Entanglement Rényi entropies in celestial holography

The authors have addressed my question in detail. Entanglement entropy in flat holography is a topic that deserves further study, and while some issues may remain unresolved, the results presented here are consistent with prior works and provide an interpretation of a certain cosmic string as the holographic dual of a twist operator. Therefore, I recommend acceptance for publication.

Publish (meets expectations and criteria for this Journal)

Echoing my previous report, I believe this paper meets SciPost Physics' general acceptance criteria and the journal expectations indicated by the authors. Moreover, it seems that the authors have addressed the suggestions made in my last report. I therefore recommend the publication of this manuscript.

As I will elaborate below, I am still not convinced that the bulk Hilbert space is isomorphic to the CFT Hilbert space (according to the conventional notion of "states" in the CFT) in dS/CFT and celestial holography. However, I do not think this is a point on which the decision to publish should critically hinge. (So, if the authors still feel confident about this claim, then I would not be strongly opposed to the authors keeping this stance in the paper.) Let me now elaborate on my confusion, so that at least I might learn something from the authors' reply or vice versa.

The authors remind us in their reply that both the bulk Hilbert space and the CFT Hilbert space give representations for the same conformal symmetry group. Firstly, I agree with this statement, but I don't think that's enough to conclude that the Hilbert spaces are isomorphic to each other.

Secondly, to establish a common language, let us consider section 4 of arXiv:2105.00331, which the authors mention in their reply. The bulk Hilbert space, I would say in this setup, is $\mathcal{H}_{in}$ or equivalently $\mathcal{H}_{out}$ --- the two are isomorphic, being related by unitary evolution. The (conventional) CFT Hilbert space, I would say, is $\mathcal{H}_N$ or equivalently $\mathcal{H}_S$ --- again, I think the two are isomorphic. Let me emphasize I really do mean "or" and not "and" in these sentences, meaning, e.g. the CFT Hilbert space is $\mathcal{H}_N$ or equivalently it is $\mathcal{H}_S$, but it is not some union, sum, or product of the two at the same time. In CFT for example, by radial quantization, one should be able to prepare all, e.g. ket, states using operator insertions in say, the southern hemisphere, or even just at the south pole --- these states form the full Hilbert space. Bra states can be prepared on the opposing hemisphere and the path integral over the whole sphere computes inner products. (The restriction of operator insertions to a hemisphere in state preparation is related to the radial-ordering mentioned further below.)

The point is, I don't think $\mathcal{H}_{in}$ (or equivalently $\mathcal{H}_{out}$) is isomorphic to $\mathcal{H}_N$ (or equivalently $\mathcal{H}_S$). I also don't think section 4 in arXiv:2105.00331 claims this isomorphism exists. (And, if I am reading the text on page 3 of v2 of the current manuscript correctly, it seems the authors here are also not claiming this isomorphism.) But what other meaning could there be to having an isomorphism between the bulk Hilbert space and CCFT Hilbert space? (As mentioned in my previous report, if one instead considers a less conventional notion "states" in the CFT as I think some papers do, then one might be able to claim an isomorphism with the bulk Hilbert space almost by definition.)

Thirdly, to further illustrate this point, let us try to write down the hypothetical isomorphism between bulk states and CCFT states (in the conventional sense). Consider the bulk states

where each $a(w)^\dagger$ is a creation operator associated with a conformal primary wavefunction (or possibly a shadow) for a bulk field, with $w$ denoting a position on the celestial sphere. Suppose $w_1,\ldots,w_n$ includes points distributed throughout the sphere (specifically, not restricted to a hemisphere) and the same is true of $w_1',\ldots,w_m'$. The question I would like to ask is: what CCFT states would the above bulk states correspond to under the hypothetical isomorphism?

An isomorphism of Hilbert spaces must be an isometry, preserving inner products, so let's consider the inner product of the above bulk states. Keeping implicit the isomorphism between $\mathcal{H}_{in}$ and $\mathcal{H}_{out}$ (i.e. the unitary evolution that involves the $S$-matrix),

From what little I know about celestial holography (so please correct me if I'm wrong), this is equated to a CCFT correlator, as expressed in the second line. This notation for the correlator means I imagine performing the Euclidean CCFT path integral with the indicated operator insertions. Now, crucially, this means that if I want to write it as a CCFT vacuum expectation value of operators acting on states in, say radial quantization on celestial space, then radial ordering $\mathcal{R}$ must be imposed: I find it difficult to believe that there is an isomorphic map of $|\psi\rangle$ and $|\psi'\rangle$ to the CCFT Hilbert space, such that the inner product of the resulting CCFT states coincides with the above. Among other things, because $\mathcal{R}$ reorders the operator insertions, the naive guess, that $|\psi\rangle$ maps isomorphically to $O_+(w_1) \cdots O_+(w_n) |0\rangle_{CCFT}$ and similarly for $|\psi'\rangle$, does not seem to work.

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