Imaginary geometry III: Reversibility of SLE
k for K ∈ (4; 8)
Suppose that D⊆C is a Jordan domain and x,y∈∂D are distinct. Fix κ∈(4,8), and let η be an SLEκ process from x to y in D. We prove that the law of the time-reversal of η is, up to reparametrization, an SLEκ process from y to x in D. More generally, we prove that SLEκ(ρ1;ρ2) processes are reversible if and only if both ρi are at least κ/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 κ∈(4,8), the so-called conformal loop ensembles CLEκ 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 SLEκ curves.