SciPost logo

SciPost Submission Page

Global symmetry and conformal bootstrap in the two-dimensional $O(n)$ model

by Linnea Grans-Samuelsson, Rongvoram Nivesvivat, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur

This Submission thread is now published as SciPost Phys. 12, 147 (2022)

Submission summary

As Contributors: Rongvoram Nivesvivat · Sylvain Ribault
Arxiv Link: (pdf)
Code repository:
Date accepted: 2022-04-06
Date submitted: 2022-03-23 11:01
Submitted by: Ribault, Sylvain
Submitted to: SciPost Physics
Academic field: Physics
  • High-Energy Physics - Theory
  • Mathematical Physics
Approach: Theoretical


We define the two-dimensional $O(n)$ conformal field theory as a theory that includes the critical dilute and dense $O(n)$ models as special cases, and depends analytically on the central charge. For generic values of $n\in\mathbb{C}$, we write a conjecture for the decomposition of the spectrum into irreducible representations of $O(n)$. We then explain how to numerically bootstrap arbitrary four-point functions of primary fields in the presence of the global $O(n)$ symmetry. We determine the needed conformal blocks, including logarithmic blocks, including in singular cases. We argue that $O(n)$ representation theory provides upper bounds on the number of solutions of crossing symmetry for any given four-point function. We study some of the simplest correlation functions in detail, and determine a few fusion rules. We count the solutions of crossing symmetry for the $30$ simplest four-point functions. The number of solutions varies from $2$ to $6$, and saturates the bound from $O(n)$ representation theory in $21$ out of $30$ cases.

Published as SciPost Phys. 12, 147 (2022)

Author comments upon resubmission

We are grateful to the Editor and reviewers for their work, and in particular to the author of Report 1 for the helpful suggestions. Following the four numbered suggestions, we have made the following changes:

  1. We have stated explicitly that the coefficients are $n$-independent.

  2. We have rewritten that discussion in order to clarify it and make the matrices more explicit.

  3. After Table (3.43), we now state that the excluded field may be chosen arbitrarily, and that we chose them by increasing values of the conformal dimension.

  4. The conjecture is now displayed as Eq. (4.5), and the argument for the conjecture is more detailed, including the new Table (4.6).

Moreover, Report 1 suggests that the paragraph on "Four-point $O(n)$ invariants" could be simplified. We do not see how to simplify it while keeping the needed results like Eq. (3.26). What we have done is to add a review of the notation Hom and its properties, see Eq. (3.20) and the preceding text. We hope that this clarifies the paragraph.

Submission & Refereeing History

Published as SciPost Phys. 12, 147 (2022)

You are currently on this page

Resubmission 2111.01106v3 on 23 March 2022

Reports on this Submission

Anonymous Report 1 on 2022-3-26 (Invited Report)


I am happy with the changes and recommend publication.

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

Login to report or comment