Connections and the Dirac operator on spinor bundles

Abstract

There are two approaches to spinor fields on a (pseudo-) Riemannian manifold ( M , g ) : the bundle of spinors is either defined as a bundle associated with the principal bundle of ‘spin frames’ or as a complex bundle Σ → M with a homomorphism τ : C ℓ ( g ) → End Σ of bundles of algebras over M such that, for every x ∈ M , the restriction of τ to the fiber over x is equivalent to a spinor representation of a suitable Clifford algebra. By Hermitian and complex conjugation one obtains the homomorphisms τ † : C ℓ ( g ) → End Σ ̄ ∗ and τ ̄ : C ℓ ( g ) → End Σ ̄ . These data define the bundles a ( τ ) and c ( τ ) of intertwiners of τ with τ † and τ ̄ , respectively. It is shown that, given sections of a ( τ ) → M and of c ( τ ) → M , any metric linear connection on ( M , g ) defines a unique connection on the spinor bundle Σ → M relative to which these sections are covariantly constant. The connection defines a Dirac operator acting on sections of Σ . As an example, the trivial spinor bundle on hypersurfaces in R m and the corresponding Dirac operator are described in detail.