General solution of an exact correlation function factorization in conformal field theory

TL;DR

This paper identifies a universal correlation function factorization in conformal field theory that applies across various models and central charges, extending previous specific cases to a general setting.

Contribution

It discovers a general correlation function factorization valid for any central charge c, broadening the scope of known factorizations in conformal field theory.

Findings

Factorization applies to FK and spin clusters in Q-state Potts models.

It also applies to dense and dilute phases of O(n) loop models.

A unique set of operators with negative dimension also exhibits this factorization.

Abstract

We discuss a correlation function factorization, which relates a three-point function to the square root of three two-point functions. This factorization is known to hold for certain scaling operators at the two-dimensional percolation point and in a few other cases. The correlation functions are evaluated in the upper half-plane (or any conformally equivalent region) with operators at two arbitrary points on the real axis, and a third arbitrary point on either the real axis or in the interior. This type of result is of interest because it is both exact and universal, relates higher-order correlation functions to lower-order ones, and has a simple interpretation in terms of cluster or loop probabilities in several statistical models. This motivated us to use the techniques of conformal field theory to determine the general conditions for its validity. Here, we discover a correlation function which factorizes in this way for any central charge c, generalizing previous results. In particular, the factorization holds for either FK (Fortuin-Kasteleyn) or spin clusters in the Q-state Potts models; it also applies to either the dense or dilute phases of the O(n) loop models. Further, only one other non-trivial set of highest-weight operators (in an irreducible Verma module) factorizes in this way. In this case the operators have negative dimension (for c < 1) and do not seem to have a physical realization.