Twist operator correlation functions in O(n) loop models

TL;DR

This paper employs conformal field theory to compute correlation functions involving twist operators in the O(n) loop model at criticality, especially focusing on the self-avoiding loop case (n=0), and derives new explicit probabilities for SLE windings.

Contribution

It introduces a novel application of LCFT operators to calculate winding probabilities in SLE processes within the O(n) model, revealing potential operator incompatibilities and their resolutions.

Findings

Derived explicit winding probabilities for SLE_{8/3}

Identified a collection of c=0 LCFT operators used in the analysis

Highlighted potential incompatibilities due to logarithmic partners in the theory

Abstract

Using conformal field theoretic methods we calculate correlation functions of geometric observables in the loop representation of the O(n) model at the critical point. We focus on correlation functions containing twist operators, combining these with anchored loops, boundaries with SLE processes and with double SLE processes. We focus further upon n=0, representing self-avoiding loops, which corresponds to a logarithmic conformal field theory (LCFT) with c=0. In this limit the twist operator plays the role of a zero weight indicator operator, which we verify by comparison with known examples. Using the additional conditions imposed by the twist operator null-states, we derive a new explicit result for the probabilities that an SLE_{8/3} wind in various ways about two points in the upper half plane, e.g. that the SLE passes to the left of both points. The collection of c=0 logarithmic CFT operators that we use deriving the winding probabilities is novel, highlighting a potential incompatibility caused by the presence of two distinct logarithmic partners to the stress tensor within the theory. We provide evidence that both partners do appear in the theory, one in the bulk and one on the boundary and that the incompatibility is resolved by restrictive bulk-boundary fusion rules.