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.

In this chapter, we describe a model of random planar maps weighted by self-dual Fortuin-Kasteleyn (FK) percolation. This can be thought of as a canonical discretisation of Liouville quantum gravity. We start with some generalities about planar maps and then introduce the FK random map model, which depends on a parameter $q\ge 0$, before explaining the conjectured connection to Liouville quantum gravity. A fundamental tool for studying such random planar maps is Sheffield’s (hamburger-cheeseburger) bijection. We first explain it carefully for tree-decorated maps (the special case of the FK model of planar maps with $q=0$), which correspond under this bijection to random walk excursions in the quarter-plane. We then explain its generalisation to $q\ge 0$ in detail. This is first used to show that the maps possess an infinite volume limit in the local topology. Then, a theorem of Sheffield gives a scaling limit result for these maps. One consequence is that a phase transition takes place at $q=4$. Furthermore, it allows one to compute some associated critical exponents when $q<4$ (which are consistent with the KPZ relation of Chapter 3). These arguments are a discrete analogue of the “mating of trees” perspective on Liouville quantum gravity described in Chapter 9.

In this appendix, we define reverse Loewner evolutions and reverse Schramm–Loewner evolutions, then going on to discuss symmetries in law with ordinary (forward) Loewner evolutions.

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.

We describe couplings between Schramm–Loewner Evolution (SLE) curves and variants of the Gaussian free field (GFF). In particular, we give a complete proof of Sheffield’s construction of $\gamma$-quantum boundary length along an ${\text{SLE}}_{{\gamma }^2}$ curve, as measured by an independent underlying GFF. The main input for this proof is a rigorous construction of the so-called quantum gravity zipper, which is a stationary dynamic on quantum surfaces (defined using a GFF) decorated by SLE. Another consequence of this construction is that drawing an SLE curve on top of an appropriate independent quantum surface splits the surface into two independent and identically distributed (sub)-surfaces, glued according to boundary length. In particular, this shows that SLE curves are solutions of natural random conformal welding problems.

In this chapter, we introduce the Liouville measures associated with the continuum two-dimensional Gaussian free field (GFF). Informally speaking, for a fixed parameter $\gamma$ (the so-called coupling constant), the $\gamma$-Liouville measure is obtained by exponentiating $\gamma$ times the GFF and taking this as a density with respect to Lebesgue measure. Since the GFF is not defined pointwise, the rigorous construction of this measure requires an approximation procedure. The bulk of this chapter is dedicated to establishing appropriate approximations, justifying their convergence, and proving uniqueness of the resulting measures. We also prove an important change-of-coordinates formula. The construction will be generalised in Chapter 3, which treats the overarching theory of Gaussian multiplicative chaos measures. These are measures of the same form discussed above, but constructed from a general underlying log-correlated Gaussian field. While the two-dimensional GFF is really just a specific example of such a field, some arguments specific to the GFF can be used to simplify the presentation and introduce relevant ideas in a clean way, without the need to introduce too much machinery.

In this appendix, we prove that convergence is a measurable event for a sequence of random variables in the space of distributions.