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. 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 → FSO on M. Given any CℓnR-module V one can construct the associated spinor bundle
where σ : Spin → GL is the representation of Spin 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 are called spinors on M.
Given a spinor bundle S there is a natural bundle map
which is given by left multiplication on each fiber. The spinor bundle S is therefore a bundle of Clifford modules over Cℓ.