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

TL;DR

This paper develops a conformal bootstrap approach for the two-dimensional $O(n)$ model, analyzing its spectrum, conformal blocks, and crossing symmetry solutions, revealing bounds and fusion rules across various correlation functions.

Contribution

It introduces a numerical bootstrap framework for the 2D $O(n)$ conformal field theory, including logarithmic blocks and representation theory bounds, for arbitrary complex $n$.

Findings

Determined conformal blocks, including logarithmic ones, for the $O(n)$ model.

Counted solutions of crossing symmetry for 30 four-point functions, with 21 saturating the bounds.

Provided fusion rules and spectrum decomposition for the $O(n)$ 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 $n\in\mathbb{C}$, 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.