Area law and OPE blocks in conformal field theory

1- Contains concrete computations and results that encode nontrivial information of CFT.

1- Some key quantities and major steps lack clear definition/explanation.
2- Overall physical significance is unclear.
3- Exposition, grammar, and clarity of writing needs major improvement.

The author studies nonlocal operators in CFT known as "OPE blocks", computes their correlators, and extracts the universal logarithmic pieces in the small cutoff limit. The key results are a UV/IR relation (3.11) and a cyclic identity (3.29). However, I have some questions about both results, and about the significance of "area law" in this story. I have listed my requested changes in order of appearance in the paper. The more important ones are 5, 13, 14, 15, 16, 18, 19.

Overall, I think the paper can be improved by making a stronger case for why the studied quantity may be physically interesting (point 19), and/or computing more examples (point 18). I cannot yet recommend this paper for publication until a major revision is made, possibly with some enrichment of the content.

Disclaimer: I am an expert in CFT but not specifically in OPE blocks.

1- line 25: Is OPE block a new topic? Maybe its detailed study as a non-local operator, its relation to information theory, or its application in the holographic context is, but references [5,6] are certainly not new. I suggest the author be more specific about what the new topic is.

2- line 33: Leading order in what?

3- line 39: I cannot understand the sentence: "This leads to the conjecture that OPE block may be related to area law as modular Hamiltonian."

4- line 44: Natural numbers do not include 0, but as noted later type-J in 3d has q=0. It is also weird to not just say "0, 1, 2", instead of "natural number no larger than 2". Also, is it clear that the expansion cannot have fractional powers of the log?

5- line 65: The author defines the "area law" of a quantity Q in a(A) in its small cutoff expansion as the leading non-universal piece being proportional to the area of A, but the key quantity of interest is actually a subleading logarithmic piece. Thus, the main slogan "area law <-> OPE block" appears a little inaccurate.

6- line 67: Normally when we say QFT, it does not include Einstein gravity as an example.

7- (2.7) and line 214: I suppose ... also contains other log pieces with smaller power, but their coefficients are not universal (say under rescaling $\epsilon \to 2\epsilon$). Maybe a slight clarification like this is helpful.

8- line 89: Calling J the "magnitude" might be a little confusing, since $\sum_{i,j} J_{ij}^2 \neq J^2$.

9- line 109: Does "the same" mean the same quantum numbers or literally the same operator?

10- line 141: I do not understand what c are, and why they are free parameters. From (2.12) and (2.14), Q is unambiguously defined if one fixes the normalization of O. Is c just the normalization constant for O?

11- (2.26): Does $\gamma$ depend on n?

12- lines 151 and 210: When did R=1 happen?

13- line 176: As noted, the unboundedness of the OPE block makes (2.30) ill-defined. The thing that is actually well-defined is the CCF which you compute. Do you expect to be able to resum the CCFs to recover (2.30)? For instance, can you do this for $Q_A[T]$ to define non-integer Renyi?

14- (2.47): What is D? Are you calling G the conformal block, or DG the conformal block? Are you talking about the 4pt conformal block or some other conformal block? Just knowing that a function is an eigenfunction of the conformal Casimir does not completely specify the conformal block, in addition, boundary conditions/singularities must be specified. Is D kinematical or dynamical? Conformal blocks should be purely kinematical. Please add more explanation of D, since it enters one of the major results, the UV/IR relation (3.11), of the paper.

15- line 293-295: I do not understand the argument here. Just because a function has a symmetry in a certain limit does not mean it has a symmetry away from the limit. Please clarify.

16- line 313: Do you expect the cyclic identity to be only true for integer weights? Or do you expect it to be true for non-integer weights, but you only checked (3.39) in the integer case? If the latter, why not check non-integers?

17- line 387: "region" seems like the wrong word.

18- line 389: Can you at least compute them in simple examples like free theory or generalized free fields (holographically dual to weakly coupled gravity)?

19- Could you add some thoughts about how the quantity (3.11) may be potentially interesting? Could it be monotonic under RG flows? Does it have some nice holographic interpretation? Does it shed light on the analytic structure of some well-defined version of (2.30)?

20- Significant improvement of the writing is necessary. I am listing some of the mistakes I noticed, but there are certainly more.

Misspellings: e.g.
line 23 and 50: vari"ous"
line 178: Tayl"o"r

Articles are missing: e.g.
line 32: "the" reduced
line 146: "the" Renyi

Singular/plural mistakes: e.g.
line 24: area"s"
line 63: degree"s"
line 85: term(s)
line 105: numbers"s"
line 320: lead(s)

Others:
line 46: far away "from"
line 82: chosen "to be"
line 113: "inserted into" -> "are inserted"
line 125: "intersects" with the" t = 0 slice "at/in" a unit ball
line 134: correspond"ing"
line 218: necessar"ily"
line 235: set"ting"
line 238: logarithmic"ally"
line 242: coincid"ing"
line 336: increases "by"
line 344: "and" -> "or"
line 396: attach "to"
(3.17): extra "log"

The explanation of quantities following certain formulae uses very fractured sentences, for example below (2.10).

Equation overflow in lines 55, (3.39), (4.11).

1- Clear review of background material

2- Technical results seem correct and important to understand the physical interpretation of the generalizations of the reduced density matrix operator described in the draft.

1- It is not clear why the exponential operators studied in the paper are important. It seems the calculations have been done just because they can be performed.

In this paper, the author motivates the study of a specific set of conformal correlators by introducing exponential operators that generalize the usual reduced density matrix of a CFT in a ball shaped region.

The results concern the connected correlator of OPE blocks that generalize the construction of the modular Hamiltonian in a conformal field theory.
Explicit results are given for d=2,3,4, and 6.
The result is a cut-off independent coefficient in the log-term divergence of the correlator, and it is obtained by explicitly computing the correlators using known conformal block expressions.

After the changes made by the authors in the second version of the draft, I recommend the draft for publication.

(Minor issue). There are problems with the grammar and vocabulary in the paper. Many indefinite articles are missing.