The decomposition of global conformal invariants

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.

Vicente Muñoz
Book details

This book proves a conjecture of Deser and Schwimmer regarding the algebraic structure of global conformal invariants (conformally invariant integrals of scalar functionals on the Riemannian metric). 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.

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