Conformal Field Theories of Stochastic Loewner Evolutions

TL;DR

This paper establishes a deep connection between stochastic Loewner evolutions (SLE) and conformal field theories (CFT), linking probabilistic growth processes with algebraic structures and correlation functions in two-dimensional conformal models.

Contribution

It introduces a group theoretical framework for SLE, identifies boundary states and zero modes, and relates SLE probabilities to CFT correlation functions, advancing the theoretical understanding of SLE-CFT correspondence.

Findings

Defined infinite SLE zero modes as martingales.

Derived linear systems for crossing probabilities using CFT correlation functions.

Proposed a conjecture for reconstructing CFTs from SLE data.

Abstract

Stochastic Loewner evolutions (SLE) are random growth processes of sets, called hulls, embedded in the two dimensional upper half plane. We elaborate and develop a relation between SLE evolutions and conformal field theories (CFT) which is based on a group theoretical formulation of SLE processes and on the identification of the proper hull boundary states. This allows us to define an infinite set of SLE zero modes, or martingales, whose existence is a consequence of the existence of a null vector in the appropriate Virasoro modules. This identification leads, for instance, to linear systems for generalized crossing probabilities whose coefficients are multipoint CFT correlation functions. It provides a direct link between conformal correlation functions and probabilities of stopping time events in SLE evolutions. We point out a relation between SLE processes and two dimensional gravity and conjecture a reconstruction procedure of conformal field theories from SLE data.