Symmetry breaking

Quantum mechanics

Function (biology)

Conformal symmetry

Linguistics

Philosophy

Pure mathematics

Mathematical physics

Biology

Correlation function (quantum field theory)

Scattering amplitude

Crossing

Fusion rules

Global symmetry

Evolutionary biology

Physics

Fusion

Central charge

Dielectric

Amplitude

Mathematics

Representation theory

Image fusion

Irreducible representation

Spontaneous symmetry breaking

Mathematical analysis

Geometry

Conformal field theory

Conformal map

Combinatorics

Charge (physics)

Logarithm

Conjecture

Symmetry (geometry)

SciPost Physics

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.

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