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 associated to the corresponding principal bundle of spin frames over M and the spin representation of its structure group on the space of spinors \Delta_n. A section of the spinor bundle is called a spinor field.

Formal definition

Let be a spin structure on a Riemannian manifold (M, g),,that is, an equivariant lift of the oriented orthonormal frame bundle with respect to the double covering of the special orthogonal group by the spin group. The spinor bundle is defined to be the complex vector bundle associated to the spin structure {\mathbf P} via the spin representation where denotes the group of unitary operators acting on a Hilbert space The spin representation \kappa is a faithful and unitary representation of the group