Bulk entanglement entropy for photons and gravitons in AdS$_3$

1) Exceptionally well written

2) Relevant topic, combines multiple strands of research

3) Reaches an intriguing result and corroborates the HLM prescription

1) A few physical points were addressed in insufficient detail - see Report

This paper was a pleasure to read. Overall it is written very clearly, which makes it very easy to go through and understand. Using rather standard techniques for calculating thermal partition functions of 2D CFTs, the authors calculated entanglement entropies of topological theories in $AdS_3$ for subregions that contain a piece of the AdS boundary (more precisely, for bulk entanglement wedges corresponding to boundary regions). Several examples were studied: vacua of $U(1)$ Chern-Simons theories, simple excited states of $U(1)$ Chern-Simons theories, and vacua of Einstein gravity (which, in $AdS_3$, is a Chern-Simons theory with a non-compact gauge group). The result indicated that the universal piece of entanglement entropy takes the form characteristic for 2D theories, though with bulk and boundary data contributing parametrically different terms. No (obvious) 3D-specific universal terms were found.

The calculation was very concrete, presented succinctly but not sloppily, and I believe it sets a good standard for the field.

I have a few comments or confusions. They can be divided into three classes, according to severity:

1) Very important conceptual issue:

When studying $U(1)$ gauge fields in $AdS_3$, the authors work with a pure Chern-Simons theory. This is a good warm-up for studying pure gravity, and this choice can be picked by fiat too. But I am less comfortable with this calculation if Chern-Simons is an effective bulk theory (as mentioned at the top of page 3). The reason is that entanglement entropy has the pesky ability to sense UV physics (see Kabat, Shenker, Strassler [hep-th/9506182]). In the context of 3D gauge theories, this Maxwell vs Chern-Simons issue was studied by Agon, Headrick, Jafferis and Kasko [arXiv:1310.4886] and by Radicevic [arXiv:1509.08478]. These papers all claim that, depending on the size of the region $L$, the universal term will either be of the form $\frac12 \log k$ or $\frac12 \log(e^2L)$, where $e$ is the Maxwell coupling. The crossover between these universal terms happens at $L \sim k/e^2$. In particular, if $k \rightarrow \infty$, as the authors assume, then the Maxwell-appropriate universal term will always be in the entropy, even though Chern-Simons dominates the IR physics at the level of correlation functions.

Of course, all these results are in flat space. In AdS, the size of the region $L$ diverges, so maybe you can pick a regime where $L \gg k/e^2$. This may be impossible if $k \sim c_{\textrm{CFT}}$, as mentioned in footnote 3. Maybe a way out is to find that the entire crossover doesn't happen in AdS. That would be an interesting finding on its own; I am not aware that anyone checked this. I think there is some interesting physics here. At worst, the authors can explicitly restrict to studying pure (not effective) Chern-Simons theories in the bulk.

2) Important clarifications:

2.1) I understand the authors take large $k$ because that is featured in brane constructions. But is that necessary for this calculation, assuming the Chern-Simons theory is pure? Maybe the assumption can be dropped altogether?

2.2) In Sec. 3.1, I would like to ask the authors to consistently use terms "states" and "operators". The state-operator correspondence does not work in the bulk, so the sentence "In particular, states..." turned out slightly confusing. I think this sentence meant "The bulk is topologically trivial, so Wilson lines in the bulk are trivial, so their boundary duals are trivial too, and hence the boundary states these boundary operators correspond to are not in the Hilbert space of interest." I would also delete references to dualities of oranges...

2.3) The conformal mapping in Sec 3.3 is stated a bit unclearly. The sentence "this surface is topologically a torus" seems to be referring to the bulk tube that was just defined - and topologically this is manifestly a cylinder, not a torus. I am pretty sure the authors were imagining a torus obtained by concatenating the bulk tube/cylinder with a corresponding boundary tube, but I admit I found this hard to infer solely from the text or from Fig. 4. I think it would be very helpful to indicate $\epsilon_F$ on Fig. 3, and perhaps to very explicitly state which surfaces we are talking about before ever conformally mapping them.

3) Minor kvetching:

3.1) It's Kac (or Кац in Russian), not Kač (or Кач). A lot of the old literature mysteriously changes his name, as was done 4 out of 9 times in this paper.

3.2) It's Virasoro, not Virasoso (2 out of 7 times in this paper)

3.3) I understand the authors meant "boundary photons" when choosing the title, but I wonder if it can be changed to more generically refer to gauge fields or gauge theories, especially in view of my comment 1). Chern-Simons doesn't really have photons on its own. Same for gravitons.

3.4) Have the authors considered comparing their result to the bulk entropy associated to an "island" in the bulk, i.e. to a region that doesn't extend all the way to the boundary? It would be simple to calculate but may be nice to present.

3.5) Speaking of universal terms on subregions with boundaries, have the authors found any inspiration in the vast literature on entropic g-functions? This was done mostly for conformal theories (Jensen and O'Bannon arXiv:1509.02160 seems like a relevant starting point for 3D theories). It would at least be nice to contrast those results with the ones obtained here.

3.6) In equation (1.5), perhaps $A$ should have been replaced by $L$?

3.7) What is the boundary dual of $\epsilon_{\textrm{bulk}}$? Less annoyingly said, how should it scale with other "small" parameters, like $\epsilon_{\textrm{CFT}}$, $G_N$...?

3.8) What symmetry are we talking about when we talk about symmetry-breaking relevant deformations at the bottom of page 10?

3.9) Fig. 3 is very nice, but for a bit I was confused by the gray line that almost (but not quite) overlaps with the red line $\gamma_A$. Maybe that gray line can be removed?