SciPost Submission Page
Area law and OPE blocks in conformal field theory
by Jiang Long
Submission summary
As Contributors:  Jiang Long 
Preprint link:  scipost_202010_00028v3 
Date submitted:  20210221 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: 

Approach:  Theoretical 
Abstract
This is an introduction to the relationship between area law and OPE blocks in conformal field theory.
Current status:
List of changes
1. Line 3031 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 3537: we rewrote the UV divergences of the entanglement entropy.
3. Line 4748: 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 8889: we discussed the possible logarithmic pieces with smaller power. They don't provide any universal information.
6. Line 9497: we added one comment on the relation between the area law and OPE blocks.
7. Line 101: we changed "so(d1) spin J_{ij} with magnitude J" to "so(d1) spin J".
8. Line 120121: we changed the "the external operators are the same" to "the external operators have the same quantum numbers".
9. Line 154155: 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 196198: we added the convergence problem of the summation of eq. (2.31).
13. Line 221226: we discussed the coefficient D and conformal block G.
14. Line 317319: the cyclic identity is a conjecture. We don't prove its validity, however, we can check it case by case.
15.Line 337339: we added the discussion on the noninteger conformal weights.
16. Line 412: we changed "we restrict to the region m\ge 3" to "we consider type(2) and type(3) CCFs".
17. Line 414415: we added the reference on the type(4) CCF for free theory.
18. We rewrote (3.39) and (4.11).
19. We improved the writing of the paper.
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 2021324 (Invited Report)
Strengths
1 Proposes new observables in CFT.
2Presents 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
1line 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."
2line 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 9697).
3lines 4243: 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."
4lines 4748: 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.
5line 86: "Natural number" excludes q=0, but typeJ in 3d has q=0.
6lines 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?
8line 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.
11lines 314316: 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.
12lines 338339: Are you saying that there is no unambiguous way to regularize the divergences for noninteger weights? Does this mean that there is no unambiguous way to define $p_q$ for general operators?
13It 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.
14Exposition 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 socalled (quasi)primary operator, and extends the study of local operators in CFT to nonlocal operators.
line 264 still contains "far away to".