This chapter provides an introduction to Liouville conformal field theory on the sphere, as developed in a series of papers starting with the work of David, Kupiainen, Rhodes and Vargas. We give an informal overview of conformal field theory in general and Polyakov’s action, before starting our rigorous presentation. For this, we first spend some time defining Gaussian free fields on general manifolds, and explaining how to construct their associated Gaussian multiplicative chaos measures via uniformisation. We then show how to construct the correlation functions of the theory under certain constraints known as the Seiberg bounds. One remarkable feature of the theory is its integrability: we demonstrate this phenomenon by expressing the k-point correlation functions as negative fractional moments of Gaussian multiplicative chaos. We conclude with a brief overview of some recent developments, including a short discussion of BPZ equations, conformal bootstrap and the proof by Kupiainen, Rhodes and Vargas of the celebrated DOZZ formula.

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.