A renormalization approach to the Liouville quantum gravity metric

Abstract

This thesis explores metric properties of Liouville quantum gravity (LQG), a random geometry with conformal symmetries introduced in the context of string theory by Polyakov in the 80’s. Formally, it corresponds to the Riemannian metric tensor “e^{γh}(dx² + dy²)” where h is a planar Gaussian free field and γ is a parameter in (0, 2). Since h is a random Schwartz distribution with negative regularity, the exponential e^{γh} only makes sense formally and the associated volume form and distance functions are not well-defined. The mathematical language to define the volume form was introduced by Kahane, also in the 80’s. In this thesis, we explore a renormalization approach to make sense of the distance function and we study its basic properties