Clifford module bundle

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.

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).

See also

• Orthonormal frame bundle
• Spin representation
• Spin geometry

Notes

1. Berline et al (2004) pp.113-115
2. Lawson & Michelsohn (1989) pp.96-97
3. Berline et al 2004, Proposition 3.35.

References

• 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 38. Princeton University Press. ISBN 978-0-691-08542-5. Zbl 0688.57001.