Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes

1) original take on the subject of edge modes in a concrete setup that emphasizes interesting and subtle features 2) technical results specific to the chosen model (CS theory on the annulus) of interest to experts

1) not always easy to tell apart what are the original contributions and what is taken from the existing literature (despite the extensive bibliography) 2) at times rather confusing presentation which lacks (conceptual) rigor, e.g. some key equations lack justification and key concepts lack an a priori definition. Even though I think the main results are sound, this lack of rigor leads to various confusing, imprecise, and possibly wrong statements throughout the manuscript which require further attention.

I find the content of the manuscript meets the criteria for publication on SciPost, however I find the weaknesses mentioned above too important to recommend publication in its current form.

With the caveat above, I would say the article meets the following expectations:

• Provide a novel and synergetic link between different research areas.
• Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work

In relation to SciPost general acceptance criteria (all required) :

• Be written in a clear and intelligible way, free of unnecessary jargon, ambiguities and misrepresentations [not met]
• Provide sufficient details (inside the bulk sections or in appendices) so that arguments and derivations can be reproduced by qualified experts [mostly met, with some important gaps]
• Provide citations to relevant literature in a way that is as representative and complete as possible [partially met]
• Contain a clear conclusion summarizing the results (with objective statements on their reach and limitations) and offering perspectives for future work. [mostly met, but could be improved]
• Contain a detailed abstract and introduction explaining the context of the problem and objectively summarizing the achievements [partially met, I find the abstract a little misleading especially in its reference to entropy and 3d gravity]

My evaluations below for validity, significance, originality, and especially that for clarity, might very well be revised for the better upon resubmission. The evaluation of clarity as "low" is probably severe, but as I said there are in my view a few too many points that are not clearly explained in the manuscript and require clarification. As far as originality is concerned, mine here is a conservative assessment. A more precise one will be formulated when the authors will better compare their work with the existing literature.

I understand that my review might sound severe. However, I find that in such technical subjects which have no experimental/observational criteria of viability, conceptual rigor and clarity are paramount -- especially when, like here, they are fully achievable.

In summary, I believe the manuscript deserves publication after major revisions in the exposition.

I refer to the attached file.

Ask for major revision

1) Thorough and technically sound presentation of a factorization map.

2) a noted step forward in understanding subregions in systems with topological invariance

2) Results are applicable for low dimensional gravity and understanding gravitating subregions.

1) easy to get lost in notation

In this article the authors describe a factorization of the phase space of Chern-Simons theory across an entangling surface using the phase space using the phase space of a particle on a quantum group as a set of edge modes. The authors claim that this factorization is minimal in the sense that it eliminates many of the degrees of freedom in the standard factorization (using a chiral WZNW theory as the edge modes) while preserving the Poisson structure. The factorization has the additional properties of preserving the topological nature of the Wilson lines crossing the entangling cut, as well as adapts naturally to non-compact groups such as SL(2,R) which is applicable for 3d gravity.

The article is fairly technical and while I did not work through all of the details, the computations that I followed were technically sound. The article is clearly written enough and the results are a step forward in understanding entanglement and subsystem phase spaces in gravity and non-compact gauge theories more generally. While the article meets the criteria for publishing in SciPost, I do have some minor questions and clarifications that I would request the authors to address:

1) The criteria the authors use for a faithful factorization is the embedding of subsystem phase spaces, P_A x P_{\bar A} -> P (equation 1.4) preserving the Poisson algebra. However in a quantum system with a Hilbert space, it is also natural that this preserves the norm of states (i.e. the map is an isometry). I realize the analysis of this paper is entirely classical, however since the authors clearly have questions of entanglement as a central motivation, can the authors comment on if and how their factorization would work with the norms on these Hilbert spaces?

2) In equation (1.8) I believe there should not be an integration over J^\mu: otherwise Z_L is not a functional of the current and is just a number. Relatedly the order of integrations / introduction of the delta function in equation (1.6) is clearly nonsensical. Unless I’m interpreting incorrectly, both Z_L and Z_R should be functionals of J and J’ (respectively) and the gluing is given by \int DJ DJ’ \delta(J-J’) Z_L[J]Z_R[J’]?

3) A minor complaint in the first sentence of Section 2.1: in a generic pure gauge theory there are also traces of polynomials of F_{\mu\nu} as local degrees of freedom. It is after specializing to a topological gauge theory that we are left with only Wilson lines.

4)Can the authors check the sign of equation (2.7) (and subsequent equations)? (Or maybe the can make explicit if they are putting a Lorentzian or Riemannian conformal structure on their boundary)

5) Regarding notation, the authors alternate between subscripts L,R and superscripts +/- to denote quantities on the different boundaries, the latter of which might be conflated with t+x and t-x (which I think the authors denote with bar and un-bar in equation (2.11)). Later they use x_1 and x_2. I think it would help the readability / presentation of the article if the authors were a bit more consistent in notating various quantities, or maybe made a guide earlier in the article.

6) Relatedly, the authors use W(x^+,x^-) to indicate a Wilson line between two points and then factor into W(x^+)W^{-1}(x^-) which is a bit of confusing notation if one imagines that W(x^\pm) are local group elements. I assume the point is that despite the dependence one coordinate the W(x^\pm)’s are not local group elements because of the monodromy condition, eq (2.15) which indicates that they are secretly attached at an unspecified fiducial point. This is explained in footnote 20 but that comes much later than this notation is introduced.

7) Can the authors check equation (2.64)? Is there no dependence on x_{12} in the sgn(n) term as in equation (2.61)?

8) Can the authors motivate the assumption that r_{12}(x_{12)) is independent of m? It is certainly not obvious from equations (2.64) and (2.61) which have explicit m_{1,2} dependences.

9) Related to point (6), I believe equation (3.3) is indicating that g(x) is not a local group element since its Poisson algebra sums over the winding of x_1 around x_2.

10) In equations (3.23) and (3.24) I believe I should treat \hbar as unitless (also so that it makes sense to subtract from the level, k). Footnote 25 is a bit confusing then; what sets the ‘units’ of \hbar?

11) Missing ) in equation (B.1).

12) Is the coefficient of \delta \rho correct in equation (B.2)? I had a hard time reproducing that it vanishes given equation (B.3).

13) The section on entanglement and anyonic entropy in the discussion is a nice ending point that ties into the motivation introduced at the beginning of the paper, however the large body of this paper being a phase space analysis does little to motivate why we need to utilize a quantum trace in equation (6.7). In some sense, this is a similar question to how does this minimal factorization treat norms (or more generally traces) that I brought up in point 1. That the inclusion of the D=q^{H/2} in the trace is necessary for reproducing gravitational entropy has been noted before, even in JT gravity (e.g. https://arxiv.org/abs/1911.10663 which isn’t cited here).

14) Related to citations, at the end of page 7 the authors might want to cite https://arxiv.org/abs/2404.03651 along with [26] who propose a factorization map (also inspired by [26]) preserving topological invariance of ribbon operators and reproducing anyonic entropy. Similarly although not phrased in the language of factorization maps (instead on subsystem algebras), https://arxiv.org/abs/2306.06158 might be relevant background as well.

See list of changes in the report. Implementing all of these changes is not mandatory for my endorsement of publishing, however it would be good for the authors to consider them and at a minimum address them in a reply if they are not implemented.

Ask for minor revision