Instanton Density Operator in Lattice QCD from Higher Category Theory

The main purpose of this work, as the referee points out, is indeed that the lattice discretization should correctly reflect the expected topological classification before taking a continuum limit---this serves both the fundamental purpose (since we want a lattice QFT with good enough properties) and the practical purpose (since in practice there is always a finite lattice length) of lattice QFT.

The first main concern of the referee is that the model construction is not explicit enough in this paper. Indeed, the purpose of this paper is to introduce the big picture, the principles and the mathematics required to solve this problem. A follow-up work, which was mentioned in the original version, has now appeared as arXiv:2411.07195, in which a more technical explicit construction is given. The follow-up paper is self-contained for an audience who only care about the model and its physical intuition but not the overall big picture and mathematics, in complimentary to the present paper. Perhaps the follow-up paper 2411.07195 is closer to what the referee has in mind, according to the report.

I will then take this opportunity to explain why I decided to have the present paper and the follow-up paper 2411.07195 serving such complimentary roles. First of all, the detailed model construction itself is highly technical and takes tens of pages to explain. On the other hand, this is just one particular construction---there is no single "canonical construction", only the principles (see below) are crucial. My prospect is that in the future, numerics will help us find better optimized constructions, different from the one being described in 2411.07195, but still following the same principles. So I decided that it is important to have a first paper (i.e. the present paper) stating the principles (see below) which are correct in a model-independent manner, and sketch the model construction, and then have a follow-up paper (i.e. the 2411.07195) presenting the highly technical details of one particular model construction.

A second key question brought up by the referee is, what does it mean for our proposal "to work"? There are two layers of requirements for our proposal "to work". Along the way we will respond to the referee's third question, on why the mathematical Sections 5&6 must not be dropped.

The first layer is to serve our motivation of (in the referee's terms) "having a lattice discretization which correctly reflects the expected topological classification before taking the continuum limit". Thus, we need to make it precise what it means for a lattice theory to "correctly reflect" the topology.

In the past, this used to be only a matter of intuition. There has been no precise principle for this (except for TQFT, but here we are working with dynamical QFT). One major contribution of the present paper is to uncover a precise mathematical principle for what this really means---this is indeed why we wrote the mathematical Sections 5&6, and why these sections must not be dropped. (On the other hand, in the complimentary paper 2411.07195 whose goal is to introduce a detailed construction, there is no discussion of these mathematics but only the physical intuition behind. So this is along the lines of what the referee suggests.)

In the updated version of the present paper, I will state it clearly at the beginning of Section 5 that the mathematics to be introduced is not only motivational; they serve the purpose of uncovering the precise meaning of ``having the lattice theory to correct reflect the topology", hence indeed making the paper convincing. In particular, in the new Section 5.5 of the updated version (which used to be the second half of the original Section 5.4), I will give a sharpened mathematical statement for what it means to ``correct capture the topology", and explain why the construction in Section 4 (and in 2411.07195 in more details) indeed fulfills this principle by design. This explains the first layer of "the theory works"---that the construction indeed correctly captures the topology.

The second layer is that the dynamics of the lattice theory, in particular including the dynamics of the topological configurations, should be right as we approach the continuum limit. Surely this ultimately can only be tested numerically, which we have not yet done, but we have some good reasons to argue that this is highly likely the case:

We introduced some new degrees of freedom and some new weights. If the new weights are chosen in certain limits, then integrating out the new d.o.f. will just reduce the theory back to the traditional Wilson's definition. So first of all, our refinement at least would not make things worse.

On the contrary, it is likely that suitably chosen new weights will make the renormalization convergence of the theory better. In the Villain model, which has been well-studied for decades, it is well-known that adding a suitable vortex fugacity term weighing the new d.o.f. actually facilities the renormalization convergence of the theory, rather than ruins the theory (see e.g. the cited Kogut's review paper). The physical idea is that, even if one does not include those higher d.o.f. and higher weights to begin with, along the coarse-graining procedure, those effects will be effectively generated anyways (since those higher d.o.f. are to capture how the fields interpolate, which is like the reminiscence of the d.o.f. on a finer lattice), so having those higher d.o.f. and weights in the action actually helps keeping track such effects along the renormalization flow. This has been analyzed in the Villain model. Since the new constructions are based on the same physical idea, it is physically intuitive to expect the same advantage (although the reasonable range of choices for the new weights should be determined numerically). Discussions along this line appeared in the final section in the original version, but in the updated version I will make the discussions more clearly throughout the text.

Therefore, this second layer of the meaning of "the theory works" is an educated expectation, that awaits for numerical checks as a next step (following our follow-up explicit construction paper 2411.07195). I believe, nonetheless, it is good to write theoretical papers introducing the important new concepts before the numerical implementation is carried out. As far as I am aware of, Wilson's original 1974 lattice gauge theory paper was motivated by the formal purpose of making good sense of what a QFT really means; only a few years later did Creutz implement it numerically and found it to be numerically useful. Also, Luscher's 1982 geometrical construction of instanton was a theoretical proposal which is only implemented a few years later. I hope this can justify that the present theoretical paper, which introduced the mathematical principles of "capturing topology onto lattice" (which are novel and physically intuitive), is of important values in its own right.

I would like to thank Referee 2 for their recognition of the originality and the potential impact of this work. Regarding their major concerns of weakness:

1 & 2: In the present paper, the construction of the new models, especially that for the refined Yang-Mills theory in Section 4.2, was indeed only a sketch. In the original version, I said more details would be contained in a follow-up paper. The follow-up paper is now completed as arXiv:2411.07195, which will be cited in the updated version of the current paper. The follow-up paper is self-contained for an audience who only care about the model and its physical intuition but not the overall big picture and mathematics, in complimentary to the present paper.

The reason why I decided to present the details in a separate paper is that the detailed construction itself is highly technical and takes tens of pages to explain. On the other hand, this is just one particular construction---there is no single "canonical construction", only the principles are crucial. My prospect is that in the future, numerics will help us find better optimized constructions, different from the one being described in 2411.07195, but still following the same principles. So I decided that the present paper should focus on the principles, while the highly technical details of one particular construction is left to a separate paper 2411.07195.

3 & 2: I will update the version so that at the beginning of Section 5, the relation to Section 4 is more clearly stated: The category theory description is not only motivational; more crucially, we kept saying our motivation is to ``capture the $\pi_3$ physics into the refined model", and what that really means can be given a precise mathematical meaning. Having such a precise mathematical principle for ``refining a lattice model to capture the topology", I believe, is very important for the development of the field. In the updated version, I will split the original Section 5.4 into 5.4 and 5.5, and in the new Section 5.5 state a more exact mathematical principle for what this ``refinement that captures topology onto lattice" means.

Before further replying to the detailed comments, I would also like to respond to the general comment that the writing of this paper is in a largely motivational style. This is indeed true and intentional. A relavant reality in the community of theoretical physics is, while some groups of people are excited about category theory, some other groups are skeptical about its usefulness in the so-called "real physics" (in the sense of describing the real physical world). Since this work is to solve a physics problem using category theory in a necessary manner, I believe it is important to convince a broad audience that the appearance of category theory here is natural, not being anything fancy or artificial. More particularly, there are ideas that used to be familiar to different groups: lattice QCD, condensed matter, TQFT and homotopy theory, and the motivational style serves the purpose of illustrating that if one digs deeper into these existing ideas, they really do come to a confluence in terms of category theory, leading to a natural and useful picture. I hope this motivational style can help with bringing in more consensuses to our community. And, finally, again, a detailed technical description is separately presented in the follow-up 2411.07195, serving the complimentary purpose.

Now I reply to the more detailed comments:

-- Typos will be corrected and various unclear or inaccurate explanations will be improved per the referee's suggestions. I would like to thank the referee for pointing them out.

-- Indeed, the higher categories involved are higher (simplicial) groupoids that appear in homotopy theory, since homotopy is indeed what we want to capture in physics. Should we use the general term "anafunctor", or more field-specific names such as bibundle, H-S morphism, (and as the referee bring up) cofibration resolution etc? My inclination is to use the more general name, because I suppose category theory might be new to a certain proportion of the audience, and I want to show the natural progression of function-->functor-->anafunctor as generally useful concepts without bringing up too much specific knowledge which might bring barriers to readers. In the end, unifying seemingly different concepts in different branches of mathematics is what category theory is good for. That said, in a footnote, I did mention that anafunctor in specific fields has specific names; in the update I will add "cofibration resolution" brought up by the referee, since it is particularly relevant.

-- Those (generalized) higher gauge redundancies in this paper are actually dealt with in a gauge fixed manner. In the Villain model, we discussed why we can fix e.g. $\theta_v\in (-\pi, \pi]$ instead of $\mathbb{R}$, in the spinon decompostion we lifted $S^2$ to $SU(2)$ in certain gauge convention based on the north pole, and likewise for our new constructions. In principle one could also use arbitrary gauge, and the redundancy will be manifestly local. But in this paper we just fixed the (generalized) higher gauge, and each time the independence of the choice of gauge has been explained in the text.

-- The m_l label in the refined non-linear sigma model is not only not a $Z_2$ group, nor is it naturally a $Z_2$ torsor, because the $W_1$ weight for the two choices of $m_l$ are not symmetric under exchange, i.e. there is no natural $Z_2$ symmetry acting on these two labels. Also, topologically, one could have in principle used more patches, which obviously then does not admit a $Z_2$ action. I am not aware of mathematical literature referring to the choice of patches that we described as naturally forming a torsor.

-- When considering "the contributions from all other continuum paths", it is unnecessary for the entirety of the path to stay within one patch. This is just a schematic way of understanding what the $m_l$ label represents, just like in Kadanoff's block spin procedure there is some choice for what a renormalized spin represents.

-- I thank the referee for stressing on this important problem: Do those new d.o.f. added along with the new weights back-react on the dynamics and ruins the ``original theory''?
The same question could have been asked to the Villain model (after the vortex fugacity weight is introduced), so in the update I will add a paragraph elaborating on this in Section 2.1, and re-emphasize this point in the later sections. In the Villain model, which has been well-studied for decades, it is well-known that adding a suitable vortex fugacity term weighing the new d.o.f. actually facilities the renormalization convergence of the theory, rather than ruins the theory (see e.g. the cited Kogut's review paper). The physical idea is that, even if one does not include those higher d.o.f. and higher weights to begin with, along the coarse-graining procedure, those effects will be effectively generated anyways (since those higher d.o.f. are to capture how the fields interpolate, which is like the reminiscence of the d.o.f. on a finer lattice), so having those higher d.o.f. and weights in the action actually helps keeping track such effects along the renormalization flow. This has been analyzed in the Villain model. Since the new constructions are based on the same physical idea, it is physically intuitive to expect the same advantage (although the reasonable range of choices for the new weights should be determined numerically). Discussions along this line appeared in the final section in the original version, but in the updated version I will have the discussion throughout the text.

-- I thank the referee for bringing up the important issue that, given that EX and (when X=|G|) its delooping BEG are naturally equivalent to the trivial category, why the theories can still have non-trivial topology.
In the original version (mainly Section 5.2), I have explained the physical reason for this: The path integral weight in a dynamical QFT does not respect natural transformation, and this is a crucial difference with TQFT. So one cannot conclude the triviality of the theory just based on the triviality of EX or BEG.
While this explanation is correct, I myself also found this not satisfactory enough, both in terms of the way it was written and in terms of the sharpness of the statement. So in the updated version I will greatly improve the explanation on this point:
First, I will move certain discussions in the original Section 5.2 to Section 5.1 where EX and BEG were first introduced, emphasizing the physical reason mentioned above, and emphasizing the crucial difference between dynamical QFT vs TQFT (where the category theory foundation of the latter is much more familiar to certain groups of readers).
Second, in the new Section 5.5 (which used to be the second half of the original Section 5.4), I will make it clear that while EX is trivial, what we are looking at is the inclusion of X into EX, because the path integral weight is able to distinguish a locally constant configuration from a non-constant one; likewise, while BEG is trivial, what we are looking at is the inclusion of BG into BEG, because the path integral weight is able to discern a flat connection from a non-flat one. This will lead to relative cohomology classification. This is a substantial improvement in the scientific content of the updated version.