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

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

SciPost Phys. 12, 147 (2022) · published 6 May 2022

- doi: 10.21468/SciPostPhys.12.5.147
- Submissions/Reports

### 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.

### Cited by 8

### Authors / Affiliations: mappings to Contributors and Organizations

See all Organizations.-
^{1}Linnea Grans-Samuelsson, -
^{1}Rongvoram Nivesvivat, -
^{1}^{2}^{3}Jesper Lykke Jacobsen, -
^{1}Sylvain Ribault, -
^{1}^{4}Hubert Saleur

^{1}L'Institut de physique théorique [IPhT]^{2}Sorbonne Université / Sorbonne University^{3}École Normale Supérieure [ENS]^{4}University of California Los Angeles [UCLA]