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

### Submission summary

 As Contributors: Rongvoram Nivesvivat · Sylvain Ribault Arxiv Link: https://arxiv.org/abs/2111.01106v3 (pdf) Code repository: https://gitlab.com/s.g.ribault/Bootstrap_Virasoro/-/tree/precision Date accepted: 2022-04-06 Date submitted: 2022-03-23 11:01 Submitted by: Ribault, Sylvain Submitted to: SciPost Physics Academic field: Physics Specialties: High-Energy Physics - Theory Mathematical Physics Approach: Theoretical

### Abstract

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)

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)

Resubmission 2111.01106v3 on 23 March 2022
Submission 2111.01106v2 on 19 January 2022

## Reports on this Submission

### Report

I am happy with the changes and recommend publication.

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