Entanglement Rényi entropies in celestial holography

The manuscript aims to calculate the entanglement R\'enyi entropy (ERE) in celestial holography, a conjecture that maps a $(d+2)$-dimensional theory in asymptotically flat spacetime to a
d-dimensional celestial sphere. The authors argue that the superrotated Minkowski spacetime serves as the holographic dual of the celestial conformal field theory (CCFT) on the replica manifold (one should extend the angular coordinates warping around the cosmic string to remove the conical singularity in the bulk). They then compute the holographic ERE following the spirit of AdS/CFT as in [109]. The single-interval ERE matches the results of [73] in the limit
$n\to 1$. The authors further extend their analysis to higher dimensions using a CHM-like coordinate transformation and identify universal coefficients from the Weyl anomaly. While the work is technically sound, I have several suggestions for improvement:

1) Figure/Table for section 4:
Figure 2 provides a helpful summary of coordinates used in Section 3. However, Section 4 introduces additional coordinates related to the CHM map. To enhance clarity, the authors should include a table or figure summarizing the key coordinates and solutions specific to this section.

2)Field theoretic interpretation of ERE:
While the holographic computation of ERE is thorough, the manuscript lacks explicit field-theoretic calculations in CCFT. Since the equivalence between holographic ERE and field-theoretic ERE (defined via the reduced density matrix and twist operator insertions) is non-trivial, the authors should clarify how the reduced density matrix is constructed in CCFT or discuss potential subtleties in interpreting their results as genuine ERE.

3)Clarification on the central charge:
The discussion of the imaginary central charge (previously noted in [73]) and its dependence on the IR cutoff $L$ requires refinement. The authors should explicitly reconcile two seemingly conflicting claims:

i)The assertion that the central charge is unphysical and vanishes after holographic renormalization.

ii)The argument (via AdS/CFT and dS/CFT analogies) that the central charge diverges in the
$L\to\infty$ limit.

A clearer stance on whether the central charge approaches zero or infinity and the physical implications of this divergence is essential.

The authors should clarify these points in the revised version.

Ask for major revision

1. This paper fills a gap in our understanding of the celestial holographic dictionary: namely, the explicit classical bulk dual of CCFT Renyi entropy.

2. This paper presents in celestial holography new and creative applications of methods for calculating Renyi entropies originally established in other contexts: uniformization conformal transformations in two-dimensional CFTs, and the Casini-Huerta-Myers transformation in higher dimensions.

3. The explanation of the calculations were clear and should be understandable by an appropriately broad audience (though some suggestions for improvement on finer details can be found in the "Requested changes" section).

1. One might argue that it's not clear why CCFT Renyi entropies are an interesting set of quantities to consider. In particular, it is not obvious yet what new lessons we will be able to learn about the bulk or boundary theories from this entry in the celestial holographic dictionary.

2. There are some minor issues with wording and typos, laid out in the "Requested changes" section.

This paper studies the Renyi entanglement entropies of celestial CFTs (CCFTs) hypothetically dual to gravitational theories in asymptotically flat spacetime. In particular, it finds creative uses in this context for existing tools in calculating Renyi entropies in two-dimensional CFTs and AdS/CFT in higher dimensions. The main results include the construction of explicit classical bulk solutions dual to CCFTs on replica manifolds and expressions for CCFT Renyi entropies calculated using these bulk solutions. There are some suggestions for minor improvements and typo corrections found in the "Requested changes" section of this referee report, but these do not significantly detract from the impact of this paper.

In general, I believe this paper meets all of SciPost Physics' general acceptance criteria and the particular journal expectations indicated by the authors.

** Section 1: Introduction and summary

*** Section 1.1: Motivation: celestial holography

1. On Page 2, in the second paragraph, it is stated that "Translations act as a constant shift of the advanced or retarded time coordinate". However, I believe this is only true for time translations. Spatial translations result in a non-constant shift of the advanced or retarded time. See penultimate sentence in paragraph around (5.2.3) in arXiv:1703.05448.

2. In the same paragraph, "Lorentz transformations acts" should be grammatically corrected to "Lorentz transformations act".

3. In the last paragraph of page 2, it is stated that "Any duality is ultimately an isomorphism between gauge-invariant states of the Hilbert spaces of two theories." For dS/CFT and celestial holography, I suspect this is not true if one adopts a conventional notion of "states" in the boundary CFT. I will now explain what I mean and why it might be worthwhile to discuss this point particularly in this paper.

A state of an asymptotically flat or dS bulk describes the configuration on a slice (say future infinity) which spans all d directions of the boundary. However, a state of a d-dimensional CFT (or more generally a QFT) describes the configuration on a (d-1)-dimensional slice --- for the sake of disambiguation, I will refer to this as the "conventional" notion of states in the CFT. It therefore seems implausible for bulk states to be related isometrically to conventional states in the boundary theory. (The second paragraph on page 3 touches on this point somewhat. This point is also emphasized in the referenced paper arXiv:2105.00331.)

For completeness (though this is somewhat beside the point) I'll mention that other authors sometimes advocate for a generalized notion of a state in the CFT, which, by definition, are isometric to states in the bulk. (For example, I believe this is the viewpoint of arXiv:2501.00462 --- see section 3.3 therein.) However, this generalized notion of states in the boundary CFT should not be confused with the conventional states of the CFT. For example, celestial CFTs are not unitary with respect to conventional states, but are unitary with respect to this generalized notion of states, since the bulk theory is unitary (presumably).

Why this point might be worth discussing further in this paper: I believe the Renyi entropies considered here are the entropies for conventional states of the CFT, in particular describing configurations on (d-1)-dimensional slices where the twist operators are inserted. Because these conventional states are not isomorphic to bulk states, the aforementioned entropies do not describe the entropies of bulk states (contrary to what a reader with a background in AdS/CFT might expect). There is a hint of this discussion in the second paragraph on page 31, but I feel this fact that CCFT entropy does not equal bulk entropy is a conceptually important point that is worth stating explicitly in the introduction.

4. On page 3, in the fourth paragraph, it may be worthwhile to remark that bulk in- and out-states do not correspond respectively to the two hemispheres, so the relation between the bulk and boundary transition matrices is not the naive equality that readers might imagine. (Each hemisphere can contain operator insertions for both in- and out-states of the bulk. This is closely related to point 3 above.)


*** Section 1.3: The replica trick in celestial holography

5. Page 7, second paragraph: Usually in holography, at least in AdS/CFT, the UV of the boundary theory corresponds to the IR of the bulk. (Admittedly, it is not obvious to me that this is true also in celestial holography.) With this intuition in mind, it seems peculiar to attribute the UV divergence of the CFT to a bulk UV divergence. In particular, I would not say that $S_{grav}^*$ diverges due to effects at short distances near the string deep in the bulk. Indeed, if the bulk did not have infinite IR extent, then (3.28) would be finite, as expressed in (3.29). It may very well be that $\delta$ has a subtle effect on the final answer for the entropy, but I am suggesting that this comes from the part of the regulator tube *near the spacetime boundary*. I would therefore have attributed all divergences of $S_{grav}^*$ to effects occurring in the IR of the bulk, near the asymptotic boundary. However, the choice of wording is subjective to some degree and it is not obvious to me that a UV-IR relation between the boundary and bulk generally holds in celestial holography anyway, so I leave it to the authors whether or not to rephrase the wording.

6. Page 7, third paragraph: It is stated that "we could Wick rotate the direction normal to the great circle defined by the two twist fields, thus producing a compact time direction." Don't we simply end up with two-dimensional Lorentzian de Sitter spacetime? We might get global de Sitter or just static patches depending on the choice of time coordinate that is Wick rotated, but in both cases the Lorentzian time direction is not compact.

(A parenthetical comment that the authors can feel free to ignore: The states one obtains from either these immediate Wick rotations from the sphere or the Wick rotation from $\mathbb{R}^2$ mentioned at the end of this paragraph are just conformal to each other. It's a choice of language, but I would have called these states the "same" in a CFT.)


*** Section 1.4: Summary of results

7. On page 8, it is stated that the universal term in the entropy for general CFTs in any odd dimension is $\propto (-1)^{\frac{d}{2}-2} Z(CS^d)$. What is the reference for this formula? Is the exponent really correct? (If so, the $-2$ seems superfluous.)


** Section 3: Holographic calculation of CCFT EREs: $d=2$

*** Section 3.1: Holographic duals of uniformisation and twist fields

8. Above (3.1): Should $\theta_s$ be ranged in $[0,\pi]$ instead?

9. Below (3.5): Should $U$ be ranged in $(-\infty,\infty)$ instead?


*** Section 3.2: Setup of holographic calculation

10. Below (3.27), in "$\mu=1,2,\ldots,d$", is $d$ really $2$ in this section? I would have thought that we want the Greek indices $\mu,\nu,\rho,\ldots$ to be bulk $(d+2)=4$-dimensional tensor indices. Since we are working with a fixed dimension in this section, it would at any rate be good to specify the range of the indices explicitly or remind the reader that we have $d=2$ in this section.

11. Below (3.29), I think it's inaccurate to say that (3.29) necessarily gives a flat entanglement spectrum (i.e. that $S_{\ell,n}$ is independent of $n$). This is because $A_{string}$ is generically $n$-dependent.

This should be evident from the calculations in section 4 starting from (4.14), but just in case, let me briefly describe two ways to see that $A_{string}$ is generically $n$-dependent. Firstly, from the perspective before the orbifold by the replica symmetry (so the bulk is everywhere smooth and has $\mathcal{M}_n$ as its celestial boundary), the bulk geometry generically depends on $n$ and the bulk codimension-two fixed surface under the replica symmetry can have an area $A_{string}$ dependent on $n$. Secondly, from the perspective after the orbifold (so the bulk is conically singular and has $\mathcal{M}_n/\mathbb{Z}_n=\mathcal{M}_1$ as its celestial boundary), the backreaction of the string on the spacetime geometry should depend on the string tension. Consequently, $A_{string}$ should generically depend on $n$. The fact that $A_{string}$ depends on $n$ is also the reason we find a non-flat entanglement spectrum in AdS/CFT, except when we fix the value of $A_{string}$ by hand (i.e. when we consider fixed-area states). I gather that we are not considering fixed-area states in this paper.

Perhaps a better set of words is that, in order to obtain a finite regularized expression for $A_{string}$ which displays its $n$-dependence, we need to regulate its divergence near the asymptotic boundary. This involves the part of the regulator tube near the asymptotic boundary, so we will keep $\delta$ around in case it plays a subtle role here.


*** Section 3.3: Holographic calculation

12. Top of page 20 and Figure 2: I believe it's inaccurate to say that the geometry depicted in the bottom right of Figure 20 is conically singular. In particular, in the top and middle rows (which are coordinate transformations of each other), the celestial geometry $\mathcal{M}_n$ has an opening angle of $2\pi n$. The phase of $w$ in the middle panel gives a *coordinate* opening angle $2\pi$, but the actual metric opening angle is $2\pi n$ if we account for the singular conformal factor. If we take an orbifold by the replica symmetry, we should then get a smooth geometry $\mathcal{M}_n/\mathbb{Z}_n=\mathcal{M}_1$ with metric opening angle $2\pi$ in the bottom row. (Again, $2\pi/n$ is just a coordinate opening angle for the phase of $w$.)

13. Eqns. (3.36) and (3.37) and Figure 3: Eqns. (3.36) and (3.37) use (3.15a) and assume that the corrections to (3.15a) are negligible. Can the profile of the red cutoff surface in Figure 3 really be trusted everywhere it is drawn? I suspect that, for any large but fixed value of $r_c$, the approximations (3.15a), (3.36), and (3.37) (as well as the profile of the red surface in Figure 3) cannot be trusted for $z$ sufficiently close to $z_1$ and $z_2$. (E.g. one might worry that the corrections to (3.15a) can become comparable to the displayed term as $z\to z_1,z_2$.) If this is the case, then to take increasingly smaller $\delta$, one must choose an increasingly large $r_c$ in order for the calculations in this section to be accurate.

If the above is true, then I suggest firstly explaining this explicitly, and secondly omitting any parts of the red cutoff surface in Figure 3 that might not be accurate.

14. Page 25, third paragraph: Again, can we really trust Figure 3b in the limit where we take $\delta\to 0$ while keeping $r_c$ finite? (See preceding point.)


** Section 4: Holographic calculation of CCFT EREs: $d\ge 2$

*** Section 4.1: Review: Casini-Huerta-Myers in AdS/CFT

**** Section 4.1.2: Casini-Huerta-Myers in AdS

15. Somewhere around (4.19), one should mention that this geometry is an AdS hyperbolic (a.k.a. topological) black hole and give appropriate references. (At least one reference that I know of which studies entanglement in AdS/CFT using hyperbolic black holes of various temperatures is arXiv:2312.06803. One might look for older references cited therein.)

16. At the bottom of page 31, it is stated that "we will perform a bulk coordinate change that produces this cosmic brane" --- I think it's inaccurate to describe the procedure as a coordinate change in $d>2$. In particular, the intrinsic geometry of each AdS slice changes, even away from the brane, in the step where we replace (4.17) with (4.19). For example, with (4.17), the geometry is locally pure AdS and is locally maximally symmetric, as can be seen if one calculates the Riemann tensor. However, I believe the hyperbolic black hole with (4.19) is not locally pure AdS even away from the brane and has a different Riemann tensor.

A comment about why $d>2$ is different from $d=2$: In $3$ dimensions, solutions to vacuum Einstein's equation with a cosmological constant are all locally pure (A)dS. This explains why we were successful in section 3 in describing everything in terms of changes in coordinates, without affecting the local geometry away from the string. The same statement is not true in $d+1>3$ dimensions.


*** Section 4.2: EREs in higher-dimensional CCFT

17. Around (4.24), it might be helpful, for sake of the reader, to state that the $t$ here and below is different from the $t$ introduced in section 4.1.

18. Figure 5: The caption states that the vertical grey arrow indicates the direction of increasing Milne time $\tau$. However, from the definition (4.28) of $\tau$, I believe that the arrow should be reversed in AdS$_-$.


**** Section 4.2.1: AdS$_\pm$ patch contributions

19. Around (4.28), one should mention that the $\tau$ here and below is different from the $\tau$ introduced in section 4.1.

20. Above (4.35), it is stated that "Eq. (4.31) remains a local solution to the vacuum Einstein equations if we replace $f(\zeta)$ in eq. (4.32) with the function in eq. (4.19)". I encourage the authors to explain why this is true.

As remarked in point 16 above, replacing (4.32) by (4.19) leads to a different local geometry on each slice of fixed $\tau$. In particular, the metric in the square brackets of (4.31) becomes a Euclidean AdS hyperbolic black hole at different temperatures set by $\zeta_0$ --- I understand that this is a solution to the $(d+1)$-dimensional Einstein equations with a cosmological constant (with the possible exception of the conical singularity at the bifurcation surface). However, it is not immediately obvious to me that this must uplift to a solution of the $(d+2)$-dimensional Einstein equations.

I do believe the claim, because I briefly checked it computationally (out of laziness) for several examples of $d\ge 3$. Nevertheless I think it would be beneficial to have a general argument presented. One might say that at least half of section 4 is devoted to deriving this solution, so it would be worthwhile to explain why it is indeed a solution.


**** Section 4.2.3: Results for EREs

21. Top of page 39: Same comment about the formula $(-1)^{\frac{d}{2}-2}Z(S^d)$ as point 7 above.


** Section 5: Summary and outlook

22. In the last paragraph of the main text, it is stated that "Carrollian conformal symmetry [...] roughly speaking is the limit of conformal symmetry in which the speed of light goes to infinity." I am no expert on this, but I thought that limit gives a Galilean theory and the opposite limit of zero speed of light gives the Carrollian theory.

Publish (easily meets expectations and criteria for this Journal; among top 50%)