Spinor bundle

In differential geometry, given a spin structure on an formula_1-dimensional orientable Riemannian manifold formula_2 one defines the spinor bundle to be the complex vector bundle formula_3 associated to the corresponding principal bundle formula_4 of spin frames over formula_5 and the spin representation of its structure group formula_6 on the space of spinors formula_7. 
A section of the spinor bundle formula_8 is called a spinor field.
Formal definition.
Let formula_9 be a spin structure on a Riemannian manifold formula_2 that is, an equivariant lift of the oriented orthonormal frame bundle formula_11 with respect to the double covering formula_12 of the special orthogonal group by the spin group.
The spinor bundle formula_8 is defined to be the complex vector bundle
associated to the spin structure formula_15 via the spin representation formula_16 where formula_17 denotes the group of unitary operators acting on a Hilbert space formula_18 It is worth noting that the spin representation formula_19 is a faithful and unitary representation of the group formula_20.