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 ${\displaystyle \rho \colon {\mathrm {Spin} }(n)\to {\mathrm {SO} }(n)}$ of the special orthogonal group by the spin group.

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
• 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