# Area law and OPE blocks in conformal field theory

### Submission summary

 As Contributors: Jiang Long Preprint link: scipost_202010_00028v3 Date submitted: 2021-02-21 14:14 Submitted by: Long, Jiang Submitted to: SciPost Physics Proceedings Proceedings issue: 4th International Conference on Holography, String Theory and Discrete Approach in Hanoi Academic field: Physics Specialties: High-Energy Physics - Theory Approach: Theoretical

### Abstract

This is an introduction to the relationship between area law and OPE blocks in conformal field theory.

###### Current status:
Editor-in-charge assigned

### List of changes

1. Line 30-31 and 40, we added the discussion on OPE blocks , emphasising that OPE block provides a novel look at the modular Hamiltonian. This is also the motivation to relate OPE block to area law in this paper.
2. Line 35-37: we rewrote the UV divergences of the entanglement entropy.
3. Line 47-48: we wrote explicitly the possible values of the degree q. There is no fractional power of the logarithm.
4. Line 72: we added a sentence to extend the area law to general field theory.
5. Line 88-89: we discussed the possible logarithmic pieces with smaller power. They don't provide any universal information.
6. Line 94-97: we added one comment on the relation between the area law and OPE blocks.
7. Line 101: we changed "so(d-1) spin J_{ij} with magnitude J" to "so(d-1) spin J".
8. Line 120-121: we changed the "the external operators are the same" to "the external operators have the same quantum numbers".
9. Line 154-155: the constant c is related to normalization of the operator . This explains why we can set c=1.
10. Eq (2.26): \gamma\to\gamma(n).
11. Line 138: we added "(R=1)".
12.Line 196-198: we added the convergence problem of the summation of eq. (2.31).
13. Line 221-226: we discussed the coefficient D and conformal block G.
14. Line 317-319: the cyclic identity is a conjecture. We don't prove its validity, however, we can check it case by case.
15.Line 337-339: we added the discussion on the non-integer conformal weights.
16. Line 412: we changed "we restrict to the region m\ge 3" to "we consider type-(2) and type-(3) CCFs".
18. We rewrote (3.39) and (4.11).
19. We improved the writing of the paper.

### Submission & Refereeing History

Resubmission scipost_202010_00028v3 on 21 February 2021
Resubmission scipost_202010_00028v2 on 30 December 2020
Submission scipost_202010_00028v1 on 30 October 2020

## Reports on this Submission

### Strengths

1- Proposes new observables in CFT.
2-Presents concrete computations and results.

### Weaknesses

1- Grammar and clarity of the exposition can be improved.

### Report

The author studied nonlocal operators in CFT known as "OPE blocks" that generalize the modular Hamiltonian, computed their correlators and extracted the universal leading logarithmic piece in the small cutoff limit. The key results were a UV/IR relation (3.16) and a cyclic identity (3.29). The former was proven by exploiting conformal symmetry, and the latter was conjectured based on evidence.

The revised manuscript has addressed many of the earlier problems, but a few issues remain. My most serious question is point 12 below, as the author seems to suggest that the key quantity, the coefficient of the leading logarithm, is not unambiguously defined for general operators. The grammar and clarity of the exposition can still be improved, but do not present a major problem.

I would recommend the manuscript for publication after the issues are addressed.

### Requested changes

1-line 26: I suggest changing to something like "...is a relatively unexplored topic in conformal field theory, though it has been defined and discussed at the early stages of conformal field theory."

2-line 42: Here I think it is important to state clearly what you mean by area law, so I suggest putting "area law" in quotation marks, and adding a footnote saying that your notion of area law includes subleading corrections, and you use the slogan "area law" following the convention of geometric entanglement entropy. (copied from lines 96-97).

3-lines 42-43: If I understand your statement correctly, you want to say "This leads to the conjecture that similar to the modular Hamiltonian, general OPE blocks exhibit area law."

4-lines 47-48: By "We don’t find fractional powers of the logarithm.", is the claim of q = 0, 1, 2 just from the specific examples you studied, or can it be argued from properties of the conformal block? If it is just from examples, I suggest writing: "In all examples we studied, we found q = 0, 1, 2, but in general we do not rule out the possibility of other values." If there is some general argument, please explain.

5-line 86: "Natural number" excludes q=0, but type-J in 3d has q=0.

6-lines 165 and 231: "inserted the radius R = 1" -> "explicitly restored the radius R that was previously set to 1".

7-(2.18) to line 155: I still fail to understand the purpose of c. I can't find where a canonical normalization for Q is given. So why can't c be removed completely?

8-line 226: This sentence seems to imply that for m>2, D is not related to the normalization of operators. If so, why?

9-(3.1) The universal constant $p_q$ in (3.1) depends on your operator normalization. How are you normalizing O?

10-(2.45)-(2.47): Going from (2.45) to (2.47) should involve some Ward identity that relates the quantum operator L^2 to an explicit differential operator in \eta. Could you provide some explanation or a reference for this step? If \eta is the cross ratio of four points, then this is a standard exercise in CFT, but here \eta is defined in terms of some spatial regions.

11-lines 314-316: I maintain my opinion that just because a function has a symmetry in a certain limit does not mean it has a symmetry away from the limit. Therefore I think the argument given in this sentence is not very sensible even as a heuristic.

12-lines 338-339: Are you saying that there is no unambiguous way to regularize the divergences for non-integer weights? Does this mean that there is no unambiguous way to define $p_q$ for general operators?

13-It seems to me that the UV/IR relation is robust and follows from conformal symmetry, whereas the cyclic identity is a conjecture. If so, whenever you say that you "check the UV/IR relation and the cyclic identity", I suggest that you separate the two, since checking a conjecture is morally different from checking a fact.

14-Exposition and grammar. Just to point out two examples:
line 29 is unnecessarily fractured. It could be combined into "It is a smeared operator which is generated from a so-called (quasi-)primary operator, and extends the study of local operators in CFT to non-local operators.
line 264 still contains "far away to".

• validity: high
• significance: good
• originality: high
• clarity: ok
• formatting: perfect
• grammar: reasonable