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

• Orthonormal frame bundle
• Spinor
• Spin manifold
• Spinor representation
• Spin geometry
• Clifford bundle
• Clifford module bundle

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

• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN 978-0-691-08542-5.
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN 978-0-8218-2055-1