Conformally invariant powers of the Laplacian, Q-curvature, and tractor calculus

TL;DR

This paper presents an elementary algorithm to explicitly express conformally invariant GJMS operators and Q-curvature using tractor calculus, facilitating calculations in conformal geometry and generalizing these operators.

Contribution

It introduces a new algorithm for expressing GJMS operators and Q-curvature explicitly in tractor calculus, based on ambient metric construction, with applications to specific operators and curvatures.

Findings

Explicit formulae for GJMS operators in tractor calculus

Direct definition and formula for Q-curvature

Examples of operators of order 4, 6, and 8

Abstract

We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et alia. These differential operators have leading part a power of the Laplacian. Conformal tractor calculus is the natural induced bundle calculus associated to the conformal Cartan connection. Applications discussed include standard formulae for these operators in terms of the Levi-Civita connection and its curvature and a direct definition and formula for T. Branson's so-called Q-curvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Q-curvature. Among examples, the operators of order 4, 6, and 8 and the related Q-curvatures are treated explicitly. The algorithm exploits the ambient metric construction of Fefferman and Graham and includes a procedure for converting the ambient curvature and its covariant derivatives into tractor calculus expressions. This is partly based on "Standard tractors and the conformal ambient metric construction" (A. Cap and A.R. Gover, math.DG/0207016), where the relationship of the normal standard tractor bundle to the ambient construction is described.