Liouville Conformal Field Theories in Higher Dimensions

TL;DR

This paper generalizes Liouville conformal field theory to higher even dimensions, exploring its properties, correlation functions, and anomalies, revealing new insights into non-unitary, conformally invariant theories beyond two dimensions.

Contribution

It introduces higher-dimensional Liouville theories, computes their conformal anomalies, and derives a generalized DOZZ formula for three-point functions.

Findings

$C_T$ is independent of $ ext{Q}$-curvature charge

Calculated the A-type Euler conformal anomaly

Derived a higher-dimensional DOZZ formula for three-point functions

Abstract

We consider a generalization of the two-dimensional Liouville conformal field theory to any number of even dimensions. The theories consist of a log-correlated scalar field with a background $\mathcal{Q}$-curvature charge and an exponential Liouville-type potential. The theories are non-unitary and conformally invariant. They localize semiclassically on solutions that describe manifolds with a constant negative $\mathcal{Q}$-curvature. We show that $C_T$ is independent of the $\mathcal{Q}$-curvature charge and is the same as that of a higher derivative scalar theory. We calculate the A-type Euler conformal anomaly of these theories. We study the correlation functions, derive an integral expression for them and calculate the three-point functions of light primary operators. The result is a higher-dimensional generalization of the two-dimensional DOZZ formula for the three-point function of such operators.