Sur l'existence d'une prescription d'ordre naturelle projectivement invariante

TL;DR

This paper proves a conjecture that a natural, projectively invariant ordering prescription exists for differential operators between density bundles, using a lift of connections and invariant symbol lifting, with special results for Ricci-flat manifolds.

Contribution

The paper constructs a projectively invariant ordering prescription for differential operators between density bundles, confirming a conjecture and extending previous work with explicit formulas for Ricci-flat cases.

Findings

Constructed a projectively invariant lift of torsion-free connections.

Proved the existence of a natural ordering prescription invariant under projective changes.

For Ricci-flat manifolds, the ordering matches known explicit formulas.

Abstract

P.Lecomte has proposed to take into account the covariant derivatives used to build ordering prescriptions for the naturality of transformation properties and has conjectured that there exists an natural ordering prescription for differential operators of any orders between density bundles which in addition is invariant under projective changes of the covariant derivatives. We prove this conjecture by constructing a projectively invariant lift of a torsion-free connexion to a torsion-free connexion on (the positive part of) the total space of the bundle of all $a$-densities for nonzero $a$, by lifting the symbols in a projectively invariant way (they turn out to be in bijection to the space of all $\real^+$-equivariant divergence-free symmetric tensor fields on the total space), and by using the standard ordering procedure (`all the covariant derivatives to the right') on the total space. For Ricci-flat manifolds we show that this ordering prescription coincides --with the appropiate replacements-- with an explicit formula in $\real^m$ obtained by Duval, Lecomte and Ovsienko.