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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.^{[1]}^{[2]} In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle.^{[3]}

The notion "Clifford module bundle" should not be confused with a Clifford bundle, which is a bundle of Clifford algebras.

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*).

**^**Berline, Getzler & Vergne 2004, pp. 113–115**^**Lawson & Michelsohn 1989, pp. 96–97**^**Berline, Getzler & Vergne 2004, Proposition 3.35.

- Berline, Nicole; Getzler, Ezra; Vergne, Michèle (2004).
*Heat kernels and Dirac operators*. Grundlehren Text Editions (Paperback ed.). Berlin, New York: Springer-Verlag. ISBN 3-540-20062-2. Zbl 1037.58015. - Lawson, H. Blaine; Michelsohn, Marie-Louise (1989).
*Spin Geometry*. Princeton Mathematical Series. Vol. 38. Princeton University Press. ISBN 978-0-691-08542-5. Zbl 0688.57001.