Clifford bundle

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.

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

$v^2 = -\langle v,v\rangle$

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:

$C\ell(E) = \coprod_{x\in M} C\ell(E_x,g_x)$

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

$\rho : \mathrm O(n) \to \mathrm{Aut}(C\ell_n\mathbb R)$

is determined by

$\rho(A)(v_1v_2\cdots v_k) = (Av_1)(Av_2)\cdots(Av_k)$

where v_i are all vectors in Rⁿ. The Clifford bundle of E is then given by

$C\ell(E) = F(E) \times_\rho C\ell_n\mathbb R$

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:

$C\ell(E) = C\ell^0(E) \oplus C\ell^1(E).$

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).

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

$C\ell(T^*M) \cong \Lambda(T^*M).$

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

$\begin{align} C\ell^0(T^*M) &= \Lambda^{\mathrm{even}}(T^*M)\\ C\ell^1(T^*M) &= \Lambda^{\mathrm{odd}}(T^*M). \end{align}$

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(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_nR-module V one can construct the associated spinor bundle

$S(M) = F_{\mathrm{Spin}}(M) \times_\sigma V\,$

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

$C\ell(T^*M) \otimes S(M) \to S(M)$

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).

Notes

1. ^ 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.

References

• Lawson, H. Blaine, Jr.; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton, N.J: Princeton University Press. ISBN 0-691-08542-0.