Conformally Invariant Powers of the Laplacian, QCurvature, and Tractor Calculus
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 LeviCivita connection and its curvature and a direct definition and formula for T. Branson's socalled Qcurvature (which integrates to a global conformal invariant) as well as generalisations of the operators and the Qcurvature. Among examples, the operators of order 4, 6 and 8 and the related Qcurvatures 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 [12], where the relationship of the normal standard tractor bundle to the ambient construction is described.
 Publication:

Communications in Mathematical Physics
 Pub Date:
 2003
 DOI:
 10.1007/s0022000207904
 arXiv:
 arXiv:mathph/0201030
 Bibcode:
 2003CMaPh.235..339G
 Keywords:

 Mathematical Physics;
 Mathematics  Differential Geometry;
 53B50 (Primary) 53A30;
 53A55;
 58J70 (Secondary)
 EPrint:
 42 pages. No figures. Record of changes: V1, 15 January 2002: Original posting. V2, 17 January 2002: Changing comment fields. Leaving abstract and text of article unchanged. V3, 1 February 2003: Minor changes and typographical corrections throughout article