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• Djay 3 Spin Bundle Pc

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${\displaystyle {\pi }_{\mathbf{S}}:\mathbf{S}\to M}$ associated to the corresponding principal bundle${\displaystyle {\pi }_{\mathbf{P}}:\mathbf{P}\to M}$ of spin frames over ${\displaystyle M}$ and the spin representation of its structure group${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)}$ on the space of spinors${\displaystyle {\mathrm{\Delta }}_n.}$.

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

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Formal definition[edit]

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 ${\displaystyle \rho :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{S}\mathrm{O}(n)}$ of the special orthogonal group by the spin group.

The spinor bundle${\displaystyle \mathbf{S}}$ is defined ^[1] to be the complex vector bundle

${\displaystyle \mathbf{S}=\mathbf{P}{\times }_{\kappa }{\mathrm{\Delta }}_n}$

associated to the spin structure${\displaystyle \mathbf{P}}$ via the spin representation${\displaystyle \kappa :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{U}({\mathrm{\Delta }}_n),}$ 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 ${\displaystyle \mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)}$.^[2]

See also[edit]

Notes[edit]

1. ^Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1 page 53
2. ^Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1 pages 20 and 24

Further reading[edit]

• Lawson, H. Blaine; Michelsohn, Marie-Louise (1989). Spin Geometry. Princeton University Press. ISBN978-0-691-08542-5.
• Friedrich, Thomas (2000), Dirac Operators in Riemannian Geometry, American Mathematical Society, ISBN978-0-8218-2055-1

Djay 3 Spin Bundle Pc