Connecting SLE and minisuperspace Liouville gravity

TL;DR

This paper establishes a connection between Schramm-Loewner Evolution (SLE) and minisuperspace Liouville gravity by relating their fundamental equations, potentially explaining the gravitational origin of SLE critical exponents.

Contribution

It demonstrates that the Fokker-Planck equation for SLE can be derived from the minisuperspace Wheeler-de Witt equation in 2D Liouville gravity, linking stochastic processes with quantum gravity.

Findings

Fokker-Planck equation for SLE maps to Wheeler-de Witt equation in Liouville gravity

Insertion of operators in SLE corresponds to matter contributions in gravity equations

Provides a potential explanation for SLE critical exponents via KPZ scaling

Abstract

We show that Fokker-Planck equation for chordal SLE process under a simple rescaling of the probability density can be traced to the minisuperspace Wheeler-de Witt equation for boundary operator in 2d Liouville gravity. Insertion of an operator, calculating SLE critical exponent, corresponds to adding matter contribution to WdW equation. This observation may be useful for understanding of why SLE critical exponents are given by KPZ gravitational scaling dimensions. Possible applications of the obtained relation are discussed.