Spinor bundle

Short description: Geometric structure

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

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

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

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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