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

· pdf
We define the twodimensional $O(n)$ 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 fourpoint 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 fourpoint
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 fourpoint 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.
SciPost Phys. 10, 021 (2021) ·
published 29 January 2021

· pdf
Using derivatives of primary fields (null or not) with respect to the
conformal dimension, we build infinite families of nontrivial logarithmic
representations of the conformal algebra at generic central charge, with Jordan
blocks of dimension $2$ or $3$. Each representation comes with one free
parameter, which takes fixed values under assumptions on the existence of
degenerate fields. This parameter can be viewed as a simpler,
normalizationindependent redefinition of the logarithmic coupling. We compute
the corresponding nonchiral conformal blocks, and show that they appear in
limits of Liouville theory fourpoint functions.
As an application, we describe the logarithmic structures of the critical
twodimensional $O(n)$ and $Q$state Potts models at generic central charge.
The validity of our description is demonstrated by semianalytically
bootstrapping fourpoint connectivities in the $Q$state Potts model to
arbitrary precision. Moreover, we provide numerical evidence for the
DelfinoViti conjecture for the threepoint connectivity. Our results hold for
generic values of $Q$ in the complex plane and beyond.