Spinor bundle

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

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

Formal definition

Let ${\displaystyle ({\mathbf {P} },F_{\mathbf {P} })}$  be a spin structure on a Riemannian manifold ${\displaystyle (M,g),\,}$ that is, an equivariant lift of the oriented orthonormal frame bundle ${\displaystyle \mathrm {F} _{SO}(M)\to M}$  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 ${\displaystyle {\mathbf {S} }\,}$  is defined ^[1] to be the complex vector bundle

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

 

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

See also

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

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

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