Spinor bundle

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In differential geometry, given a spin structure on a -dimensional 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 definition[edit]

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

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 also[edit]

Notes[edit]

  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 reading[edit]