Conformal invariant powers of the Laplacian, Fefferman-Graham ambient metric and Ricci gauging

TL;DR

This paper rederives conformally invariant powers of the Laplacian using the ambient space approach, clarifies their holographic structure, and explores the ambient space origin of Ricci gauging for Weyl invariance.

Contribution

It introduces a new ambient space perspective on conformally invariant operators and connects Ricci gauging with the Fefferman-Graham expansion and ambient metric techniques.

Findings

Rederived conformally invariant Laplacian powers using ambient space.

Clarified the holographic structure of these operators.

Proposed a gauged ambient metric and analyzed Penrose-Brown-Henneaux transformations.

Abstract

The hierarchy of conformally invariant k-th powers of the Laplacian acting on a scalar field with scaling dimensions $\Delta_{(k)}=k-d/2$, k=1,2,3 as obtained in the recent work [1] is rederived using the Fefferman-Graham d+2 dimensional ambient space approach. The corresponding mysterious "holographic" structure of these operators is clarified. We explore also the d+2 dimensional ambient space origin of the Ricci gauging procedure proposed by A. Iorio, L. O'Raifeartaigh, I. Sachs and C. Wiesendanger as another method of constructing the Weyl invariant Lagrangians. The corresponding \emph{gauged} ambient metric, Fefferman-Graham expansion and extended Penrose-Brown-Henneaux transformations are proposed and analyzed.