Clifford module bundle

[1][2] In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle.

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 FSpin(M) → FSO(M) on M. Given any CℓnR-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.