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 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.

arXiv comment32 pages. Minor corrections, presentation improved

References in corpus (11)

• The Large N Limit of Superconformal Field Theories and Supergravity
• Structure Constants and Conformal Bootstrap in Liouville Field Theory
• Two and three-point functions in Liouville theory
• Liouville theory revisited
• Uniqueness and universality of the Brownian map
• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• On the Liouville three-point function
• A lecture on the Liouville vertex operators
• Two perspectives of the 2D unit area quantum sphere and their equivalence
• Polyakov formulas for GJMS operators from AdS/CFT
• Liouville Conformal Field Theories in Higher Dimensions

Cited by in corpus (2)

• Q-curvature and Path Integral Complexity
• Conformally invariant random fields, quantum Liouville measures, and random Paneitz operators on Riemannian manifolds of even dimension