Given a Riemannian manifold $(M,g)$, consider a formal polynomial expression $L(g)$ on the coefficients of the metric $g_{ij}$, its partial derivatives, and $(\det g)^{-1}$. If $L(g)$ is invariant by changes of charts, then it can be written as a linear combination $\sum_{l\in L} a_l C^l(g)$ of some complete contractions of covariant derivatives of the Riemannian curvature tensor. A global conformal invariant is a functional of the form $\int_M L(g) dV_g$ which remains invariant under conformal rescalings of the metric. A conjecture or the physicists Stanley Deser and Adam Schwimmer in 1993 asserts that

$$

L(g)=W(g)+ \mathrm{div}_i T^i(g)+C \cdot \mathrm{Pfaff}(R_{ijkl}),

$$

where $W(g)$ is a local conformal invariant, $T^i(g)$ is a vector field, $C$ is a constant, and $\mathrm{Pfaff}(R_{ijkl})$ is the Pfaffian of the curvature tensor. The Chern-Gauss-Bonnet theorem says that this last term integrates to a multiple of the Euler-Poincaré characteristic of $M$.

This book provides a proof of this conjecture. The method consists on an iterative procedure to reduce the expression $\sum_{l\in L} a_l C^l(g)$ to another expression of a similar type for a reduced subset $L'\subset L$ of "good" complete contractions, trading them by local conformal invariants and divergences. The second step is a lengthy algebraic proof that the expression $\sum_{l\in L'} a_l C^l(g)$ satisfies the required property.