Clifford module bundle explained

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. In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle.

The notion "Clifford module bundle" should not be confused with a Clifford bundle, which is a bundle of Clifford algebras.

Spinor bundles

See main article: Spinor bundle.

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 _{Spin} (M) x _{\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)} ⊗ 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

References

• Book: Berline . Nicole . Nicole Berline . Getzler . Ezra . Ezra Getzler . Vergne . Michèle . Michèle Vergne . Heat kernels and Dirac operators . Paperback . Grundlehren Text Editions . Berlin, New York . . 3-540-20062-2 . 2004 . 1037.58015 .
• Book: Lawson . H. Blaine . H. Blaine Lawson . Michelsohn . Marie-Louise . Marie-Louise Michelsohn . Spin Geometry . . 978-0-691-08542-5 . 1989 . Princeton Mathematical Series . 38 . 0688.57001.