Spinor bundle

Summary

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding principal bundle of spin frames over and the spin representation of its structure group on the space of spinors .

A section of the spinor bundle is called a spinor field.

Formal definitionEdit

Let   be a spin structure on a Riemannian manifold  that is, an equivariant lift of the oriented orthonormal frame bundle   with respect to the double covering   of the special orthogonal group by the spin group.

The spinor bundle   is defined [1] to be the complex vector bundle

 
associated to the spin structure   via the spin representation   where   denotes the group of unitary operators acting on a Hilbert space   It is worth noting that the spin representation   is a faithful and unitary representation of the group  [2]

See alsoEdit

NotesEdit

  1. ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 page 53
  2. ^ Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1 pages 20 and 24

Further readingEdit

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