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Liouville Conformal Field Theory on even-dimensional spheres

Baptiste Cerclé

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.

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 -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.

Paper Structure

This paper contains 32 sections, 20 theorems, 132 equations, 1 table.

Table of Contents

  1. Introduction
  2. Liouville Conformal Field Theory in higher dimension
  3. The formalism of Liouville Conformal Field Theory
  4. Acknowledgements
  5. Liouville Conformal Field Fheory in higher dimension: the classical theory
  6. Conformal transformations in higher dimension
  7. Conformal operators in higher dimension: GJMS operators and the $\mathcal{Q}$-curvature
  8. Definition of the operators
  9. Green's kernels for GJMS operators
  10. Classical LCFT on the sphere and Uniformization
  11. The Classical Liouville action functional
  12. The Liouville functional with conical singularities
  13. Quantization of the action: Liouville Conformal Field Theory on the higher-dimensional sphere
  14. Probabilistic background
  15. Log-correlated fields
  16. ...and 17 more sections

Key Result

Lemma \oldthetheorem

Let $\psi:\mathbb{R}^d\cup\lbrace\infty\rbrace\mapsto\mathbb{R}^d\cup\lbrace\infty\rbrace$ be a Möbius transform, and $x,y$ be any two points in $\mathbb{R}^d$ (not mapped to $\infty$). Then

Theorems & Definitions (39)

  • Lemma \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • proof
  • Lemma \oldthetheorem
  • ...and 29 more