Spinor bundle

In differential geometry, given a spin structure on an $n$-dimensional orientable Riemannian manifold $(M,g),$ one defines the spinor bundle to be the complex vector bundle ${\pi }_{𝐒}:𝐒\to M$ associated to the corresponding principal bundle ${\pi }_{𝐏}:𝐏\to M$ 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 ${\Delta }_n$.

A section of the spinor bundle $𝐒$ is called a spinor field.

Formal definition

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

The spinor bundle $𝐒$ is defined to be the complex vector bundle $$𝐒=𝐏{\times }_{\kappa }{\Delta }_n$$ associated to the spin structure $𝐏$ via the spin representation $\kappa :\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n)\to \mathrm{U}({\Delta }_n),$ where $\mathrm{U}(𝐖)$ denotes the group of unitary operators acting on a Hilbert space $𝐖.$ The spin representation $\kappa$ is a faithful and unitary representation of the group $\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}(n).$

See also

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

Further reading

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