Liouville Quantum Gravity as a Mating of Trees

There is a simple way to ``glue together'' a coupled pair of continuum random trees (CRTs) to produce a topological sphere. The sphere comes equipped with a measure and a space-filling curve (which describes the ``interface'' between the trees). We present an explicit and canonical way to embed the sphere in $\C \cup \{ \infty \}$. In this embedding, the measure is a form of Liouville quantum gravity (LQG) with parameter $\gamma \in (0,2)$, and the curve is space-filling SLE$_{\kappa'}$ with $\kappa' = 16/\gamma^2$. Achieving this requires us to develop an extensive suite of tools for working with LQG surfaces. We explain how to conformally weld so-called ``quantum wedges'' to obtain new quantum wedges of…