SciPost Submission Page
Area law and OPE blocks in conformal field theory
by Jiang Long
This is not the current version.
Submission summary
As Contributors:  Jiang Long 
Preprint link:  scipost_202010_00028v1 
Date submitted:  20201030 05:24 
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:
Submission & Refereeing History
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Reports on this Submission
Anonymous Report 1 on 20201222 Invited Report
Strengths
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.
Weaknesses
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.
2 All of the results involve at most correlators involving three operators. All results are thus fixed by conformal invariance, which makes the calculations almost trivial.
Report
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 cutoff independent coefficient in the logterm divergence of the correlator, and it is obtained by explicitly computing the correlators using known conformal block expressions.
In its current state I am not comfortable recommending the draft for publication. Even though the technical results of the paper seem correct and might prove important in the future, I do not think the calculations are justified physically.
Requested changes
1 I recommend that a physical interpretation of equation 2.29 is formulated. The objects studied in the paper are exponential operators constructed from OPE blocks. They are denoted by \rho_A. (this is confusing given that \rho_A is how the reduced density matrix is also denoted in the paper.
It is not clear why this object is interesting in any way. It is not clear whether it can be interpreted as a reduced density matrix. The author mentions this issue in the discussion, giving the impression that the calculations have been performed simply because they can be done, but there does not seem to be much physics justifying the work.
2  All of the results in the paper involve at most m=3, which implies that the correlators playing a role in the calculation are fully fixed by conformal invariance. This either should be expanded on with more cases, or should be mentioned in the abstract, as currently there seems to be a disconnect between what is implied in the abstract/introduction, and the actual scope of the paper.
3  (Minor issue). There are problems with the grammar and vocabulary in the paper. Many indefinite articles are missing, and some words are used incorrectly (evolution vs evaluation for example).
(in reply to Report 1 on 20201222)
Our response: In the new manuscript, we added reference [15] below equation 2.29 as an interpretation of the exponential object discussed in this paper. Our definition of the deformed reduced density matrix is a direct generalization of the operator in the context of charged Renyi entropy.
Our response: We discussed several cases for m=4 in two dimensions in the paragraph above equation (3.40). We could check all the points in this paper for two dimensional massless free scalar theory.
Our response: We improved the manuscript on the grammar and vocabulary in the new version.