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

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