Deser–Schwimmer conjecture on global conformal invariants

Let $M^n$ be a Riemannian manifold with $n$ even, and let $P(g)$ be a Riemannian invariant of weight $-n$ such that

$$\int_{M^n}P(g)\,dV_g$$

is unchanged under conformal rescalings $g\mapsto e^{2\phi}g$ for every $\phi\in C^\infty(M^n)$. A local conformal invariant is an invariant $W(g)$ with the corresponding conformal transformation property, and let $T^i(g)$ be a Riemannian vector field. Deser–Schwimmer conjecture. There exist a local conformal invariant $W(g)$, a Riemannian vector field $T^i(g)$, and a constant $(Const)$ such that

$$P(g)=W(g)+\operatorname{div}_i T^i(g)+(Const)\cdot\operatorname{Pfaff}(R_{ijkl}).$$

The final term is the Chern–Gauss–Bonnet integrand, while the divergence integrates to zero on a closed manifold. The paper states that this conjecture is confirmed by the series of works of Alexakis, with the present paper proving the algebraic proposition at the heart of the resolution.