SciPost Submission Page
Minimal Factorization of Chern-Simons Theory -- Gravitational Anyonic Edge Modes
by Thomas G. Mertens, Qi-Feng Wu
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Qi-Feng Wu |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2505.00501v3 (pdf) |
| Date submitted: | Oct. 21, 2025, 1:57 p.m. |
| Submitted by: | Qi-Feng Wu |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
One approach to analyzing entanglement in a gauge theory is embedding it into a factorized theory with edge modes on the entangling boundary. For topological quantum field theories (TQFT), this naturally leads to factorizing a TQFT by adding local edge modes associated with the corresponding CFT. In this work, we instead construct a minimal set of edge modes compatible with the topological invariance of Chern-Simons theory. This leads us to propose a minimal factorization map. These minimal edge modes can be interpreted as the degrees of freedom of a particle on a quantum group. Of particular interest is three-dimensional gravity as a Chern-Simons theory with gauge group SL$(2,\mathbb{R}) \times$ SL$(2,\mathbb{R})$. Our minimal factorization proposal uniquely gives rise to quantum group edge modes factorizing the bulk state space of 3d gravity. This agrees with earlier proposals that relate the Bekenstein-Hawking entropy in 3d gravity to topological entanglement entropy.
Author indications on fulfilling journal 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
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Strengths
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
Weaknesses
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.
Report
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.
Requested changes
I refer to the attached file.
Recommendation
Ask for major revision
Strengths
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.
Weaknesses
1) easy to get lost in notation
Report
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.
Requested changes
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.
Recommendation
Ask for minor revision
We thank the referee for their report. We address the comments one by one.
1) The factorization map should be an isometry at the quantum level, in order to implement shrinkability of a microscopic hole in the Euclidean path integral calculation. We would claim that our factorization map precisly implements exact shrinkability (the WZW factorization map would lead to a spurious divergence) (by construction). In fact, in earlier work 2210.14196, it was shown that if we could factorize the Hilbert space in such a 'minimal' way, then we would have exact shrinkability and the factorization map would be an isometric map.
2) We have rewritten that part of our paper to improve readability. Our previous statements we believe were correct: the one-sided partition function itself would include a path integral over all surface charges (they becomes phase space variables of the one-sided system). We have rephrased this discussion to clarify.
3) We can obtain F_{\mu\nu} (in any gauge theory) by considering an infinitesimal Wilson loop, so we believe our statement is correct.
4) We are considering Lorentzian signature manifolds. We added this specification right after eq. (2.1).
5) We used the notation x^+ and x^- to denote chiral coordinates, where one can interpret this either as a ligthcone coordinate on the respective boundaries, or just as a spatial coordinate on the boundary circles.
6) We agree, and have moved the earlier footnote 20 up as our new footnote 5.
7) We assumed |x_12| < 2\pi to simplify these equations.
8) r_{12} is the integration constant, and we hence choose it to be independent of m. This is a choice, but otherwise the factorization map would presumably be consistent but much more complicated. This fits into our narrative where we aim to find the 'simplest possible' factorization map.
9) Our statement is that g(x) is locally defined on the circle, but is not a single-valued function on the circle. W(x) on the other hand, contains degrees of freedom that are spatially separated from the circle. So in a sense, W(x) is even more non-local than g(x).
10) We agree, and changed the footnote to avoid confusion.
11) Thanks!
12) We added some extra details on this calculation, but emphasize that this is essentially a summary of older literature, and not meant to be a new derivation.
13) The appearance of the quantum trace is natural from the quantum group structure that we uncover, but we agree that it would be highly valuable to obtaining a better physical understanding in this context of its appearance, which we would postpone to future work. Within JT gravity, the element D should not be literally interpreted as a Drinfeld element, since the group structure is not q-deformed in that case. So we believe these cases are not fully related, but agree this is an interesting aspect to understand better.
14) We added the reference here.
Author: Qi-Feng Wu on 2026-02-03 [id 6296]
(in reply to Report 3 on 2026-01-01)We thank the referee for their detailed report. We go through the different subsections following the referee's report.
1.1
We have cleaned up the language in the introduction to avoid confusion.
1.2.
Our statement is that the gauge-invariant degrees of freedom one would ordinarily have in any gauge theory, get drastically reduced by imposing topological invariance. We agree that gauge symmetry and topological invariance go hand in hand in Chern-Simons theory, but would not argue that one implies the other. (Stating that gauge symmetry implies topological invariance is obviously misleading since YM theories do not have this.)
We thank the referee for this interesting comment and suggestion for finding a more technical description of when this minimal set of edge states can be found. The intrinsic description of the phase space of the Chern-Simons theory only depends on the homotopy type of the underlying manifold. In this sense, there are no bulk Kac-Moody modes to begin with.
1.3
Since both referees raised good points about this subsection, we have substantially improved it, by adding clarifications, definitions and additional information to make this subsection self-contained and readable. In the process, we addressed the comments made by the referee.
2.1
We have added an extensive comparison to the literature at the end of section 1.
Also, when discussing "minimality", the referee is actually referring to the lattice independence of the physical
subspace of the topological lattice gauge theory. What we are aiming for in this work, is a better understanding on whether one can choose a minimally consistent factorization map.
2.2
a) We indeed have Lorentzian signature throughout our work, and have added this clarification at various places in the text.
b) We have added a ref. there towards the original paper that derived chiral WZW at the boundaries of Chern-Simons theory.
c) We have added the clarification just below eq. (2.5) that $W$ is not single-valued on our topology.
d) We have added an explanation after eq (2.11), explaining how our phase space is defined. Concerning presentation of this phase space, we find our eq (2.17) and the following bullet points more intuitive to read for the physics audience so we chose to stick with it.
2.3
a) We rewrote this sentence to improve readability.
b) Eq. (2.23) is a definition for the Kac-Moody current. It is defined in this way because
it is the Noether charge corresponding to the Kac-Moody symmetry.
c) In principle, all of our coordinates are in the universal cover of $S^1$, although we do not always make this clear in the notation. However, we believe adding more notation to make this distinction would deteriorate the overall readability of the paper, so we decided to allow this slight imprecision.
d) They are derived as the action of the current on the Wilson lines.
e) Because y can in principle be in the bulk. We added a sentence to clarify this.
f) Yes, we assume we are working with a semi-simple Lie algebra for which the Cartan-Killing form is non-degenerate. We added footnote 9 to explain this.
2.4
a) We agree. We added footnote 13 to explain this point, refering to a later Fig. 8 for an example we need later on.
b) We choose not to change notation here as before.
c) We changed notation here emphasizing it is not a dirac delta function that appears here when gluing. We are interested precisely in (compact and non-compact) Lie groups, so dealing with the consequences of continuity is a necessity in our framework. We also refer to our introduction for a deeper comparison with the literature.
d) We have rewritten parts of the footnote to address the referee's comments. By invisible, we mean that a one-sided observer views the edge degrees of freedom as living on a horizon, which this observer cannot access in a finite amount of his time.
e) We have indeed already encountered this gauge redundancy in eq (2.21) earlier. The discussion here is meant to restate this in terms of the one-sided phase spaces we defined in this subsection. We are not sure precisely how we could improve the readability of this discussion.
2.5
a) We added footnote 14 to warn the reader about the change in the notation in this and the previous subsection.
b) y can be in the bulk.
c) This was a more "philosophical" comment, but we removed the footnote. We have explained this notation earlier in the introduction above eq (1.5), and the footnote was hence superfluous.
d) Both referees asked about this point, and we agree with the referee's comments about this. We can interpret either $r$ or $\tilde{r}$ as the integration constant (w.r.t. $x$-integration). We added this comment after eq (2.64).
e) We removed the confusing footnote, and for our purposes we can take the definition of the $r$-matrix as a solution to the YB equation.
f) We added a footnote refering to the equation that can be used to readily derive this identity.
3.1
a) we agree that the map is a priori many-to-one. We impose additionally locality to distinguish g(x) from h. Moreover, g(x) is the WZNW field which is known to satisfy the given Poisson bracket in the literature. The associated r-matrix is in general not necessarily the same as the one associated with W (x). Different choices will lead to the affine Poisson structure discussed in Appendix D.
By locality, we mean here that spatially separated degrees of freedom commute with each other. Our argument of locality is physically and mathematically well-motivated. E.g. in combinatorial quantization, it is imposed that U(i) and U(j) commute with each other if the links i and j do not share a common vertex.
The earlier literature did not arrive at the algebra for h, nor the dressing bracket relation.
b) This has nothing to do with whether G is complexified or split-real. h and m are
both valued in the same manifold G. We never specify the group multiplication of m. As
manifolds, G and G∗ are locally indistinguishable.
3.2
We inserted factors of $i$ to indeed correspond to the compact group U(1).
3.3
a) 3d $\Lambda \lt 0$ gravity in Lorentzian signature.
b) We added an additional clarification in the sentence. It is the algebra generated by a,b,c,d with the relations (3.23)-(3.24) imposed.
c) We need to make a distinction between the linear limit $q\to 1$ (or $k\to\infty$), and the classical limit $\hbar \to 0$. We added a summarizing scheme of the difference after the paragraph following eq (3.24).
3.4
a) This ambiguity exists generally in the chiral WZW model. The current subsection is meant to explain this directly for the simple U(1) model in terms of constrained quantization.
b) Indeed, we modified the sentence below eq (3.39).
c) This footnote is optional; and our only goal in it was to say that extension of this argument to the non-abelian case might not be as straightforward.
4
a) We removed this sentence as it was superfluous in any case.
b) We believe we have reinforced the interpretation of h as a genuine phase space degree of freedom earlier, and therefore stand behind our comment here.
5
We changed the word factorization map into gluing map
5.1
We agree that this is not a wanted construction, but feel it is worth mentioning for completeness.
5.2
a) One can go between interpreting $r$ or $\tilde{r}$ as the integration constant, since neither depends on the $x$-coordinates.
b) We do not have an interpretation of these factorization maps directly in a Lagrangian language (which would be very interesting to achieve). We also remark that none of these are WZW model factorizations.
c) We have changed the notation of eq (5.5). We only mean that the element can only be glued if the endpoints match, otherwise it is implicitly assumed to map to zero. This is similar to being only able to glue edge modes with matching charges.
d) Indeed, we changed the word into the Kac-Moody gluing map.
e) We included a more detailed reference to what was done in ref. 19.
7 (was section 6 in the earlier version)
The comments on "towards an action principle" on the black hole horizon are meant for the application of this factorization map to 3d gravity. In that case, the entangling surface is a black hole horizon, and the one-sided observer sees degrees of freedom on the horizon as infinitely redshifted and frozen. In particular they cannot evolve in time and hence no interesting dynamics is associated to them. This is found in particular when including a brick wall cutoff to entanglement entropy calculations of edge states as described in e.g. Donnelly and Wall's 2014 papers.
We have reinforced our discussion and application to 3d gravity in the last paragraph of the conclusion.
Appendix B
a) The Poisson structure is globally defined and degenerate. So the phase space is foliated
into symplectic leaves on which there are symplectic forms. The foliation is singular. The symplectic form given here is defined on the largest leaf whose complement is of measure
zero in the phase space.
b) We simply mean here the idea of adding degrees of freedom to the symplectic form to allow it to factorize is heavily utilized in this literature, not the precise calculation we in chiral WZW we presented in this appendix.