Connections and the Dirac operator on spinor bundles

There are two approaches to spinor fields on a (pseudo-) Riemannian manifold (M,g)(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→MΣ→M with a homomorphism τ:Cℓ(g)→EndΣτ:Cℓ(g)→EndΣ of bundles of algebras over MM such that, for every x∈Mx∈M, the restriction of ττ to the fiber over xx is equivalent to a spinor representation of a suitable Clifford algebra. By Hermitian and complex conjugation one obtains the homomorphisms τ†:Cℓ(g)→EndΣ̄∗ and τ̄:Cℓ(g)→EndΣ̄. These data define the bundles a(τ)a(τ) and c(τ)c(τ) of intertwiners of ττ with τ†τ† and τ̄, respectively. It is shown that, given sections of a(τ)→Ma(τ)→M and of c(τ)→Mc(τ)→M, any metric linear connection on (M,g)(M,g) defines a unique connection on the spinor bundle Σ→MΣ→M relative to which these sections are covariantly constant. The connection defines a Dirac operator acting on sections of ΣΣ. As an example, the trivial spinor bundle on hypersurfaces in RmRm and the corresponding Dirac operator are described in detail.