Clifford Bundle - General Construction

General Construction

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. 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_nR 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 Cℓ_nR. The homomorphism

is determined by

where v_i 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 Cℓ_nR 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.