The decomposition of global conformal invariants (Q2887104)

Consider a formal expression in the terms of the coordinates of an \(n\)-dimensional metric tensor and its inverse, the partial derivatives of these coordinates and the volume form. If such an expression behaves well with respect to coordinate changes, then it associates to each Riemannian metric \(g\) on an \(n\)-dimensional manifold \(M\) a smooth function \(M\to\mathbb R\). This is the classical definition of a local Riemannian invariant, and the weight of such invariants is defined via their behavior under constant rescalings of the metric. There is a a classical theorem of Hermann Weyl giving an explicit description of all local Riemannian invariants: they can be built from complete contractions of tensor products of iterated covariant derivatives of the Riemann curvature tensor.NEWLINENEWLINENow suppose that \(g\mapsto P(g)\) defines a local Riemannian invariant of weight \(-n\) in dimension \(n\). Then, for a compact Riemannian manifold \((M,g)\) of that dimension, the number \(\int_MP(g)\text{vol}_g\) is unchanged if \(g\) is replaced by a constant rescaling. One says that \(P\) defines a ``global conformal invariant'' if this number is actually invariant under arbitrary conformal rescalings of the metric.NEWLINENEWLINEThere are three rather trivial ways to obtain such invariants. Firstly, \(P\) itself may already be invariant under conformal rescalings, i.e., a local conformal invariant. Then, of course, the integral is conformally invariant too. Secondly, one may form a Riemannian invariant vector field (with an obvious meaning) and \(P\) its divergence. In this case, the integral vanishes and thus is conformally invariant. Finally, one may take \(P\) to be the Pfaffian of the Riemann curvature tensor. Then, by the Gauss-Bonnet-Chern theorem, the integral equals the Euler characteristic of \(M\), and thus is completely independent of the metric. Motivated by considerations from physics, S.~Deser and A.~Schwimmer conjectured in 1993 that any Riemannian invariant \(P\) whose integral is a global conformal invariant can be written as a linear combination of invariants of these three types.NEWLINENEWLINEThis book contains the results of a long term project of the author. In a short introduction, he outlines the problem and the structure of the book, puts the conjecture into perspective, and discusses possible generalizations. The rest of the book is devoted to providing (together with two recent articles of the author containing several technical proofs) a complete proof of the Deser-Schwimmer conjecture. As one can imagine from the nature of the conjecture, this is highly technical and very demanding. Given a Riemannian invariant \(P\) which defines a global conformal invariant, the basic strategy is to iteratively split off divergences and local conformal invariants, with each step ``improving'' the form of the remaining invariant in a certain sense. Assuming several technical results on local Riemannian invariants, it is shown in chapters 2 and 3, that this procedure can be used to reduce the full Deser-Schwimmer conjecture to a partial case, which was solved in earlier works of the author. Chapters 4 to 7 contain the proof of the above-mentioned technical results on Riemannian invariants, thus completing the proof of the Deser-Schwimmer conjecture.