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Conformal dispersion relations for defects and boundaries

by Lorenzo Bianchi, Davide Bonomi

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Submission summary

Authors (as registered SciPost users): Lorenzo Bianchi · Davide Bonomi
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-06-01
Date submitted: 2023-04-03 17:12
Submitted by: Bonomi, Davide
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
Approach: Theoretical


We derive a dispersion relation for two-point correlation functions in defect conformal field theories. The correlator is expressed as an integral over a (single) discontinuity that is controlled by the bulk channel operator product expansion (OPE). This very simple relation is particularly useful in perturbative settings where the discontinuity is determined by a subset of bulk operators. In particular, we apply it to holographic correlators of two chiral primary operators in $\mathcal{N}= 4$ Super Yang-Mills theory in the presence of a supersymmetric Wilson line. With a very simple computation, we are able to reproduce and extend existing results. We also propose a second relation, which reconstructs the correlator from a double discontinuity, and is controlled by the defect channel OPE. Finally, for the case of codimension-one defects (boundaries and interfaces) we derive a dispersion relation which receives contributions from both OPE channels and we apply it to the boundary correlator in the $O(N)$ critical model. We reproduce the order $\epsilon^2$ result in the $\epsilon$-expansion using as input a finite number of boundary CFT data.

Author comments upon resubmission

Dear Editor, we would like to thank again the referees for the reports and for the comments. We implemented all the corrections requested by the referees. In the section below we address the issues raised by the reviewers point-by-point and list the changes we made.

List of changes

We addressed the issues raised by Reviewer 1 in the following way:
1. We modified section 5.3 on N=4 SYM at weak coupling to address the issue pointed out by the reviewer.
We addressed the issues raised by Reviewer 2 in the following way:
2. In the new version we referred to the analysis of 1712.08185 above equation (3.1), as suggested by the reviewer.
3. We added a comment on the alternative approach of subtracted dispersion relations under equation (3.7).
4. In the new version we have hopefully expressed more clearly that we checked equation (3.13) simply by explicitly computing the integral.
7. We have added a comment above equation (5.13) , which was equation (5.12) in the previous version, explaining that it can be derived from localization and holds only for single trace (1/2 BPS) operators at large N.
8. We have added the definition of the one point function in equation (5.6).
9. We have added a note on page 21 where we comment on what happens when we need to improve the dispersion relation.
Finally, we have corrected all the typos pointed out by the reviewer.

Published as SciPost Phys. 15, 055 (2023)

Reports on this Submission

Anonymous Report 2 on 2023-4-20 (Invited Report)


In my opinion, the paper can be published as is.
However, the current explanation of equation (3.13) is not satisfactory. The authors changed from "one can check" in the previous version, to "one can check by explicit computation" in the current version. It is not clear what is meant by explicit calculation. Is it numerical? Is it an expansion in r?

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Author:  Lorenzo Bianchi  on 2023-04-27  [id 3620]

(in reply to Report 2 on 2023-04-20)

We thank the referee for the additional comment. The integral in (3.13) has been computed analitically in the regime discussed in the text after a simple change of variable w->x=(r-w)(r-1/w) and given the simplicity of the computation we thought it would not be necessary to include the details in the paper.

Report 1 by Aleix Gimenez-Grau on 2023-4-5 (Invited Report)


The authors have addressed my concerns regarding section 5.3, so the paper is ready for publication.

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