Imaginary geometry III: reversibility of $\mathrm{SLE}_\kappa$ for $\kappa \in (4,8)$

Abstract

Suppose that $D \subseteq \mathbf{C}$ is a Jordan domain and $x,y \in \partial D$ are distinct. Fix $\kappa \in (4,8)$, and let $\eta$ be an $\mathrm{SLE}_\kappa$ process from $x$ to $y$ in $D$. We prove that the law of the time-reversal of $\eta$ is, up to reparametrization, an $\mathrm{SLE}_\kappa$ process from $y$ to $x$ in $D$. More generally, we prove that $\mathrm{SLE}_\kappa(\rho_1;\rho_2)$ processes are reversible if and only if both $\rho_i$ are at least $\kappa/2-4$, which is the critical threshold at or below which such curves are boundary filling.
Our result supplies the missing ingredient needed to show that for all $\kappa \in (4,8)$, the so-called conformal loop ensembles $\mathrm{CLE}_\kappa$ are canonically defined, with almost surely continuous loops. It also provides an interesting way to couple two Gaussian free fields (with different boundary conditions) so that their difference is piecewise constant and the boundaries between the constant regions are $\mathrm{SLE}_\kappa$ curves.