Conformal covariance of the Liouville quantum gravity metric for $\gamma \in (0,2)$

Abstract

For $\gamma \in (0,2)$, $U\subset \mathbb{C}$, and an instance $h$ of the Gaussian free field (GFF) on $U$, the $\gamma$-Liouville quantum gravity (LQG) surface associated with $(U,h)$ is formally described by the Riemannian metric tensor $e^{\gamma h}(d{x}^2+d{y}^2)$ on $U$. Previous work by the authors showed that one can define a canonical metric (distance function) $D_h$ on $U$ associated with a $\gamma$-LQG surface. We show that this metric is conformally covariant in the sense that it respects the coordinate change formula for $\gamma$-LQG surfaces. That is, if $U,\tilde{U}$ are domains, $\varphi :U\to \tilde{U}$ is a conformal transformation, $Q=2/\gamma +\gamma /2$, and $\tilde{h}=h\circ {\varphi }^{-1}+Q\log |({\varphi }^{-1}{)}^{\prime }|$, then $D_h(z,w)=D_{\tilde{h}}(\varphi (z),\varphi (w))$ for all $z,w\in U$. This proves that $D_h$ is intrinsic to the quantum surface structure of $(U,h)$, i.e., it does not depend on the particular choice of parameterization.

arXiv comment21 pages, 2 figures. To appear in AIHP. arXiv admin note: text overlap with arXiv:1905.00383