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Global symmetry and conformal bootstrap in the twodimensional $O(n)$ model
by Linnea GransSamuelsson, Rongvoram Nivesvivat, Jesper Lykke Jacobsen, Sylvain Ribault, Hubert Saleur
This Submission thread is now published as
Submission summary
Authors (as registered SciPost users):  Linnea GransSamuelsson · Jesper Lykke Jacobsen · Rongvoram Nivesvivat · Sylvain Ribault 
Submission information  

Preprint Link:  https://arxiv.org/abs/2111.01106v3 (pdf) 
Code repository:  https://gitlab.com/s.g.ribault/Bootstrap_Virasoro//tree/precision 
Date accepted:  20220406 
Date submitted:  20220323 11:01 
Submitted by:  Ribault, Sylvain 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
We define the twodimensional $O(n)$ 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 fourpoint 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 fourpoint 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 fourpoint 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.
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:

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

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

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.

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 "Fourpoint $O(n)$ 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.
Published as SciPost Phys. 12, 147 (2022)