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

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}n∈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.