Gluing random surfaces: conformal bootstrap in Liouville theory via Segal’s axioms

Abstract: The law of Markov processes indexed by time (the real line) take a simple form when expressed in terms of the action of the associated semigroup. The generalization of this question to the case when the process is indexed by higher dimensional manifolds is more intricate and this question is particularly relevant in the study of quantum field theories. A general proposal was formalized by G. Segal in the eighties in this direction. Yet, concrete examples of QFTs where the Segal axioms are indeed valid are extremely limited (beyond trivial cases). We treat here the case of a specific Conformal Field Theory (CFT), called the Liouville theory which is a probabilistic model of 2D random surfaces. The outcome is the validity of the conformal bootstrap, i.e. a bridge between probability and representation theory: correlation functions are expressed in terms of (universal) holomorphic functions of the moduli parameters of the Riemann surface, called conformal blocks which have a strong representation theoretical content, and the structure constants of the CFT, here the DOZZ formula. Conformal bootstrap was conjectured in physics in the eighties to be the universal structure of CFTs and Liouville theory is perhaps the first non trivial example where it can be shown to hold mathematically. This talk will be introductory to these topics.

analysis of PDEsprobability

Audience: researchers in the discipline

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Stochastic PDEs and their friends

Series comments: We are organizing a three day online workshop devoted to recent developments in SPDEs and related topics. Please complete the registration form at forms.gle/G9Xw942iVNNgB4SLA if you would like to take part in the conference.

Confirmed speakers:

Yuri BAKHTIN (NYU)

Ajay CHANDRA (Imperial)

Dan CRISAN (Imperial)

Nina HOLDEN (ETH Zurich)

Kostantin KHANIN (U Toronto)

Davar KHOSHNEVISAN (University of Utah)

Nicolai KRYLOV (University of Minnesota)

Jonathan MATTINGLY (Duke)

Leonid MYTNIK (Technion)

Nicolas PERKOWSKI (Free U Berlin)

Ellen POWELL (Durham University)

Jeremy QUASTEL (U Toronto)

Daniel REMENIK (Universidad de Chile)

Armen SHIRIKYAN (University of Cergy-Pontoise)

Gigliola STAFFILANI (MIT)

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