Abstract:
Liouville field theory was introduced by Polyakov in the eighties in the context of string theory. Liouville theory appeared there under the form of a 2D Feynman path integral, describing fluctuating metrics over Riemann surfaces. Since then, this theory has been extensively studied in physics and this interest has more recently spread to the probabilistic community where it appears as a natural model of random Riemann surfaces. Liouville theory is a conformal field theory and, as such, the quantities of interest are the correlation functions. In this talk, I will explain some joint works with C. Guillarmou, A. Kupiainen and V. Vargas where we prove the conformal bootstrap conjecture. Conformal bootstrap can be seen as the quantum analog of the pair of pants decomposition of Riemann surfaces. It states that the correlation functions of Liouville conformal field theory on Riemann surfaces can be expressed in terms of products of 3-point correlation functions on the sphere and the conformal blocks, which are holomorphic functions on the moduli space of punctured Riemann surfaces.
Biography:
Rémi Rhodes is a professor at the University of Aix-Marseille and a researcher at the Institute of Mathematics of Marseille in the Probability team of the Mathematics of Randomness group, since 2018. After graduating from ENS Cachan , he defended his thesis in 2006 at the University of Provence. In 2007, he became a lecturer at Paris-Dauphine University. Then, in 2014, he was assigned to the Laboratory of Analysis and Applied Mathematics at the University of Paris-Est Marne la Vallée until 2018, when he returned to his hometown. In 2019, he was appointed junior member of the Institut Universitaire de France. As a probabilist, his research themes are concerned with Gaussian multiplicative chaos, Liouville field theory, and the probabilistic approach to quantum field theories. Jointly with Vargas in 2019, he received the Marc Yor Prize from the Société mathématique de France and the French Academy of Science for his construction of Liouville conformal field theory. Then and jointly with A. Kupiainen and V. Vargas in 2022, he received the Polya Prize in Mathematics from the SIAM for the rigorous derivation of the DOZZ formula for three-point structure constants in Liouville Conformal Field Theory.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai