Speaker: Remi Rhodes (Univ. Paris)
Title: Towards quantum Kahler geometry
Abstract:
In this talk, we construct the quantum theory of the coupling of the
Liouville action and of Mabuchi's K-energy. Both functionals play a
prominent role respectively in Riemannian geometry and Kähler geometry.
As an output, we obtain a path integral whose Weyl anomaly presents the
standard Liouville anomaly plus an additional K-energy term. Motivations
come from theoretical physics where these type of path integrals have been
proposed by A. Bilal, F. Ferrari, S. Klevtsov and S. Zelditch as a model
for fluctuating metrics arising when coupling (small) massive
perturbations of conformal field theories to quantum gravity. Our
probabilistic construction
relies on a variant of Gaussian multiplicative chaos (GMC), the Derivative
GMC (DGMC for short). The main technical backbone of our construction is
twofold and consists in two estimates on (derivative and standard) GMC
which are of independent interest. First, we show that these DGMC random
variables possess negative exponential moments; second, we derive optimal
small deviations estimates for GMC (associated to a recentered GFF).