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Diagonal fields in critical loop models

by Sylvain Ribault

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Submission summary

Authors (as registered SciPost users): Sylvain Ribault
Submission information
Preprint Link:  (pdf)
Date accepted: 2023-01-09
Date submitted: 2022-12-12 21:17
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


In critical loop models, there exist diagonal fields with arbitrary conformal dimensions, whose $3$-point functions coincide with those of Liouville theory at $c\leq 1$. We study their $N$-point functions, which depend on the $2^{N-1}$ weights of topologically inequivalent loops on a sphere with $N$ punctures. Using a numerical conformal bootstrap approach, we find that $4$-point functions decompose into infinite but discrete linear combinations of conformal blocks. We conclude that diagonal fields belong to an extension of the $O(n)$ model.

Author comments upon resubmission

I have tried to clarify the text, following the referee's suggestions.

List of changes

1. I have made the use of interchiral symmetry more explicit, by replacing conformal blocks with interchiral blocks in the ansatz (10). I have also added explanations, including the new reference [9]. And interchiral symmetry now appears as a third assumption for the four-point functions $Z_4(P_s,P_t,P_u)$.

2. I have added a numerical example, with parameter values in (12), (13), and results in (14).

3. I have made the solution unique by fixing the normalization.

4. I removed the claim about shift equations in the Potts model. The claim could have been made more precise, by explaining that the shift equations have extra factors compared to what we would expect from a degenerate field. The existence of these modified shift equations surely begs for an explanation. However, for the present paper's purpose, it is probably enough to mention the fact that the three-point connectivity coincides with a Liouville structure constant.

There are other small changes and clarifications. There used to be three sections, there are now five: the section that was called "Bootstrap results for four-point functions" is now split into three sections. With so many sections, it became more reasonable to have a table of contents.

Published as SciPost Phys. Core 6, 020 (2023)

Reports on this Submission

Anonymous Report 1 on 2022-12-15 (Invited Report)


The author has addressed the suggestions of changes I made in the previous report.

It is interesting that once the overall normalization is fixed, there is a unique solution to the bootstrap problem under the given assumption. It seems to me that one interesting question would be to make a physical choice of the normalization (perhaps the particular solution of (12) with proper physical interpretation) and study how the structure constants for the non-diagonal fields in this case are related to that of the $O(n)$ model. This however may require an extensive amount of numerical work and could be left for future work.

I recommend the paper for publication.

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

Author:  Sylvain Ribault  on 2022-12-15  [id 3139]

(in reply to Report 1 on 2022-12-15)
suggestion for further work

The referee's suggestions are spot on. Yes, (12) is surely the "correct" normalization, although its interpretation from the lattice is not too clear. And yes, studying structure constants of non-diagonal fields is very interesting, but requires quite a lot of extra work.

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