SciPost logo

SciPost Submission Page

Conformal dispersion relations for defects and boundaries

by Lorenzo Bianchi, Davide Bonomi

This is not the latest submitted version.

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Lorenzo Bianchi · Davide Bonomi
Submission information
Preprint Link:  (pdf)
Date submitted: 2023-01-24 12:16
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 the referee for the thorough report and for the comments. We modified sections 5.3 and 6 according to the requests and implemented the minor corrections. In the section below we address the issues raised by the reviewer point-by-point and list the changes we made.

List of changes

We would like to thank the referee for the thorough report and for the comments. We address the issues raised by the reviewer point-by-point and list the changes we made:
1.1 and 1.2 - We have improved section 5.3 to address the issues pointed out by the reviewer. In particular, we have expanded the discussion on which operators contribute to the dispersion relation at first order in the weak coupling expansion and explicitly shown that only the short operators are necessary to reconstruct the full result. In brief, at this order the discontinuity receives contributions only from the operators that generate negative powers in (w-r) in the bulk OPE. To identify them, we rearranged the superblocks expansion as a power series around $z=\bar{z}=1$ and we found that the only negative powers come from a short multiplet and a long one. After we computed the result from the dispersion relation, by demanding a consistent bulk OPE expansion, we found out that the long multiplet contribution is related to the one from the short multiplet. Therefore, at the end of the day, the result depends only on the short multiplet. We hope that this improved version will be much clearer.
1.3 – We agree that in principle one would need to solve a mixing problem and we added this remark to the new version.
1.4 – We agree on this point, that was indeed our intention. Hopefully we have made it clearer in the new version.
2.1 and 2.2 – We agree with the comment and we added two notes in the new version to clarify these points.
2.3 and 2.4 – We have partially rewritten section 6 to implement the suggestions of the reviewer. In particular, we have computed the one loop and two loops correlators using only the bulk anomalous dimensions from the theory without defect and fixed the remaining coefficients from crossing, as the reviewer suggested.
Finally, we implemented all the minor corrections suggested by the reviewer.

Current status:
Has been resubmitted

Reports on this Submission

Anonymous Report 2 on 2023-3-28 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:2205.09775v2, delivered 2023-03-28, doi: 10.21468/SciPost.Report.6968


1. Many examples worked out in details.


1. The proposed dispersion relation is applied to relatively simple examples and it is not clear how powerful it can be to determine higher order perturbative corrections or for a non-perturbative analysis.

2. In my opinion the presentation can be improved.


The paper introduces a dispersion relation for two-point correlation functions of bulk operators in defect conformal field theories. The authors show that this relation allows to derive very efficiently these correlation functions in perturbation theory, at least at low orders, by applying it to a number of examples and obtaining (essentially) known results.

In my opinion the collection of results and examples presented in the paper can be of use to the community. For this reason it can be published after minor revision.

Requested changes

1. Footnote 5 page 9 is not visible.
2. Above equation (3.1) the authors state: "we see from the bulk..".
I think they should refer to the analysis of 1712.08185 concerning the analytic structure of F(z,w) as a function of w.
3. The authors propose an improved version of the dispersion relation.
Can the author compare this to the "standard" strategy of subtracted dispersion relations?
4. Above (3.13) the authors write: "one can check that". How is the check actually performed?
5. At the end of page 15 "defect identity" should probably be replaced by "bulk identity".
6. In (5.3) and (5.4) the equal sign should be replaced with "proportional to".
7. Equation (5.12) requires some explanations. In particular, it holds only for single trace (1/2 BPS) operators at large N.
8. The one point function in the RHS of (5.13) are not defined.
9. Above (5.19) the authors state that "Indeed, in this holographic setup only short operators contribute to the discontinuity in all the cases where we do not need to improve the dispersion relation". What happens when we need to improve the dispersion relation?
10.Equations (6.20) and (6.21) contain a typo.

  • validity: good
  • significance: good
  • originality: low
  • clarity: ok
  • formatting: reasonable
  • grammar: reasonable

Report 1 by Aleix Gimenez-Grau on 2023-2-1 (Invited Report)


The authors have addressed most of my observations satisfactorily. However, I still believe the section on N=4 SYM at weak coupling needs further rewriting. The reason is that in my opinion, it is not yet clear how the authors deal with the infinite family of long operators below the double-twist threshold. I believe that when this minor issue is addressed, the paper will be ready for publication.

  • validity: -
  • significance: -
  • originality: -
  • clarity: -
  • formatting: -
  • grammar: -

Author:  Lorenzo Bianchi  on 2023-02-08  [id 3333]

(in reply to Report 1 by Aleix Gimenez-Grau on 2023-02-01)

We would like to thank the referee for this additional comment, which we have addressed in a revised version.

Login to report or comment