Showing papers by "Rémi Rhodes published in 2020"

Integrability of Liouville theory: proof of the DOZZ Formula

Abstract: Dorn and Otto (1994) and independently Zamolodchikov and Zamolodchikov (1996) proposed a remarkable explicit expression, the so-called DOZZ formula, for the 3 point structure constants of Liouville Conformal Field Theory (LCFT), which is expected to describe the scaling limit of large planar maps properly embedded into the Riemann sphere. In this paper we give a proof of the DOZZ formula based on a rigorous probabilistic construction of LCFT in terms of Gaussian Multiplicative Chaos given earlier by F. David and the authors. This result is a fundamental step in the path to prove integrability of LCFT, i.e. to mathematically justify the methods of Conformal Bootstrap used by physicists. From the purely probabilistic point of view, our proof constitutes the first rigorous integrability result on Gaussian Multiplicative Chaos measures.

Conformal bootstrap in liouville theory

Abstract: Liouville conformal field theory (denoted LCFT) is a 2-dimensional conformal field theory depending on a parameter γ ∈ R and studied since the eighties in theoretical physics. In the case of the theory on the 2-sphere, physicists proposed closed formulae for the n-point correlation functions using symmetries and representation theory, called the DOZZ formula (for n = 3) and the conformal bootstrap (for n > 3). In a recent work, the three last authors introduced with F. David a probabilistic construction of LCFT for γ ∈ (0, 2] and proved the DOZZ formula for this construction. In this sequel work, we give the first mathematical proof that the probabilistic construction of LCFT on the 2-sphere is equivalent to the conformal bootstrap for γ ∈ (0, √ 2). Our proof combines the analysis of a natural semi-group, tools from scattering theory and the use of the Virasoro algebra in the context of the probabilistic approach (the so-called conformal Ward identities).