KPZ formula derived from Liouville heat kernel

TL;DR

This paper derives the KPZ formula within Liouville quantum gravity using the heat kernel of Liouville Brownian motion, offering a more intrinsic geometric approach and analyzing measure properties.

Contribution

It provides a novel derivation of the KPZ formula based on heat kernel methods and studies Liouville measure properties, enhancing the geometric understanding of quantum gravity.

Findings

KPZ formula derived from Liouville heat kernel

Analysis of Liouville measure doubling behavior

Provides a geometric interpretation of quantum gravity dimensions

Abstract

In this paper, we establish the Knizhnik--Polyakov--Zamolodchikov (KPZ) formula of Liouville quantum gravity, using the heat kernel of Liouville Brownian motion. This derivation of the KPZ formula was first suggested by F. David and M. Bauer in order to get a geometrically more intrinsic way of measuring the dimension of sets in Liouville quantum gravity. We also provide a careful study of the (no)-doubling behaviour of the Liouville measures in the appendix, which is of independent interest.