Robert Mills (physicist)

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

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 $\rho :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{S}\mathrm{O}(n).$

The spinor bundle $\mathbf{S}$ is defined to be the complex vector bundle

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

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

See also

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

Notes

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Further reading

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