Clifford module bundle
In
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
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
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
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
- ^ Berline, Getzler & Vergne 2004, pp. 113–115
- ^ Lawson & Michelsohn 1989, pp. 96–97
- ^ Berline, Getzler & Vergne 2004, Proposition 3.35.
References
- Zbl 1037.58015.
- Zbl 0688.57001.