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In mathematics, a **Clifford bundle** is an algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. There is a natural Clifford bundle associated to any (pseudo) Riemannian manifold *M* which is called the Clifford bundle of *M*.

Let *V* be a (real or complex) vector space together with a symmetric bilinear form <·,·>. The Clifford algebra *Cℓ*(*V*) is a natural (unital associative) algebra generated by *V* subject only to the relation

for all *v* in *V*.^{[1]} One can construct *Cℓ*(*V*) as a quotient of the tensor algebra of *V* by the ideal generated by the above relation.

Like other tensor operations, this construction can be carried out fiberwise on a smooth vector bundle. Let *E* be a smooth vector bundle over a smooth manifold *M*, and let *g* be a smooth symmetric bilinear form on *E*. The **Clifford bundle** of *E* is the fiber bundle whose fibers are the Clifford algebras generated by the fibers of *E*:

The topology of *Cℓ*(*E*) is determined by that of *E* via an associated bundle construction.

One is most often interested in the case where *g* is positive-definite or at least nondegenerate; that is, when (*E*, *g*) is a Riemannian or pseudo-Riemannian vector bundle. For concreteness, suppose that (*E*, *g*) is a Riemannian vector bundle. The Clifford bundle of *E* can be constructed as follows. Let *Cℓ*_{n}**R** be the Clifford algebra generated by **R**^{n} with the Euclidean metric. The standard action of the orthogonal group O(*n*) on **R**^{n} induces a graded automorphism of *Cℓ*_{n}**R**. The homomorphism

is determined by

where *v*_{i} are all vectors in **R**^{n}. The Clifford bundle of *E* is then given by

where *F*(*E*) is the orthonormal frame bundle of *E*. It is clear from this construction that the structure group of *Cℓ*(*E*) is O(*n*). Since O(*n*) acts by graded automorphisms on *Cℓ*_{n}**R** it follows that *Cℓ*(*E*) is a bundle of **Z**_{2}-graded algebras over *M*. The Clifford bundle *Cℓ*(*E*) can then be decomposed into even and odd subbundles:

If the vector bundle *E* is orientable then one can reduce the structure group of *Cℓ*(*E*) from O(*n*) to SO(*n*) in the natural manner.

If *M* is a Riemannian manifold with metric *g*, then the Clifford bundle of *M* is the Clifford bundle generated by the tangent bundle *TM*. One can also build a Clifford bundle out of the cotangent bundle *T***M*. The metric induces a natural isomorphism *TM* = *T***M* and therefore an isomorphism *Cℓ*(*TM*) = *Cℓ*(*T***M*).

There is a natural vector bundle isomorphism between the Clifford bundle of *M* and the exterior bundle of *M*:

This is an isomorphism of vector bundles *not* algebra bundles. The isomorphism is induced from the corresponding isomorphism on each fiber. In this way one can think of sections of the Clifford bundle as differential forms on *M* equipped with Clifford multiplication rather than the wedge product (which is independent of the metric).

The above isomorphism respects the grading in the sense that

**^**There is an arbitrary choice of sign in the definition of a Clifford algebra. In general, one can take*v*^{2}= ±<*v*,*v*>. In differential geometry, it is common to use the (−) sign convention.

- 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.