Complementing the presentation of the Gaussian free field (GFF) with zero boundary conditions in Chapter 1, and on manifolds in Chapter 5, we devote this chapter to studying further variants of the field. The main example is the GFF in two dimensions with Neumann (or free) boundary conditions. We give a rigorous definition of this field as both a stochastic process indexed by suitable test functions and as a random distribution modulo constants. As in Chapter 1, we show the equivalence of these two viewpoints; however, in this case, further analytical arguments are required. We describe the covariance function of the field, prove key properties such as conformal covariance and the spatial Markov property, and discuss its associated Gaussian multiplicative chaos measures on the boundary of the domain where it is defined. We also cover the definition and properties of the whole plane Gaussian free field and the Gaussian free field with Dirichlet–Neumann boundary conditions, building on the construction of the Neumann GFF. We further prove that the whole plane GFF can be decomposed into a Dirichlet part and a Neumann part. Finally, we show that the total mass of the GMC associated to a Neumann GFF on the unit disc is almost surely finite.

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 provides a self-contained and thorough introduction to the continuum Gaussian free field (GFF) with zero (or Dirichlet) boundary conditions. We start by describing its discrete counterpart, before presenting two constructions of the continuum object: one as a stochastic process, and the other as a random generalised function. We explain the equivalence of these two perspectives, and in the remainder of the chapter, draw on both viewpoints to prove various important properties. In particular, we prove that the GFF satisfies a certain domain Markov property and exhibits precise scaling behaviour. In two dimensions, this is a special case of its (more general) conformal invariance. We go on to study the so-called thick points of the GFF in two dimensions, which are fractal sets of points where the field is atypically “high” and are particularly useful for understanding the Gaussian multiplicative chaos measures associated with the GFF in later chapters. We close the initial chapter with a rigorous scaling limit result, justifying that the continuum GFF is indeed the scaling limit of its discrete counterpart.

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.