A characterization of Dirac morphisms (Q842488)

In analogy with the notion of harmonic morphism, the authors define the notion of Dirac morphism, by linking the Dirac operators on the total space and on the base manifold of a horizontally conformal submersion. First the pull-back of a local spinor field on the codomain of a Riemannian submersion between spinor manifolds is defined. The construction needs the target to have even dimension. When the submersion has fibres of dimension at least two, the definition depends on the choice of a fixed non-zero section of the vertical bundle (endowed with the induced spin structure). The pull-back of a local spinor field on the codomain of a horizontally conformal submersion between spinor manifolds is defined by using the associated Riemannian submersion. The authors then define Dirac morphisms, as horizontally conformal submersions which pull back local harmonic spinor fields to local harmonic spinor fields. Though the definition of the pull-back of a spinor depends on the choice (if any) of a non-zero section, that of the Dirac morphism is independent of this choice. The goal of the paper is the characterization of Dirac morphisms. According to the definition of the pull-back of a local spinor, the case of Dirac morphisms with fibres of dimension at least two and that of one-dimensional fibred Dirac morphisms are treated separately. It is shown that a horizontally conformal submersion between spin manifolds is a Dirac morphism if and only if its horizontal distribution is integrable and a certain relation between the mean curvature of its fibres and the dilation factor holds. Then, examples of Dirac morphisms from 3- to 2-dimensional and from 4- to 2-dimensional Euclidean spaces are presented. The paper concludes with an example based on projectable spinors.