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# 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 FSpin(M) → FSO(M) on M. Given any CℓnR-module V one can construct the associated spinor bundle

${\displaystyle S(M)=F_{\mathrm {Spin} }(M)\times _{\sigma }V\,}$

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

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

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. ^ 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.