Th\'eorie de Liouville et gravitation quantique: du couplage faible au couplage fort

TL;DR

This thesis explores Liouville quantum gravity, detailing algebraic structures, strong coupling behaviors, and deriving N-point functions, advancing understanding of non-perturbative regimes in quantum gravity.

Contribution

It provides a comprehensive analysis of the algebra of chiral components and proves a truncation theorem in strong coupling regimes, including fractional spins.

Findings

Complete elucidation of chiral algebra in different bases

Proof of truncation theorem for specific central charges

Derivation of N-point functions in strongly coupled models

Abstract

The first chapters introduce briefly conformal theories, Moore and Seiberg polynomial equations and Gervais-Neveu quantization of Liouville theory. The next chapters present the original results of this thesis. First, the algebra of the chiral components is completely elucidated, both in the Bloch wave and in the Quantum Group basis. Then, in the strong coupling regime, the proof of the truncation theorem of Gervais to real weight operators for C = 7, 13, 19 is completed, especially including fractionnal spins. In this strong coupling regime, the new cosmological term yields a real string susceptibility equal to the real part of the KPZ formula. And eventually the N-point functions of a strongly coupled topological model are obtained.