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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.{{harvnb|Berline|Getzler|Vergne|2004|pp=113—115}}{{harvnb|Lawson|Michelsohn|1989|pp=96-97}} In fact, on a Spin manifold, every Clifford module is obtained by twisting the spinor bundle.{{harvnb|Berline|Getzler|Vergne|2004|loc=Proposition 3.35.}}

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

Spinor bundles

{{main|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 FSpin(M) → FSO(M) on M. Given any CℓnR-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

Notes

{{Reflist}}

References

  • {{cite book | last1=Berline | first1=Nicole | author1-link=Nicole Berline | last2=Getzler | first2=Ezra | author2-link=Ezra Getzler | last3=Vergne | first3=Michèle | author3-link=Michèle Vergne | title=Heat kernels and Dirac operators | edition=Paperback | series=Grundlehren Text Editions | location=Berlin, New York | publisher=Springer-Verlag | isbn=3-540-20062-2 | year=2004 | zbl=1037.58015 }}
  • {{cite book | last1=Lawson | first1=H. Blaine |author1-link=H. Blaine Lawson | last2=Michelsohn | first2=Marie-Louise | author2-link=Marie-Louise Michelsohn | title=Spin Geometry | publisher=Princeton University Press | isbn=978-0-691-08542-5 | year=1989 | series=Princeton Mathematical Series | volume=38 | zbl=0688.57001}}

Category:Riemannian geometry

Category:Structures on manifolds

Category:Clifford algebras

Category:Vector bundles

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