Clifford module bundle

In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras. The canonical example is a spinor bundle. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle.
The notion "Clifford module bundle" should not be confused with a Clifford bundle, which is a bundle of Clifford algebras.
Spinor bundles.
Given an oriented Riemannian manifold "M" one can ask whether it is possible to construct a bundle of irreducible Clifford modules over "Cℓ"("T"*"M"). In fact, such a bundle can be constructed if and only if "M" is a spin manifold.
Let "M" be an "n"-dimensional spin manifold with spin structure "F"Spin("M") → "F"SO("M") on "M". Given any "Cℓ""n"R-module "V" one can construct the associated spinor bundle
where σ : Spin("n") → GL("V") is the representation of Spin("n") given by left multiplication on "S". Such a spinor bundle is said to be "real", "complex", "graded" or "ungraded" according to whether on not "V" has the corresponding property. Sections of "S"("M") are called spinors on "M".
Given a spinor bundle "S"("M") there is a natural bundle map
which is given by left multiplication on each fiber. The spinor bundle "S"("M") is therefore a bundle of Clifford modules over "Cℓ"("T"*"M").