Journée "Cartes Aléatoires"

Liouville quantum gravity on the Riemann sphere

by Prof. Vincent VARGAS (CNRS, ENS Paris)

Centre de conférences Marilyn et James Simons (I.H.E.S.)

Centre de conférences Marilyn et James Simons


Le Bois-Marie 35, route de Chartres 91440 Bures-sur-Yvette
In this talk, I will present a rigorous probabilistic construction of Liouville Field Theory on the Riemann sphere with positive cosmological constant, as considered by Polyakov in his 1981 seminal work "Quantum geometry of bosonic strings". Then, I will explain some of the fundamental properties of the theory like conformal covariance under PSL$_2(\C)$-action, Seiberg bounds, KPZ scaling laws, the KPZ formula and the Weyl anomaly (Polyakov-Ray-Singer) formula. If time permits, I will also explain the construction in the disk. This is based on joint works (some on arxiv and others in progress) with F. David, Y. Huang, A. Kupiainen, R. Rhodes.