Spinor bundle

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