We present a (mathematically rigorous) probabilistic and geometrical proof of the Knizhnik-Polyakov-Zamolodchikov 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 the Lebesgue measure on $D$, $\gamma$ is a real parameter, $0\le \gamma <2$, 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 $d{\mu }_{{\gamma }^{\prime }}$, ${\gamma }^{\prime }<2$, by the fundamental duality $\gamma {\gamma }^{\prime }=4$.

• Received 4 January 2009

DOI:https://doi.org/10.1103/PhysRevLett.102.150603