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.

57 pages. Accepted for publication in the journal "the Annals of Mathematics"

References in corpus (7)

• Lecture notes on Gaussian multiplicative chaos and Liouville Quantum Gravity
• Lecture notes on Liouville theory and the DOZZ formula
• Constructive Liouville Conformal Field Theory
• From Quarks to Strings
• Renormalizability of Liouville Quantum Gravity at the Seiberg bound
• The distribution of Gaussian multiplicative chaos on the unit interval
• Fusion asymptotics for Liouville correlation functions

Cited by in corpus (6)

• The complex Liouville string: the worldsheet
• Multiplicative chaos and the characteristic polynomial of the CUE: the $L^1$-phase
• The distribution of Gaussian multiplicative chaos on the unit interval
• On the regularity of complex multiplicative chaos
• Tail universality of critical Gaussian multiplicative chaos
• Path integral approach to analytic continuation of Liouville theory: the pencil region