Duality and KPZ in Liouville Quantum Gravity

TL;DR

This paper provides a rigorous probabilistic proof of the KPZ relation connecting Euclidean and Liouville quantum gravity scaling exponents, extending to boundary geometry and exploring duality for singular cases.

Contribution

It offers a mathematically rigorous proof of the KPZ relation in Liouville quantum gravity, including boundary cases and duality for b3 > 2.

Findings

Proof of KPZ relation for planar domains

Extension of results to boundary geometries

Duality relation for b3 > 2 cases

Abstract

We present a (mathematically rigorous) probabilistic and geometrical proof of the KPZ relation between scaling exponents in a Euclidean planar domain D and in Liouville quantum gravity. It uses the properly regularized quantum area measure d\mu_\gamma=\epsilon^{\gamma^2/2} e^{\gamma h_\epsilon(z)}dz, where dz is Lebesgue measure on D, \gamma is a real parameter, 0\leq \gamma <2, and h_\epsilon(z) denotes the mean value on the circle of radius \epsilon centered at z of an instance h of the Gaussian free field on D. The proof extends to the boundary geometry. The singular case \gamma >2 is shown to be related to the quantum measure d\mu_{\gamma'}, \gamma' < 2, by the fundamental duality \gamma\gamma'=4.