# Spinor bundle

In differential geometry, given a spin structure on a ${\displaystyle n}$-dimensional 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.

## Formal definition

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

${\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n}\,}$

## Notes

1. {{#invoke:citation/CS1|citation |CitationClass=citation }} page 53
2. {{#invoke:citation/CS1|citation |CitationClass=citation }} pages 20 and 24