Conformally covariant differential operators acting on spinor bundles and related conformal covariants

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Conformal powers of the Dirac operator on semi Riemannian spin manifolds are investigated. We give a new proof of the existence of conformal odd powers of the Dirac operator on semi Riemannian spin manifolds using the tractor machinery. We will also present a new family of conformally covariant linear differential operators on the standard spin tractor bundle. Furthermore, we generalize the existence proof of conformal power of the Dirac operator on Riemannian spin manifolds [GMP12] to semi Riemannian spin manifolds. Both proofs concering the existence of conformal odd powers of the Dirac operator are constructive, hence we also derive an explicit formula for a conformal thirdand fifth power of the Dirac operator. Due to explicit formulas, we show that the conformal thirdand fifth power of the Dirac operator is formally self-adjoint (anti self-adjoint), with respect to the L2−scalar product on the spinor bundle. Finally, we present a new structure of the conformal first-, thirdand fifth power of the Dirac operator: There exist linear differential operators on the spinor bundle of order less or equal one, such that the conformal first-, thirdand fifth power of the Dirac operator is a polynomial in these operators.