Dirac bundle and spinor bundle What is the difference between Dirac bundle and spinor bundle?Morever, every spinor bundle is Dirac bundle, is it true?
 A: A Dirac bundle over a manifold $(M, g_{M})$ is a quadruple $(S, \langle \cdot, \cdot \rangle, \nabla, c)$, satisfying the conditions:


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*$V \longrightarrow M$ is a complex/real vector bundle over M endowed with a Hermitian/Riemannian metric $\langle \cdot, \cdot \rangle$

*The connection $\nabla: C^{\infty}(M, S) \longrightarrow C^{\infty}(M, T^{\ast}M \otimes S)$ is compatible with the Hermitian/Riemannian structure

*$S$ is a bundle of Clifford modules. Namely, $c: Cl(M,g_{M}) \longrightarrow Hom(S)$ is a real bundle homomorphism between the bundle of Clifford algebras and real bundle of complex homomorphisms of $S$ and the Clifford multiplication $c$ by tangent vectors is fiberwise skew-adjoint with respect to the Hermitian (Riemannian) structure. Additionally, we require that $\nabla c = 0$; i.e $c$ is covariantly constant with respect to $\nabla$.
Assuming that $M$ admits a Spin-structure, one can see that the Spinor bundle is a Dirac bundle. But in general, a Dirac bundle need not be a Spinor bundle. 
Example: The Hodge-De Rham operator $d + d^{\ast}: \Omega^{\bullet}(M) \longrightarrow \Omega^{\bullet}(M)$ is Dirac operator, but not spin Dirac operator. 
Refer to Nicolaescu's book Lectures on Geometry of Manifolds, Chapter 11.
