Noncritical Conformal Gravity and Four-Dimensional Liouville Theory

TL;DR

This paper investigates the quantum properties of four-dimensional conformal gravity, resolving beta function discrepancies through gauge-fixing and ghost terms, and derives a consistent Liouville action inspired by 2D quantum gravity.

Contribution

It introduces a BRST-invariant gauge-fixing approach that accounts for scalar modes and constructs a 4D Liouville theory from trace anomaly integration.

Findings

Resolved beta function mismatch with gauge-fixing and ghost terms

Derived a consistent 4D Liouville action from trace anomaly

Established conditions for anomaly-free BRST symmetry

Abstract

We study the quantum aspects of the conformal gravity in four dimensions, specifically addressing a known discrepancy in beta functions between general quadratic curvature theories and conformal gravity, which corresponds to two scalar degrees of freedom. We demonstrate that this mismatch is resolved by carefully introducing gauge-fixing and ghost terms via the BRST symmetry, which effectively adds the two scalar modes. Drawing lessons from two-dimensional quantum gravity and Liouville theory, we proceed to integrate the four-dimensional trace anomaly to derive a consistent Liouville action, which is given by a free-field action for the conformal mode with a consistent conformal anomaly. Finally we give the condition that the BRST transformation is anomaly free.