Conformally invariant powers of the Laplacian, \(Q\)-curvature, and tractor calculus.

*(English)*Zbl 1022.58014Summary: We describe an elementary algorithm for expressing, as explicit formulae in tractor calculus, the conformally invariant GJMS operators due to C.R. Graham et al. 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 [A. Čap and A. R. Gover, Ann. Global Anal. Geom. 24, 231-259 (2003; Zbl 1039.53021], where the relationship of the normal standard tractor bundle to the ambient construction is described.

##### MSC:

58J50 | Spectral problems; spectral geometry; scattering theory on manifolds |

53A30 | Conformal differential geometry (MSC2010) |

53C27 | Spin and Spin\({}^c\) geometry |

81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |

81T40 | Two-dimensional field theories, conformal field theories, etc. in quantum mechanics |