Topological Data Analysis of Monopoles in $U(1)$ Lattice Gauge Theory

1. Very well written.
2. Clear and sound presentation of the problem and the results.
3. High level of interpretability of the observables defined through topological data analysis.

In my opinion, this work does not present any major weaknesses, and I consider that all the suggested changes can be addressed at this stage.

This work explores the use of topological data analysis (TDA) to characterise topological aspects of a 4-dimensional pure compact U(1) lattice gauge theory (LGT), relating them to the phase structure of the model. The main contribution of this work is to show that relatively simple and highly interpretable topological observables, built from the homology groups of monopole current networks, allow for a quantitative estimation of the critical inverse coupling of the theory via finite-size scaling analysis. This is possible as monopole current networks can be regarded as directed graphs and, therefore, described using notions of algebraic topology within the framework of TDA. While topological characterisations of this problem have been proposed previously in the literature, the present work illustrates in a novel way the usefulness of TDA for problems relevant to LGT and beyond, with the potential for follow-up studies. As this type of data analysis is attracting growing attention in the physics community, I believe that this work will be of interest to a broad readership of the SciPost Physics.

The paper generally satisfies the acceptance criteria of SciPost Physics. However, before I can recommend for its publication, I would ask the authors to consider the changes mentioned below.

1. It is mentioned in the abstract (and briefly in the introduction) that the present approach can be generalised to study Abelian monopoles in non-Abelian LGT. However, there is no comment throughout the rest of the manuscript on how this could be done and which major technical difficulties one would need to overcome in this case (if any). I think it would be nice if the authors could elaborate on this point, for example, in Sec. 4, as it seems to be a very interesting outlook.

2. The authors correctly acknowledge and summarise previous works exploring topological aspects of monopoles currents and Dirac sheets in Abelian LGT. In particular, Ref. 10 by Kerler et al., which is mostly based on the topological analysis of Dirac sheets, is taken as a reference point for the present work. However, a previous work also by Kerler et al., namely Ref. 22, provides a topological characterisation of networks of current lines, which is based on their fundamental homotopy group. This can be regarded as a further application of TDA to characterise monopole current networks of the system under study. Therefore, I think it would be relevant to mention more explicitly the work in Ref. 22. This could emphasise possible advantages of the present approach (e.g., it is my understanding that properties based on homology are, in general, easier to compute than those based on homotopy, etc.).

3. I appreciate that the authors try not to obscure the discussion in the main text with technical aspects. However, since Betti numbers play a central role in their analysis, I think that the authors should consider to include Appendix C as a part of Sec. 3 in the main text.

4. In Sec. 3, when introducing Betti numbers, it is mentioned that "We use the coefficient field $\mathbb{Z}_2$ but, since this is a 1-dimensional complex, the resulting Betti numbers are independent of the coefficient field." I think that adding a relevant reference here is in order.

5. In Sec. 3, the authors talk about the reweighted variance curves for the observables $\rho_{b_0}$ and $\rho_{b_1}$. Including examples of such curves (as figures, for instance, in Appendix B.1) could help to understand better the explanation provided there.

6. I would suggest to include in the title the wording "Monopole Current Networks" instead of just "Monopoles", since this would be a more accurate description.

7. In Appendix B.2, I believe that the samples used in the bootstrap analysis are of size $N$. This could be mentioned explicitly.

Ask for minor revision

1. Interesting application of topological data analysis (TDA) to lattice gauge theory.
2. Clear quantitative predictions using homology.
3. State-of-the-art lattice methods.

1. TDA part not on the same quality level as the lattice part.

The present work provides an interesting application of TDA, a relatively new field of data-driven applied topology, to the deconfinement phase transition of U(1) lattice gauge theory. Building on older results of [Kerler et al., Phys. Lett. B 348, 1995], the authors employ Betti numbers to characterize monopole current networks. This fresh approach allows them to obtain estimates for the pseudo-critical inverse coupling at infinite volume, which agree with those computed from average plaquette actions and have error bars of similar size.

The actual lattice gauge theory part of the paper is profound and well written. In particular, this applies to the introduction, to the non-TDA descriptions of the phase transition in the studied U(1) lattice gauge theory and the related lattice computations (i.e., Secs. 1 and 2).

The authors publish the utilized software and the investigated data using the platform Zenodo. After a superficial inspection, the code and data provision for the work seems exemplary (I did not re-run the simulations and scripts).

This brings me to my major points of critique:

1. Towards the end of the abstract and in the introduction, the authors claim that their approach can be generalized to investigate monopoles in non-Abelian gauge theories. Yet, the authors do not provide any further discussion of this. At least in the conclusions, the authors should provide a paragraph on the envisioned generalization of their analysis pipeline to the non-Abelian case.
2. At the beginning of Sec. 2.2, the authors introduce the nomenclature that the low-$\beta$ phase is called hot and the large-$\beta$ phase is called cold, based on the analogy of $\beta$ with the role of the inverse temperature. Even if this argument has been employed in older papers [e.g., Kerler et al., Phys. Lett. B 348, 1995], I find it misleading: if physical temperatures for lattice gauge theories are regarded, the inverse nomenclature is more appropriate. Indeed, even U(1) lattice gauge theory is confining in the low physical temperature regime (low $\beta$) and deconfines at a larger physical temperature [see, e.g., Svetitsky & Yaffe, Nucl. Phys. B 210, 1982]. I recommend that throughout their paper, the authors change the nomenclature to a solely $\beta$-based one, entirely refraining from calling the regimes hot and cold, but potentially including a sentence on the correlation between $\beta$ and the physical temperature.
3. At multiple points in their manuscript, the authors mention how beneficial it is to consider the investigated monopole current networks as directed graphs. Yet barely any details are provided in this regard: in the beginning of Sec. 3, the authors provide a short paragraph on the definition of the monopole current graph $X_j$, which is complemented by another brief paragraph on the definition of the Betti numbers of $X_j$. In the acknowledgements, they mention that the software package giotto-tda is utilized for the computation of the Betti numbers, and Appendix C provides some background on homology.
To me, it would appear natural to introduce a separate subsection 3.1, which is devoted to the introduction of the monopole current graph and to the introduction of its Betti numbers, potentially accompanied by intuitive illustrations to improve the readability for the readers not acquainted with TDA. This subsection should also include more details on how specifically giotto-tda is used to compute the Betti numbers, regarding the underlying monopole current graph as a directed graph. The latter is not clear to me who is accustomed with TDA, and likely even much less clear to the general readership of SciPost Physics. Including such a subsection and spending more time to carefully introduce the employed TDA observables and computational pipeline could substantially improve the readability of the TDA-related part of the manuscript as well as emphasize the promise of TDA for lattice simulations.
4. The description of the TDA-based observables shown in Figs. 3-5 with the inferred critical inverse couplings given in Tables 2 and 3 is described on only a bit more than half a page in Sec. 3. I recommend making it a longer subsection 3.2, in which further details are provided, and if possible, also more intuitive explanations for the behavior of the Betti number densities. For instance, the authors write that the maximum of $\rho_{b_0}$ is realized when the large percolating network breaks up into smaller networks. Can the authors provide arguments underlying this claim, at least mentioning that this can be a posteriori justified by the agreement of the pseudo-critical inverse couplings? Furthermore, the authors write that $\rho_{b_0}$ declines for larger $\beta$-values, since the energy cost for generating a monopole current loop is expensive. Can this be described less vague? Fig. 4 is described only in two sentences, which certainly deserves a longer discussion including a probably intuitively accessible interpretation based on the behavior of the (percolating) monopole current networks (referring also to Sec. 2).

Smaller remarks:

1. In Sec. 2.2, it is written that monopole current conservation and the periodic, untwisted boundary conditions imply the total charge of the current loops is zero. I assume there is a topological argument for this, based on homotopy classes of maps from the lattice 4-torus to the set $\{0,\pm 1, \pm 2\}$ or a variant of the latter (maybe I’m missing a piece of knowledge here). For better accessibility to their work also for non-lattice experts, can the authors provide the argument for this (potentially in a footnote)?
2. I’m wondering about the prefactors in Eq. (8): while the $1/(4\pi)$ prefactor in the first equality is consistent with [Kerler et al., Phys. Lett. B 348, 1995], the next line should contain a prefactor $-1/2$. Then Eq. (8) would be also consistent with Eqs. (13) and (14).
3. Even if the extrapolated infinite-volume, pseudo-critical inverse couplings computed for zeroth and first Betti numbers agree, I’m curious to learn whether the authors have an interpretation for the pseudo-critical inverse coupling computed from $\rho_{b_1}$ being consistently smaller than the one computed from $\rho_{b_0}$ for finite lattices. Given that this is a very small effect, I would not expect this to be discussed in the manuscript.
4. In Figs. 3-5 ensemble averages for the densities $\rho_{b_k}$ are shown. In the main text, no ensemble averages are indicated. Can the notation be made consistent in this regard?
5. In Sec. 3 and Appendix C.2 it is stated that the Betti numbers are independent from the coefficient field. Targeting the non-mathematician readership of SciPost Physics, I believe that a corresponding reference for this fact or a rough explanation in a footnote would be beneficial.
6. Throughout Figs. 3-5, subfigures (b) show zoom-ins of subfigures (a). I am confident that the authors can highlight this fact visually better, so that it is more intuitively clear, e.g., by means of diagonal lines and a square in subfigures (a), indicating the displayed zoom-in.
7. The authors mention in Sec. 3 twice that the reweighted variance curves for $\rho_{b_k}$, $k=0,1$, peak at the critical point (partially obscured). Can the authors maybe provide these figures as distinct plots, potentially shrinking the prominent zoom-ins of subfigures (b) and showing the reweighted variance curves in addition?

To conclude, the main results of the work concerning the application of TDA to monopole current networks in U(1) lattice gauge theory are indeed valid and new, and moreover provide another fruitful study that showcases the potential of TDA for physics applications. Yet, I feel that a few points deserve more attention and in particular Sec. 3 on the actual topological data analysis requires a major rewriting and expansion. For this reason, even if the paper and its results fit into SciPost Physics, in its current form I recommend a major revision of the manuscript.

The requested changes have been mentioned in the report and concern mostly but not exclusively Sec. 3.

Ask for major revision