Constraining braneworlds with entanglement entropy

In this paper the authors use consistency conditions on entropy to rule out certain values for DGP couplings in a particular bottom-up model of wedge holography. The paper is well written, clearly structured and suggests a nice way for how to link Quantum information theory, the Swapland programme and holography and I recommend its publication.

In addition to the other referees' comments I only have one point I would like to stress:

1- As already mentioned by another referee, the discussion of positivity of EE needs to be improved. In particular, 1. subregion entanglement entropy does not generally count microstates (as e.g. mentionted on p.3 or p.20). 2. positivity of EE in subregions is a scheme-dependent statement and relies on specifying a regularization and renormalization procedure (The statement about "typical AdS calculations" on p.20 is very vague and should be made more precise). 3. The fact that the brane construction gives UV finite answers should be discussed in a bit more detail. After all, since the region $\mathcal R$ under consideration is does not split a connected CFT region I would think that all the discussions about UV divergences are immaterial, since the authors work with a type I algebra whose EE is manifestly finite.

The article studies braneworld models in AdS where the theory that describes the brane contains, in addition to the brane tension, a DGP term. The aim is to obtain constraints on these brane parameters from the requirement that (i) the entanglement entropy on the brane is positive, and (b) that causal wedge inclusion is satisfied. Since these are constraints on the coefficients of the low-energy effective theory (LEFT) of the brane, they are regarded as similar to swampland constraints.

The idea of the article is new and interesting, and the article is well written, so I recommend its publication in SciPost after the points I raise are addressed.

1-In p.3, $\mathcal R$ appears without having defined it before.

2-In the last paragraph in p.4, it seems to me that (I) and (III) are interchanged relative to their definition two paragraphs above.

3-Positivity of entropy is justified in eq.(2.8) using the microcanonical concept of entropy as a coarse-graining over a number of microstates. However, the concept of entropy that is used in this paper is not this one but rather that of entanglement entropy, whose positivity follows from a different condition. In general, entanglement entropy is not directly related to a microstate counting, and even if in many circumstances such an interpretation turns out to be possible, this requires some elaboration. A reworking of the argument, in one direction or another, should be made in the paper.

4-Above eq.(2.12), $A_h$ and $S_h$ are referred to as distinct quantities (one upper-bounds the other) while in (2.12) they are set equal. This should be clarified.

5-The argument for CWI in black holes in p.12 is based on the claim that the $t=0$ section of a one-sided eternal black hole has no interior geometry. This does not seem correct to me. The section $t=0$ of the two-sided eternal black hole does not have any interior since the ER bridge connects two different exteriors, but in an eternal one-sided geometry, the ER bridge opens to a compact interior. Indeed it can contain an arbitrary interior (as in bag-of-gold examples). I would like the authors to clarify this point.

6-At the top of p.15 it says that with a $K^2$ term "a KR brane with only a redefined tension is not as obviously a solution." This sounds ambiguous. Is it never a solution, or can it be a solution in some cases? If it's the latter, then what argument can be used to exclude this term from the brane LE?

7-In my opinion fig.5 would be clearer if only non-redundant information were presented, ie the region below the diagonal, and only the strongest of the two constraints for cases of the same colour (the dashed line). This is a mere suggestion, and the authors may or not agree with it. Similarly for fig.6.

This paper studies certain consistency constraints on braneworld scenarios in AdS. The constraints come from two properties of the holographic entanglement entropy: it is positive and must obey causal wedge inclusion (CWI). The authors consider a system of two branes, anchored on a shared surface at the boundary of AdS, in the background of a black string. The action of the branes is taken to be the Randall-Sundrum one (\emph{i.e.} the tension term) augmented by an Einstein-Hilbert term, known as the Dvali-Gabadadze-Porrati (DGP) term. The authors show that the entropy constraints rule out certain regions in coupling space.

Presently, the main motivation of interest in these models is in relation to the black hole information paradox. Indeed, these branes support a massless graviton, and the entropy exchange between the two branes can be shown to follow the expected time-dependent Page curve, for appropriate choices of couplings. The most consequential claim of the authors is that such choices of couplings are in the swampland, thus strengthening the conclusion that non-trivial islands, which lead to the Page curve, only exist in theories with a massive graviton.

Both the approach and the conclusions of this work are interesting. The paper is well written and, as far as I can see, correct. I will be happy to recommend it for publication after the authors have addressed the following observations.

1. In the paragraph before eq. 1.2, two sentences contain the unnecessary repetition of the same concept, with the same exact words: "[the branes are] coarse descriptions of warped compactifications in string theory".
2. More importantly, it is not known---at least to me---what generality can be attributed to the statement itself. The available UV complete construction of boundary conditions for holographic CFTs often do not look like Karch-Randall branes at low energy, and might for instance involve extra dimensions shrinking smoothly instead. The authors do not claim differently, but it would be better to add a paragraph which points this out explicitly, since the reader might get the impression that the theories considered here are universal low energy effective field theories for CFTs with defects. Some literature in this context should be cited as well, including for instance 2108.10345, 2110.05491, 2212.14058 and some additional references to the appropriate string theory constructions.
3. At the bottom of page 4, there is a typo: ''the Von Neumann entropy of the entropy of the defect''.
4. In the outline---section 1.2---the authors summarize their finding that a ''range of combinations of DGP couplings are disallowed''. This allows me to state the only criticism about the exposition of the results. The region in coupling space which ends up in the swampland is never fully explained. It is a region in the four dimensional space spanned by the two tensions and the DGP couplings (or equivalently by $\theta_i$ and $\lambda_i$), so it cannot be drawn on paper. Part of this region is analytically known, though: the one excluded by the criterion of positivity of the entropy. The authors draw a projection on this region in figure 5: it would be useful to refer to this figure, and to the equations that determine the full region in $(\theta_i,\lambda_i)$, already in the introduction. Vice versa, the region which is put in the swampland by CWI is only explored numerically, and results are reported only for a few values of the brane tensions. Nevertheless, figure 6 also deserves a reference in the introduction, so that the reader can easily decipher the main results of the paper.
5. In section 2.1, the modification of the RT prescription for the brane setup is explained. When the authors declare to use Neumann boundary conditions for the end-points of the RT surface on the brane, 1105.5165 (ref. 67) should be cited, since this prescription was proposed there. Furthermore, it would be useful to comment on its status. To the best of my knowledge, no complete proof à la Lewkowycz-Maldacena is available in this case. Vice versa, arguments in favor of its validity were presented in appendix A of 2006.04851 and in section 5.1 of 2202.11718.
6. On page 10, the ideas behind CWI are outlined. In view of the importance of this fact for the results on the paper, it would be advisable to comment explicitly on the extension of the theorem of reference 36 to the case of wedge holography.
7. On page 15, the authors discuss the possibility of adding other two derivative couplings beyond the DGP term. They disregard this possibility because of the unclear effect of these terms on the prescription for the computation of entanglement entropy. This is an important caveat, and should be mentioned in the introduction or in the conclusions. It also solicits a more general question about the robustness of the swampland constraints. Reliable EFT constraints are statement about low energy Wilson coefficients which are independent of all higher derivative couplings: they state, for instance, that no UV completion exists for certain values of these coefficients. Is this the case for the entropy constraints? The constraints only seem to rule out theories with precisely the considered Lagrangian, since further higher derivative modifications might change the entropy functional and the brane embeddings. Is there an argument why higher derivative couplings are expected to give parametrically small corrections to the bounds obtained here? In other words, is the entanglement entropy a quantity perturbatively computable in the derivative expansion? Either way, the authors should comment about this issue in the paper.
8. On page 19, the proof of non-extremality for non-horizon surfaces in the case of positive DGP coupling on the branes is explained. I find the phrasing of this proof slightly inaccurate, despite the conclusion being correct. In eq. 3.23, the inequalities should not be strict. Having both limits to be zero (\emph{i.e.} a smooth minimum for the curve) is allowed. More to the point, this equation is not the relevant one to prove a contradiction with $u''(\mu_0)<0$. Indeed, the value of (the limit of) the first derivative of a function is not constraining for the value of the second derivative. What the authors probably meant is that in a left neighborhood of $\mu_0$ $u'(\mu_0)<0$ and in a right neighborhood $u'(\mu_0)>0$, which indeed is incompatible with $u''(\mu_0)<0$. Finally, the proof leaves open the logical possibility of a finite discontinuity in the derivative (rather than the infinite one considered in the paper). In this case, $u''(\mu_0)$ is not defined, while the left and right limits $u''(\mu_0^\pm)$ exist and might have any value in principle. This case should be excluded on general grounds: the Euler-Lagrange equations should have smooth solutions if the metric is non-singular. The authors might want to point this out explicitly.
9. Above eq. 3.32, the authors write that ''it is natural to'' place points which disobey positivity of the entropy in the swampland. This kind of vague phrasing is used in multiple occasions throughout the paper. Why is this? Theories with precisely these values of the coupling (with the caveat of higher derivative corrections raised above) are rigorously excluded by the analysis, so the authors should make plain statements: these points are in the swampland. If, on the other hand, other caveats make the results of the paper non-rigorous, those concerns should be discussed.
10. In section 3.4, the requirement of CWI is explored numerically. Figure 6 contains a few examples of points which are put in the swampland. Are these the only values of the tension for which the numerics was run? It would be interesting and more complete to have an idea of the shape of the excluded region as a function of $\theta_i$ as well. Could the authors for instance draw a $3d$ shape of the boundary of the green region in $(\lambda_1,\lambda_2,\theta_1)$ at fixed $\theta_2$?
11. On a related note, it is important to emphasize that the values of $\lambda_i$ which obey 3.51 are \emph{not} automatically in the swampland. While implied in the paper, I feel that this is not stated clearly enough, for instance around eqs. 3.49-3.51. A point in the $\lambda_i$ plane is only excluded if it lies in the swampland for all values of the tensions. Besides being relevant to understand the boundaries of the swampland, this consideration has consequences on the main application of the paper's results, \emph{i.e.} the (non-)existence of a time dependent Page curve.
12. At the bottom of page 24, it is said that the CWI-violating points lie in $\left\{\lambda_1<-1\right\} \cap \left\{\lambda_1<-1\right\} \cap \left\{\lambda_1+\lambda_2<0\right\}$. This looks like a typo: should it be $\left\{\lambda_1>-1\right\} \cap \left\{\lambda_1>-1\right\} \cap \left\{\lambda_1+\lambda_2<0\right\}$?
13. On page 25, the paragraph which starts with ''How does this story depends on the combination of angles?'' contains rather vague statements, like excluded points being ''close'' to the bounding line and the CWI condition becoming ''weaker'' in a certain loosely defined region in parameter space, where the wedge is ''wider'' (the quotation marks are from the authors!). In accordance with a previous comment, a more quantitative characterization of the swampland would go a long way: for instance, the authors could say if in the limits $\theta_1,\,\theta_2 \to 0$ the swampland reduces to the basic region of negative effective Newton constants on the branes. This limit is especially interesting with regards to the previous comment on higher derivative corrections to the action. Indeed, it corresponds to weakly coupled gravity on the brane. In this regime, higher derivative terms are expected to give parametrically suppressed contributions to on-shell quantities, and the swampland criteria of this paper should become precise. The authors could also sharpen the loose discussion in this paragraph with a plot of, say, the intersection of the green points with the $\lambda_1=\lambda_2$ line as a function of $\theta_1=\theta_2$.
14. On page 26, the authors point out that the CWI-violating couplings do not bound the swampland. Again, this would be an occasion to comment more explicitly on the topology of the swampland, as defined in this paper. It is not convex and its projection onto the fixed $\theta_i$ plane has two disconnected components. Considerations on the $\theta_1,\,\theta_2 \to 0$ limit should also clarify if it is a disconnected manifold also in the full four-dimensional space.
15. In the conclusions, the unphysical nature of two-brane models giving rise to a non-trivial Page curve is affirmed. However, it is not clearly stated if \emph{all} of the parameter space which gives results like in ref. 60 is in the swampland. The authors again seem to avoid sharp statements (''we would assert'', ''we would put [these models in the swampland]''), but more importantly, two features of these models are mentioned but their relation is not fully clarified. On one hand, a time-dependent Page curve arises from time dependent RT surfaces. On the other hand, ''one may also get static RT surfaces which are not the horizon''. The latter feature automatically violates CWI and are in the swampland. But it is the former feature the important one for the black hole information paradox, and violation of CWI for time dependent surfaces is not explored in the present paper. Are there values of the couplings which allow for time dependent Page curves, but do not lie in the swampland as defined by the authors' results?
16. A final question, which the authors could clarify in the paper---or just answer in their response to this report---pertains locality. The boundaries of the swampland show a non-trivial dependence of the couplings of one brane on those of the other. This is a natural outcome of the procedure, but is surprising on general grounds: the UV physics, which the swampland is supposed to uncover, should be local. Is this situation due to the branes touching on the boundary of AdS? Would the swampland criteria become local, if the branes were to be separated, thus defining the dual of a CFT on a slab?