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
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:
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 CℓnR be the Clifford algebra generated by Rn with the Euclidean metric. The standard action of the orthogonal group O(n) on Rn induces a graded automorphism of CℓnR. The homomorphism
is determined by
where vi are all vectors in Rn. 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 CℓnR it follows that Cℓ(E) is a bundle of Z2-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.
Clifford bundle of a Riemannian manifold
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).
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
- Orthonormal frame bundle
- Spin manifold
- Spinor representation
- Spin geometry
- Spin structure
- Clifford module bundle
- There is an arbitrary choice of sign in the definition of a Clifford algebra. In general, one can take v2 = ±<v,v>. In differential geometry, it is common to use the (−) sign convention.
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- Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton Mathematical Series. 38. Princeton University Press. ISBN 978-0-691-08542-5. Zbl 0688.57001.