Séminaire Physique mathématique ICJ

Ward and Belavin-Polyakov-Zamolodchikov identities for Liouville quantum field theory on the Riemann sphere

par Vincent Vargas (DMA, ÉNS Paris)

Europe/Paris
Fokko du Cloux (Institut Camille Jordan)

Fokko du Cloux

Institut Camille Jordan

Université Lyon 1, Bât. Braconnier, 21 av. Claude Bernard, 69100 Villeurbanne
Description
The foundations of modern conformal field theory (CFT) were introduced in a 1984 seminal paper by Belavin, Polyakov and Zamolodchikov (BPZ). Though the CFT formalism is widespread in the physics literature, it remains a challenge for mathematicians to make sense out of it. Liouville CFT (or quantum field theory) is an important class of CFTs which can be seen as a random version of the theory of Riemann surfaces. In a recent work, we constructed the correlation functions of Liouville CFT in the Feynman path formalism using probabilistic techniques. In this talk, I will present a rigorous derivation of the so-called Ward and BPZ identities for Liouville CFT. These identities are the basis to compute the correlations and to establish the correspondence between the Feynman path formalism of Liouville CFT and the algebraic formalism based on the Virasoro algebra. Based on joint works with F. David, A. Kupiainen and R. Rhodes.