Spinor bundle

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In differential geometry, given a spin structure on a n-dimensional Riemannian manifold (M,g),\, one defines the spinor bundle to be the complex vector bundle \pi _{{{\mathbf S}}}\colon {{\mathbf S}}\to M\, associated to the corresponding principal bundle \pi _{{{\mathbf P}}}\colon {{\mathbf P}}\to M\, of spin frames over M and the spin representation of its structure group {{\mathrm {Spin}}}(n)\, on the space of spinors \Delta _{n}.\,.

A section of the spinor bundle {{\mathbf S}}\, is called a spinor field.

Formal definition

Let ({{\mathbf P}},F_{{{\mathbf P}}}) be a spin structure on a Riemannian manifold (M,g),\, that is, an equivariant lift of the oriented orthonormal frame bundle {\mathrm F}_{{SO}}(M)\to M with respect to the double covering \rho \colon {{\mathrm {Spin}}}(n)\to {{\mathrm {SO}}}(n).\,

The spinor bundle {{\mathbf S}}\, is defined [1] to be the complex vector bundle

{{\mathbf S}}={{\mathbf P}}\times _{{\kappa }}\Delta _{n}\,

associated to the spin structure {{\mathbf P}} via the spin representation \kappa \colon {{\mathrm {Spin}}}(n)\to {{\mathrm U}}(\Delta _{n}),\, where {{\mathrm U}}({{\mathbf W}})\, denotes the group of unitary operators acting on a Hilbert space {{\mathbf W}}.\, It is worth noting that the spin representation \kappa is a faithful and unitary representation of the group {{\mathrm {Spin}}}(n).[2]

See also

Notes

  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


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