Schramm Loewner Evolution and Liouville Quantum Gravity

TL;DR

This paper explores the connection between Schramm-Loewner Evolution (SLE) and Liouville Quantum Gravity (LQG) by developing a theory of quantum fractal measures and analyzing their evolution through conformal welding, leading to new quantum boundary measures.

Contribution

It introduces a novel framework linking SLE and LQG via quantum fractal measures and quantum welding, providing tools for measuring quantum lengths and intersections on SLE curves.

Findings

Development of quantum fractal measures consistent with KPZ relation

Construction of quantum length and boundary intersection measures on SLE curves

Analysis of measure evolution using SLE martingales

Abstract

We conformally weld (via "quantum zipping") two boundary arcs of a Liouville quantum gravity random surface to generate a random curve called the Schramm-Loewner evolution (SLE). We develop a theory of quantum fractal measures (consistent with the Knizhnik-Polyakov-Zamolochikov relation) and analyze their evolution under welding via SLE martingales. As an application, we construct the natural quantum length and boundary intersection measures on the SLE curve itself.