In differential geometry, given a spin structure on an n {\displaystyle n} -dimensional orientable Riemannian manifold, {\displaystyle,\,} one defines the spinor bundle to be the complex vector bundle π S : S → M {\displaystyle \pi _{\mathbf {S} }\colon {\mathbf {S} }\to M\,} associated to the corresponding principal bundle π P : P → M {\displaystyle \pi _{\mathbf {P} }\colon {\mathbf {P} }\to M\,} of spin frames over M {\displaystyle M} and the spin representation of its structure group S p i n {\displaystyle {\mathrm {Spin} }\,} on the space of spinors Δ n {\displaystyle \Delta _{n}}. Wikipedia