Template:Short description In differential geometry, given a spin structure on an <math>n</math>-dimensional orientable Riemannian manifold <math>(M, g),\,</math> one defines the spinor bundle to be the complex vector bundle <math>\pi_{\mathbf S}\colon{\mathbf S}\to M\,</math> associated to the corresponding principal bundle <math>\pi_{\mathbf P}\colon{\mathbf P}\to M\,</math> of spin frames over <math>M</math> and the spin representation of its structure group <math>{\mathrm {Spin}}(n)\,</math> on the space of spinors <math>\Delta_n</math>.
A section of the spinor bundle <math>{\mathbf S}\,</math> is called a spinor field.
Formal definitionEdit
Let <math>({\mathbf P},F_{\mathbf P})</math> be a spin structure on a Riemannian manifold <math>(M, g),\,</math>that is, an equivariant lift of the oriented orthonormal frame bundle <math>\mathrm F_{SO}(M)\to M</math> with respect to the double covering <math>\rho\colon {\mathrm {Spin}}(n)\to {\mathrm {SO}}(n)</math> of the special orthogonal group by the spin group.
The spinor bundle <math>{\mathbf S}\,</math> is defined <ref>Template:Citation page 53 </ref> to be the complex vector bundle <math display=block>{\mathbf S}={\mathbf P}\times_{\kappa}\Delta_n\,</math> associated to the spin structure <math>{\mathbf P}</math> via the spin representation <math>\kappa\colon {\mathrm {Spin}}(n)\to {\mathrm U}(\Delta_n),\,</math> where <math>{\mathrm U}({\mathbf W})\,</math> denotes the group of unitary operators acting on a Hilbert space <math>{\mathbf W}.\,</math> The spin representation <math>\kappa</math> is a faithful and unitary representation of the group <math>{\mathrm {Spin}}(n).</math><ref>Template:Citation pages 20 and 24</ref>
See alsoEdit
- Clifford bundle
- Clifford module bundle
- Orthonormal frame bundle
- Spin geometry
- Spinor
- Spinor representation