Liouville conformal field theory on even-dimensional spheres

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 few decades. Recent progress in the understanding of conformal geometry in dimension higher than two has naturally led to a generalization of the Polyakov formalism to higher dimensions based on conformally invariant operators: Graham–Jenne–Mason–Sparling operators and the Q-curvature. This article 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 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.