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 $μ^h = e^{γh(z)} dz$ 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 $h_ε(z)$ be the average value of $h$ on the circle of radius $ε$ centered at $z$ (or an analogous average defined using a bump function supported inside that circle) and then write $μ^h = \lim_{ε\to 0} ε^{\frac{γ^2}{2}} e^{γh_ε(z)} dz$. If $ϕ: \tilde D \to D$ is a conformal map, one can write $\tilde h = h \circ ϕ+ Q \log |ϕ'|$, where $Q = 2/γ+ γ/2$. The measure $μ^{\tilde h}$ on $\tilde D$ is then a.s.\ equivalent to the pullback via $ϕ^{-1}$ of the measure $μ^h$ on $D$. Interestingly, although this a.s.\ holds for each \textit{given} $ϕ$, nobody has ever proved that it a.s.\ holds \textit {simultaneously} for all possible $ϕ$. 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 $(D,μ^{h})$ 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.