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

Understand

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 $e^{\gamma h} (dx^2 + dy^2)$ 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 $U,\widetilde{U}$ are domains, $\phi \colon U \to \widetilde{U}$ is a conformal transformation, $Q=2/\gamma+\gamma/2$, and $\widetilde h = h\circ\phi^{-1} + Q\log|(\phi^{-1})'|$, then $D_h(z,w) = D_{\widetilde{h}}(\phi(z),\phi(w))$ for all $z,w \in U$.