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.

In this chapter, we provide a comprehensive exposition of the theory of Gaussian multiplicative chaos (GMC), which generalises the construction of Liouville measures (discussed in Chapter 2) to the setup of logarithmically correlated Gaussian fields in arbitrary dimension and reference measures satisfying an energy condition. We first construct the Gaussian multiplicative chaos, which can be viewed as the measure obtained by exponentiating this logarithmically correlated field against the reference measure. We show that this measure can be characterised axiomatically (Shamov’s theorem). We then present a number of key tools for the study of GMC, including Girsanov’s lemma, Kahane’s convexity inequality and the explicit construction of certain fields enjoying a notion of exact scale invariance. Together, these two tools can be used to perform a multifractal study of GMC. This allows us to characterise the positive and negative moments that are finite. Finally, we apply these results to describe a rigorous version of the so-called KPZ (named after Knizhnik, Polyakov and Zamolodchikov) scaling relation.

In this appendix, we define radial Loewner chains and radial Schramm–Loewner evolutions (SLE). This includes the case of radial SLE with force points. We prove some of their main properties and their connection to chordal SLE. We conclude by discussing the particular version of radial SLE that satisfies invariance with respect to its target point.

In this appendix, we give an overview of the (deterministic) theory of chordal Loewner chains. We then define (chordal) Schramm–Loewner evolutions, including the case where force points are added, and describe some of their key properties.

We provide an overview of some recent results, relating holomorphic symmetric differentials, semiampleness of vector bundles, and various kind of characterizations of parallelizable manifolds.

These notes grew out of a mini-course given by the second-named author at Casa Matemática Oaxaca in the Fall of 2022. Their purpose is to provide an exposition, directed at graduate students, of the basic properties of complex analytic group bundles and torsors under them, including the flat case.

We study the injectivity of the cycle class map with values in Jannsen’s continuous étale cohomology, by using refinements that go through étale motivic cohomology and the “tame” version of Jannsen’s cohomology. In particular, we use this to show that the Tate and the Beilinson conjectures imply that its kernel is torsion in positive characteristic, and to revisit recent counterexamples to injectivity.