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.

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 ${\displaystyle {\mathbf {W} }.\,}$ The spin representation ${\displaystyle \kappa }$ is a faithful and unitary representation of the group ${\displaystyle {\mathrm {Spin} }(n).}$²