On Conformal Field Theory and Stochastic Loewner Evolution

TL;DR

This paper integrates Conformal Field Theory with Stochastic Loewner Evolution on Riemann surfaces, proposing a CFT-based construction of probability measures on paths and connecting Loewner evolutions to moduli space dynamics.

Contribution

It introduces a novel CFT framework for SLE on Riemann surfaces, linking stochastic processes to geometric structures and Virasoro algebra actions.

Findings

Proposes a CFT construction for path probability measures on Riemann surfaces.

Connects Loewner evolutions to random walks in moduli space.

Shows the stochastic diffusion equation corresponds to Virasoro algebra action.

Abstract

We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with boundary using Conformal Field Theory methods. We propose in particular a CFT construction for a probability measure on (clouded) paths, and check it against known restriction properties. The probability measure can be thought of as a section of the determinant bundle over moduli spaces of Riemann surfaces. Loewner evolutions have a natural description in terms of random walk in the moduli space, and the stochastic diffusion equation translates to the Virasoro action of a certain weight-two operator on a uniformised version of the determinant bundle.