Liouville Conformal Field Theory on even-dimensional spheres

TL;DR

This paper constructs a rigorous framework for Liouville Conformal Field Theory on even-dimensional spheres, extending classical and quantum aspects using conformal geometry and probabilistic methods, aligning with physics expectations.

Contribution

It provides the first rigorous construction of Liouville CFT on higher even-dimensional spheres at both classical and quantum levels.

Findings

Classical uniformization problem solved for higher dimensions.

Quantum Liouville CFT constructed via probabilistic methods.

Results align with theoretical physics predictions.

Abstract

Initiated by Polyakov in his 1981 seminal work, the study of two-dimensional Liouville Conformal Field Theory has drawn considerable attention over the past decades. Recent progress in the understanding of conformal geometry in dimension higher than two have naturally led to a generalization of Polyakov formalism to higher dimensions, based on conformally invariant operators: Graham-Jenne-Mason-Sparling operators and the $\mathcal{Q}$-curvature. This document is dedicated to providing a rigorous construction of Liouville Conformal Field Theory on even-dimensional spheres. This is done at the classical level in terms of a generalized \textit{Uniformization} problem, and at the quantum level thanks to a probabilistic construction based on log-correlated fields and Gaussian Multiplicative Chaos. The properties of the objects thus defined are in agreement with the ones expected in the physics literature.