Conformal Decomposition of the Effective Action and Covariant Curvature Expansion

TL;DR

This work develops a conformal decomposition of the four-dimensional one-loop effective action by separating the anomalous part responsible for the conformal anomaly from a conformal-invariant remainder. By fixing a conformal gauge, the authors define a unique representative $\overline{g}_{\mu\nu}=e^{\Sigma(g)}g_{\mu\nu}$ and show that the anomaly is reproduced exactly by the anomalous term $W_{\rm A}$ while the remaining physics is captured by $\overline{W}$. They reformulate the covariant curvature expansion up to cubic order in curvatures in a new basis based on $C_{\mu\nu}$, achieving substantial simplifications of the form factors and enabling absorption of noninvariant pieces into $W_{\rm A}$; explicit $\alpha$-representations and connections to FV/R gauge choices are provided. The framework offers a path to apply conformal methods in higher dimensions and to study quantum backreaction in black hole spacetimes, where the near-horizon geometry is conformally related to $R\times H^3$ and the curvature expansion may fail in its naive form. Overall, the paper furnishes a practical, gauge-fixed decomposition that clarifies the role of conformal invariance in 4D quantum gravity and supplies tools for analyzing nonlocal effective actions.

Abstract

The class of effective actions exactly reproducing the conformal anomaly in 4D is considered. It is demonstrated that the freedom within this class can be fixed by the choice of the conformal gauge. The conformal invariant part of the generic one-loop effective action expanded in the covariant series up to third order in the curvature is rewritten in the new conformal basis. The possible applications of the obtained results are discussed.