This chapter is devoted to the study of so-called quantum surfaces, which are fields defined on a parameterising domain, viewed up to an equivalence relation corresponding to the conformal change of coordinates formula of Chapter 2. We construct various special quantum surfaces enjoying scale-invariance properties, including quantum spheres, discs, wedges and cones. These objects are the conjectured scaling limits of families of random planar maps, as in Chapter 4 for example, depending on the imposed discrete topology. We conclude the chapter by explaining how these quantum surfaces are related in a rigorous way to the Liouville conformal field theory developed in Chapter 5.

In this chapter, we take forward the ideas developed in Chapter 8 and show that if one explores a $\gamma$-quantum cone via a certain space-filling SLE with parameter $\kappa =16/{\gamma }^2$ this results in a (stationary) decomposition of the cone into two independent quantum wedges, which are glued along the boundary. Furthermore, as we discover the curve, the relative changes in the boundary lengths evolve like a pair of correlated Brownian motions, where the correlation coefficient depends explicitly on the coupling constant $\gamma$ (equivalently, on the parameter $\kappa$ of the SLE). This gives a representation of the quantum cone as a glueing (“mating”) of two correlated continuous random trees, which is a direct continuum analogue of the results on random planar maps obtained in Chapter 4. This connection provides a rigorous justification that decorated random planar map models converge to Liouville quantum gravity in a certain precise sense. In order to explain the main results, we give an extensive description and treatment of whole-plane space-filling SLE, although we do not prove the essential but complex fact that it can be defined as a continuous curve.