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We construct a conformal welding of two Liouville quantum gravity randomsurfaces and show that the interface between them is a random fractal curvecalled the Schramm-Loewner evolution (SLE), thereby resolving a variant of aconjecture of Peter Jones. We also demonstrate some surprising symmetries ofthis construction, which are consistent with the belief that (path decorated)random planar maps have (SLE-decorated) Liouville quantum gravity as a scalinglimit. We present several precise conjectures and open questions.

Conformal weldings of random surfaces: SLE and the quantum gravity zipper