The decomposition of global conformal invariants II: The Fefferman-Graham ambient metric and the nature of the decomposition

TL;DR

This paper proves a key part of a conjecture on the structure of global conformal invariants by using the Fefferman-Graham ambient metric to decompose integrands into local invariants, divergences, and the Chern-Gauss-Bonnet term.

Contribution

It demonstrates how to separate local conformal invariants from divergence terms using the ambient metric, completing the proof of the conjecture.

Findings

Successfully decomposes conformal invariants into fundamental components

Utilizes Fefferman-Graham ambient metric for invariant construction

Completes the proof of the Deser-Schwimmer conjecture

Abstract

This is the second in a series of papers where we prove a conjecture of Deser and Schwimmer regarding the algebraic structure of ``global conformal invariants''; these are defined to be conformally invariant integrals of geometric scalars. The conjecture asserts that the integrand of any such integral can be expressed as a linear combination of a local conformal invariant, a divergence and of the Chern-Gauss-Bonnet integrand. The present paper addresses the hardest challenge in this series: It shows how to {\it separate} the local conformal invariant from the divergence term in the integrand; we make full use of the Fefferman-Graham ambient metric to construct the necessary local conformal invariants, as well as all the author's prior work [1,2,3] to construct the necessary divergences. This result combined with [3] completes the proof of the conjecture, subject to establishing a purely algebraic result which is proven in [6,7,8].