Liouville Conformal Field Theories in Higher Dimensions

TL;DR

This work formulates a higher-dimensional Liouville CFT in even dimensions using a log-correlated scalar with a background ${\cal Q}$-curvature charge and an exponential potential. It establishes classical Weyl invariance, analyzes the semiclassical limit, and studies quantum aspects via a higher-dimensional Coulomb gas, showing $C_T$ is $Q$-independent and computing the A-type anomaly. The paper provides integral expressions for Liouville correlators, connects them to a free-field description, and derives a DOZZ-like three-point function for light operators, signaling a tractable path to solving the theory in higher dimensions. It also outlines rich future directions, including boundaries, Lorentzian signatures, and possible links to random geometry and AGT-like structures.

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.