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Area law and OPE blocks in conformal field theory

by Jiang Long

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Submission summary

Authors (as registered SciPost users): Jiang Long
Submission information
Preprint Link: scipost_202010_00028v4  (pdf)
Date submitted: 2021-05-21 03:50
Submitted by: Long, Jiang
Submitted to: SciPost Physics Proceedings
Proceedings issue: 4th International Conference on Holography, String Theory and Discrete Approach in Hanoi (STRHAN2020)
Ontological classification
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.

Author comments upon resubmission

1) The referee says: 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?

Our response: the constant c can be removed in principle. However, sometimes the normalization of the operator O is given, then Q still has the freedom to choose its own normalization. For example, usually, the stress tensor in integral of the modular Hamiltonian is defined unambiguously, then one should choose c=2\pi such that tr_A \rho_A=1.

2) The referee says: This sentence seems to imply that for m>2, D is not related to the normalization of operators. If so, why?

Our response: I rewrote the sentence. D also depends on the normalization of the operators.

3) The referee says: The universal constant $p_q$ in (3.1) depends on your operator normalization. How are you normalizing O?

Our response: I agree the constant $p_q$ depends on the operator normalization. I don't choose a definite normalization when I state general results. The normalization doesn't affect the validity of my results.

4) The referee says: Are you saying that there is no unambiguous way to regularize the divergences for non-integer weights...

Our response: No. I rewrote the sentence. I can't regularize the integral by straightforward computation for non-integer weights, but this doesn't rule out the possibility that the constant $p_q$ is still defined unambiguously by other means.

List of changes

1-line 25-26: I have changed the sentence to "...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 41-42: I put "area law" in quotation marks and add a footnote to explain it.

3-line 42-43: I changed the sentence as the referee suggested.

4-line 47-48: I changed the sentence to "In all examples we studied, we found q = 0, 1, 2, but in general we do not rule out the possibility of other values."

5-line 85: I changed "nature number" in the original version to "nonnegative number".

6-line 163 : I rewrote the sentence as " we have restored the radius R that was previously set to 1". There are similar modification in line 229-230.

7-line 223-224: I changed the sentence to "For $m\ge 2$, the coefficients $D^{(d)}[\mathcal{O}_1,
cdots,\mathcal{O}_m]$ are related to the normalization of the primary operators. For any $m\ge 3$, it also contains dynamical information of the theory."

8-(3.1): I added a footnote 3 to address the point that $p_q$ also depends on the normalization of the operators.

9-(2.47): I added a footnote 2 to explain equation (2.47).

10-line 312: I deleted original argument on cyclic identity, just mention that we could read out a cyclic identity (3.29) from (3.28).

11-line 334-336: I rewrote a comment on the regularization of the integral for non-integer conformal weight.

12-line 337: I changed the sentence to "For m\ge 4, the cyclic identity are ..."

13: Iine 28-30: I changed the sentence as the referee suggested.

14: line 262: I change "far away to" to "far away from".

Current status:
Has been resubmitted

Reports on this Submission

Report #1 by Anonymous (Referee 5) on 2021-6-29 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202010_00028v4, delivered 2021-06-29, doi: 10.21468/SciPost.Report.3065

Strengths

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

Weaknesses

1- It is unclear whether the key quantity $p_q$ is unambiguously defined for general operators.

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.

I am unconvinced by the statement that "there should be an unambiguous way to define $p_q$ for general operators" on lines 335-336. The author does not propose any regularization scheme for general conformal weights, let alone showing that the result is independent of the regularization.

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

Requested changes

(2.18): In response to the author's reply about the meaning of the normalization constant $c$, while I understand the author's point, I personally find it very confusing. The author is saying that $Q_A$ has no predefined normalization, and one just chooses $c$ to suit the context. Due to this context-dependent $c$, $Q_A$ is not a linear functional of $O$. Besides, $Q_A$ is a scalar number, so without a predefined normalization, it has no unambiguous meaning. I personally think it would be much more satisfying if $c$ is set to be $2\pi$ for any $O$, in which case $Q_A$ is a linear functional.

lines 335-336: See report. I suggest that the author softens the statement or presents evidence for this claim.

I kindly ask the author to review the next revision more carefully, as new errors and typos have appeared in every past revision, e.g. for the current one,
line 223- cdots -> \cdots
line 337- identity -> identities

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

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Comments

Anonymous on 2021-06-07  [id 1492]

comment to the Editors included