The quasilocal degrees of freedom of Yang-Mills theory

1. Innovative perspective on gluing and emergence of physical degrees of freedom, with interesting consequence and ramifications.
2. Clear and solid mathematical construction (modulo details as reported below)
3. Self contained presentation of the framework and clear structure of the paper, its objectives and results.

I do however feel the paper falls short on a number of issues, mostly related to preciseness of statements and (sometimes) mathematical clarity. I have detailed my concerns below. Summarising and highlighting some:

1. I would like to suggest the authors pay attention to the comments related to their Section 2, where I believe their mathematical setup is imprecise (but fixable). This is especially important when discussing generalisations of this formalism, a few comments on which should probably be given somewhere.

2. I strongly suggest that the authors curb the paper from as much jargon as possible, and that keep mathematical statements clearly separate from physical interpretation. This is a recurring weakness that I feel should be addressed. There are a number of instances of undefined terminology used to draw conclusions. Speculative interpretations can be perhaps condensed at the end of sections. In particular the redundancy of specifiers like "horizontal", "physical" and "radiative" or "vertical", "gauge", "Coulombic" does not help the reader.

3. The issue of smoothness of fields in Section 4 requires more attention. See my specific comments about it.

4. Lastly, more words on comparisons with other approaches, especially when there are contradictory conclusions, should be given.

This paper builds upon previous work of the authors on a field-space geometric approach to gauge theories, aimed at an analysis of regional decompositions of field perturbations and a splitting of degrees of freedom into "gauge" and "physical". In this context this manuscript extends previous work in the context of Lorentzian manifolds and focuses mainly on the behaviour of relevant data upon gluing of subregions in phase space, keeping the symplectic structure under consideration throughout.

The manuscript (as part of a program that the authors admittedly started elsewhere) succeeds in presenting an interesting an innovative point of view on a purely geometric approach to Yang--Mills theory on a (d+1) dimensional Lorentzian manifold, and present a consistent framework to deal with gluing of data associated to subregions in the phase space of the system.

Additionally, by means of a choice of a connection defining a notion of horizontality in field space, the authors propose to match horizontality with the notion of physicality of degrees of freedom and explore the consequences of this choice upon gluing subregions. This choice appears to be guided by the existence of a natural metric on the space of field configurations, and hence a unique connection that is metric compatible. This is called Singer-deWitt connection, and it is used throughout the paper.

The splitting into physical and gauge degrees of freedom induced by the connection brings up a natural question when it comes to gluing: whether this decomposition of fields is preserved upon gluing. The authors show that this is generally the case and argue what are the limitations of this statement, by a careful analysis of the gluing procedure, which is sensitive of global topological data and singularities in the group action (stabilisers). This comes with a nontrivial interplay of physical and gauge fields, that the authors argue can be taken as an explanation of several claims in the literature regarding the emergence of physical modes from gauge theoretic considerations.

The authors propose an analysis of reducible configurations that is interesting per se, and provides a good handle on this issue. All in all, the perspective on the problem that the authors propose is (to my knowledge) innovative and highly detailed, in that it provides a concrete method to explore the gluing of subregions.

I do think that the paper contains new material relevant to the understanding of regional properties of field theories (i.e. on bounded regions), and it provides an intriguing perspective that, to the best of my knowledge, has not been tested before (save on the author's previous works).

In its own, this paper is self contained and clear, making it a capstone summary of this program that, in principle, I could suggest for publication in Scipost, after addressing a few issues.

1. The paper explores an important aspect of gauge theories, namely the degrees of freedom in a local region $R$ of a Cauchy slice $\Sigma$. In particular, they discuss the splitting of the dof into radiative and Coulombic modes. The results are important for the discussion of entanglement entropy of subregions in gauge theories and in the study of asymptotic symmetries where such splittings are studied often.

2. The paper also discusses how dof in two separate regions $R_1$ and $R_2$ are related to those in $R_1 \cup R_2$ via a gluing procedure. This result is important for the study of entropy inequalities in the context of gauge theories.

3. The discussion is very thorough. A lot of new formalism has to be used to derive the results of this work and most of the details are presented cleanly.

1. The paper discusses the idea of "configuration space" and regularly works with vectors and forms on this space. This is a fairly non-standard manifold and it would be useful to provide explicit formulae when possible - particularly in section 1.3 where a large part of the formalism is introduced (for instance, for $\varpi$). The authors refer to another paper where this formalism has been explicitly spelled out, but where possible it would be good to add additional equations for self-containedness.

2. The overall set of ideas presented in this paper are closely related to the study of asymptotic symmetries at null infinity that are being studied now, especially in the discussion of the symplectic form and charges. In particular, how do the discussions of this paper change if $\Sigma$ is null?

3. The authors state that the observable associated to the electric flux $f$ (the pure gauge mode) on the boundary is superselected. This is known to be untrue in the context of "memory effects" where most dynamical processes causes a change in the boundary pure gauge mode. Such a change is also experimentally measurable (in theory!). I suspect that the essential difference between the two is that the authors study a compact $\Sigma$. It might be nice to include this feature in the outlook in section 5.2.

4. It is emphasized throughout this paper that the discussion differs from others in that no gauge choice is made. However, isn't choosing the SdW form to distinguish between horizontal and vertical modes precisely equivalent to working in the Coulomb gauge, $D^i A_i = 0$? It might be good to clarify this point (if I am confused about this, other readers might also be).

In this paper, the authors discuss the (important) issue of degrees of freedom in gauge theories, especially in the presence of boundaries. The discussion relies on a heavy (but ultimately very useful) mathematical tool of "configuration space. This formalism is introduced in section 2 and then applied to gauge theories in section 3. Section 4 is the most important section in that it describes precisely how dof in disjoint subspaces of $\Sigma$ are glued together and its uniqueness (up to topological modes).

The results of the paper are new and interesting and have applications to several interesting physical problems. I recommend publication of this paper.

I have mentioned the potential changes in the previous sections, but I will summarize them here.

1. Provide explicit formula for some of the configuration space ideas introduced in section 2 (for example, for $\varpi$).

2. How does the discussion change if $\Sigma$ is taken to be an asymptotic Cauchy slice, $\Sigma \to {\mathscr I}^\pm\cup i^\pm$. Similar discussions on null infinity are of recent interest so relating the two would be very interesting.

3. Superselection sectors vs memory effects.

4. Is choosing the SdW connection the same as working in Coulomb gauge?