Spin 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 ${\displaystyle \Delta _{n}.\,}$.

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