On Gaussian multiplicative chaos and conformal field theory
This thesis is concerned with conformally invariant stochastic processes in two dimensions and their applications to conformal field theory (CFT). The main probabilistic objects are the Gaussian free field (GFF) and the random geometries associated to it. Especially, we are interested in Gaussian multiplicative chaos (GMC), Schramm-Loewner evolution (SLE) and Liouville CFT, which can be understood as theories of random surfaces.
From the point of view of physics, the idea of a ``summing over surfaces" can be traced back to Polyakov's work on bosonic string theory. Indeed, the starting point of string theory is to replace a point particle by a one dimensional manifold (a string), so that one must replace the worldline by a worldsheet, i.e. an embedding of a surface into space-time. The path integral that Polyakov wrote down features a random conformal factor that should be described by the quantisation of the Liouville action. Therefore, this probability measure should describe random fluctuations around the uniform metric.
Polyakov also suggested that the resulting quantum field theory should exhibit conformal invariance. This means that the Hilbert space of the theory should carry a projective unitary representation of the group of local conformal transformations, i.e. a unitary representation of the Virasoro algebra. Since it is an infinite dimensional Lie algebra, this is a huge constraint to put on a system and this led Belavin, Polyakov & Zamolodchikov to give an axiomatic framework for CFT based on the representation theory of the Virasoro algebra. Here, the game is somehow reversed: one tries to exhibit and classify all theories fitting in this framework. In particular, it is not even clear in the first place that such algebraic structures exist.
In this context, Liouville theory is a success story in the interaction of algebra, geometry and probability. On the one hand, the algebraic point of view was successful in finding a theory fitting in the BPZ framework. On the other hand, it was unclear that this theory should correspond to the actual path integral written down by Polyakov, let alone the fact that this path integral was not a rigorously defined mathematical object. Only recently was this path integral constructed using a rigorous probabilistic framework and shown to satisfy all the properties predicted by the algebraic formulation.
The construction of the Liouville path integral relies on Gaussian multiplicative chaos, a theory pioneered by Kahane in the context of turbulence, allowing one to exponentiate a logarithmically correlated Gaussian field such as the two-dimensional GFF. The resulting object is a random multifractal measure which has found many applications in modern probability theory. In Liouville theory, partition functions and correlation functions are expressed as expected values of observables associated with GMC. The fact that the path integral fits in the BPZ framework has two important consequences. First, the algebraic constraints coming from the BPZ framework give a better understanding of the law of GMC. Second, having a concrete representation of the axiomatic structure allows one to perform additional computations and answer some algebraic questions, such as the convergence of conformal blocks.
Apart from Liouville theory, CFT has a wide scope and is conjectured (in a few cases proved) to describe the scaling limits of many statistical mechanics models at criticality. On the probabilistic side of the story, a major step was performed by Schramm with the introduction of stochastic Loewner evolutions (SLE). He was able to classify all conformally invariant probability measures on paths joining two points on the boundary of a planar domain, therefore describing all possible scaling limits of interfaces of spin clusters of critical models (provided they are conformally invariant in the limit). These measures are indexed by a real parameter κ>0 (understood as Planck's constant) and are related to the central charge of the theory (i.e. the universality class of the model considered). Sheffield later showed that SLE is the solution to a problem of conformal welding involving GMC, a deep result which was considerably generalised in the mating-of-trees approach to Liouville theory. Roughly speaking, Sheffield's result means that Liouville theory is stable under gluing of boundary components, and that the interface curve arising from the gluing is an SLE. Another corollary is the existence of a natural parameterisation of SLE known as the quantum length.
In this thesis, we tried to explore the above-mentioned connections between probability, geometry and algebra. In Chapters 2 and 3, we study the asymptotic behaviour of Liouville correlation functions in two specific geometric cases: the once-puncture torus and the four-punctured sphere. This is a purely probabilistic statement, which can be interpreted physically as the factorisation of the partition function on the boundary of the moduli space. The two cases considered constitute the two degeneration paradigms (self-gluing and gluing of disconnected components) and the methods could generalise easily to other moduli spaces of stable curves.
The data of a conformal structure on a surface can be understood as the data of Brownian motion up to reparametrisation. In this context, Liouville Brownian motion (LBM) is the Brownian motion with the parameterisation induced by GMC viewed as a random conformal factor. The existence of such a process is not clear due to the irregularity of GMC but was carried out by Garban, Rhodes & Vargas, and independently by Berestycki. In Chapter 4, we introduce the boundary version of (LBM), which is a Cauchy process reparameterised by GMC. Using Sheffield's result, an interesting consequence is the existence of a diffusion process on SLE, which is a Cauchy process parameterised by quantum length. The subsequent Chapter 5 studies the regularity of the welding homeomorphism of SLE for κ=4, which is a critical situation from many points of view.
One advantage of conformal welding is that we can view SLE as a probability measure on the group Homeo(S^1) of orientation preserving homeomorphisms of the circle. Thus, it is natural to ask whether SLE is related in a certain sense to the quantisation of Diff(S^1). More interestingly, the homogeneous space S^1\Diff(S^1) arises as a coadjoint orbit of Diff(S^1) and was shown to possess a two-parametric family of homogeneous Kähler forms, for which there is a globally defined potential. Among the various formulae known for this potential, the universal Liouville action of Takhtajan & Teo suggests a link between SLE and the geometric quantisation of S^1\Diff(S^1). This connection will be made more precise in an ongoing work described in Section 1.5, where we use the canonical action of Diff(S^1) to define a unitary representation of the Virasoro algebra on the L^2-space of SLE endowed with its quantum length. Using conformal welding, this action can be expressed in terms of the so-called universal period mapping of Nag & Sullivan. Interestingly, the integration by parts formula from Malliavin calculus can be interpreted in the context of symplectic geometry and the stress-energy tensor emerges in connection with the momentum map for the Diff(S^1)-action.
Engineering and Physical Sciences Research Council (1946490)