Field-measure correspondence in Liouville quantum gravity almost surely commutes with all conformal maps simultaneously

Abstract

In Liouville quantum gravity (or $2d$-Gaussian multiplicative chaos) one seeks to define a measure $\mu^h = e^{\gamma h(z)} dz$ where $h$ is an instance of the Gaussian free field on a planar domain $D$. Since $h$ is a distribution, not a function, one needs a regularization procedure to make this precise: for example, one may let $h_\epsilon(z)$ be the average value of $h$ on the circle of radius $\epsilon$ centered at $z$ (or an analogous average defined using a bump function supported inside that circle) and then write $\mu^h = \lim_{\epsilon \to 0} \epsilon^{\frac{\gamma^2}{2}} e^{\gamma h_\epsilon(z)} dz$. If $\phi: \tilde D \to D$ is a conformal map, one can write $\tilde h = h \circ \phi + Q \log |\phi'|$, where $Q = 2/\gamma + \gamma/2$. The measure $\mu^{\tilde h}$ on $\tilde D$ is then a.s.\ equivalent to the pullback via $\phi^{-1}$ of the measure $\mu^h$ on $D$. Interestingly, although this a.s.\ holds for each \textit{given} $\phi$, nobody has ever proved that it a.s.\ holds \textit {simultaneously} for all possible $\phi$. We will prove that this is indeed the case. This is conceptually important because one frequently defines a \textit{quantum surface} to be an equivalence class of pairs $(D, h)$ (where pairs such as the $(D,h)$ and $(\tilde D, \tilde h)$ above are considered equivalent) and it is useful to know that the set of pairs $(D,\mu^{h})$ obtained from the set of pairs $(D,h)$ in an equivalence class is itself an equivalence class with respect to the usual measure pullback relation.