gawedzki and Reis and the tautological WZW bundle gerbe constructed on the path space. Then I deciphered that the stable isomorphism can be described by manifest choices of representative paths.

5) The multiplicative bundle gerbe is not yet what we finally need, unless we are considering flat gauge configurations only. Rather, a multiplicative bundle gerbe over $G$ appears as a sub-category of actual the target category of a refined $|G|$ NLsM, obtained by fixing the source (or target) object. This point is most easily seen by comparing to the more well-known case of Villain model, as has been particularly emphasized in Section 5.5 in relation to relative cohomology. And for refined $G$ gauge theory, we need to further deloop the target category of the $G$ NLsM.

Unwinding this line of logic, I hope the referee can appreciate the facts that:
-- The multiplicative bundle gerbe on which my construction is based is well-defined in the "actual" mathematical literature.
-- On the other hand, the procedure to explicitly construct a finite dimensional multiplicative bundle gerbe has not been written explicitly in any single mathematical paper, but rather stretched over several papers over the time span of a decade.

I think it would be great if someone writes up a pure mathematics paper that explicitly combines the contents from the several papers mentioned above into one rigorous explicit description of a finite dimensional multiplicative bundle gerbe over $SU(N)$. However, I don't think I should be obligated to do this job in this physics paper in order for it to be just "potentially acceptable", especially considering that this will take tens of pages of rigorous mathematical notions (diffeological structure, Deligne cohomology etc) that are unfamiliar to the even the formal community of theoretical physicists. Over the past, physics papers which introduced important new mathematical concepts or ideas into physics were mostly not subjected to such kind of obligation, including the aforementioned 1307.4793 and 1308.2926 which directly inspired me. I don't understand why this is asked for for my paper to be "potentially acceptable". I believe properly citing (as I did) the mathematical literatures should be appropriate.

Now let's see how I presented the material.

-- From Eq(110) to Eq(114), I started by explaining how the well-known Villainization process is naturally understood from the anafunctor point of view. This serves two purposes: first, it makes precise the connection between Villainization and groupoid or 2-group that was only intuitively seen but not formalized in the aforementioned seminal works 1307.4793 and 1308.2926; second, it sets the stage for how we want to generalize towards our main problem of interest. Of particular importance is the relation between the principle bundle Eq(110) and the actual Villainization Eq(111) (or more generally, between Eq(113) and Eq(114)), which we explained at length---and this is nothing but point 5) in the above. For here we arrived at the key idea that lattice topological configurations for continuous d.o.f. (rather than the well-studied discrete d.o.f.) are characterized by *relative* cohomology, which as far as I know is novel. Then, from Eq(115) to Eq(118) we generalized the formalized ideas to the also well-known spinon decomposition.

-- Based upon the previous step, in Eq(119) I introduced the notion of bundle gerbe from the anafunctor perspective, and it is clear that it is a generalization of Eq(115). In particular, the stable isomorphism (ananatural isomorphism) between the infinite dimensional tautological bundle gerbe (which will make good connection to the continuum field theory) and the finite dimensional realization of the bundle gerbe is discussed. The mathematical literatures that contain rigorous discussions are properly cited.

-- Then I introduced the notion of multiplicative bundle gerbe in Eq(123). This serves two purposes: First, while it is not what we finally need, it is closely related to what we need, and it has been well-studied---the mathematical literatures that contain rigorous discussions are properly cited. Second, we want to see later in the text that the Dijkgraaf-Witten twist for discrete gauge group can be understood in the same manner, establishing the connection between our construction for continuous d.o.f. and the previously well-established formal literatures for discrete d.o.f..

-- Then we introduced Eq(127) and Eq(128), which corresponds to what we have in continuum NLsM and what we want for the lattice NLsM, respectively. The relation between Eq(127) and the tautological bundle gerbe, and the relation between Eq(128) and the finite dimensional bundle gerbe, are parallel to the relation between the aforementioned Eq(118) and Eq(115), and that between the aforementioned Eq(114) and Eq(110). So the physical motivation for these structures is clear.

-- The final task is to construct the desired finite dimensional span, in particular the $\Lambda$, of Eq(128), which is supposed to be ananaturally isomorphic to that in Eq(127). This is explained by Eq(129), i.e. by finding representative paths for each element of $Y$. This procedure is parallel to the idea in point 4) above in the context of multiplicative bundle gerbe. The representative paths are nothing but those introduced in Section 4.1. (Footnote 136 provides a discussion for a simpler and more familiar situation.) With these representative paths, the $\Lambda$ in Eq(128) is already well-defined via the discussion below Eq(129). The $W_2$ weight and in particular its $\mu$ function in Section 4.1 serve the purpose of assigning weight of elements in $\Lambda$. While this is straightforward to see, I can revise the manuscript and explain more about this point. That said, from the discussion in the current manuscript, the correspondence is already plausible, and can be verified straightforwardly--especially given the straightforward comparison to the more familiar Eq(41) whose geometric meaning has been explained at length in Section 2.4.

-- Eq(130) and below discusses how to deloop the structure needed for NLsM to that needed for Yang-Mills. I can explain the connection to Section 4.2 in better details in the revision.

Summarizing the above, I really do find the material (given that it is intrinsically quite non-trivial) has been presented in a well-motivated and well-organized manner, and the relevant mathematical literatures that contain more rigorous discussions have been suitably cited whenever needed. Major revision has already been made after the first report, that led to the current Section 5.5, and on top of that I don't see what must be further substantially revised, were it not for making the mathematical discussions rigorous---but as I said above, this should not be the job of a physics paper, and such requirement was indeed not seen in similar physics papers in the past, not even in most of those seminal ones. As I said, I am happy to further improve my wordings, but any further explanation will just be an increment of an extra page or so, because in the current version the connection between Section 5.5 and Section 4 can be readily seen. So, I really find it hard to understand the referee's assertions that "the advertised higher categorical construction is nowhere to be seen", "there is nothing like this", and that the Section 5 on categories "is not remotely close to"/"has no tangible relation to" the actual construction presented. And I find it equally hard to understand why the referee implied those suggested further references are the "actual" mathematical papers needed while they are actually not directly helpful (though relevant), meanwhile overlooking the many mathematical papers that I cited which are actually directly helpful.