Dirac bundle

Summary

A Dirac bundle over a Riemannian manifold is a bundle of Clifford modules with a metric and a connection that are compatible with Clifford multiplication. These data are exactly what is needed to define a Dirac operator that is formally self-adjoint and satisfies a general Bochner–Weitzenböck formula. Spinor bundles, the bundle of differential forms and the Clifford bundle itself are Dirac bundles. The notion gives one framework for the de Rham, signature, Dolbeault and spin Dirac operators used in index theory.