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

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].