Geometric structure
In differential geometry, given a spin structure on an
-dimensional orientable Riemannian manifold
one defines the spinor bundle to be the complex vector bundle
associated to the corresponding principal bundle
of spin frames over
and the spin representation of its structure group
on the space of spinors
.
A section of the spinor bundle
is called a spinor field.
Formal definition
Let
be a spin structure on a Riemannian manifold
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 [1] to be the complex vector bundle
![{\displaystyle {\mathbf {S} }={\mathbf {P} }\times _{\kappa }\Delta _{n}\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78d56e06aa7dd2ee40f3fb6e77d1d614437a95b8)
associated to the spin structure
via the spin representation
where
denotes the group of unitary operators acting on a Hilbert space
It is worth noting that the spin representation
is a faithful and unitary representation of the group
[2]