In differential geometry, given a spin structure on a
n
-dimensional Riemannian manifold
(
M
,
g
)
,
one defines the spinor bundle to be the complex vector bundle
π
S
:
S
→
M
associated to the corresponding principal bundle
π
P
:
P
→
M
of spin frames over
M
and the spin representation of its structure group
S
p
i
n
(
n
)
on the space of spinors
Δ
n
.
.
A section of the spinor bundle
S
is called a spinor field.
Let
(
P
,
F
P
)
be a spin structure on a Riemannian manifold
(
M
,
g
)
,
that is, an equivariant lift of the oriented orthonormal frame bundle
F
S
O
(
M
)
→
M
with respect to the double covering
ρ
:
S
p
i
n
(
n
)
→
S
O
(
n
)
of the special orthogonal group by the spin group.
The spinor bundle
S
is defined to be the complex vector bundle
S
=
P
×
κ
Δ
n
associated to the spin structure
P
via the spin representation
κ
:
S
p
i
n
(
n
)
→
U
(
Δ
n
)
,
where
U
(
W
)
denotes the group of unitary operators acting on a Hilbert space
W
.
It is worth noting that the spin representation
κ
is a faithful and unitary representation of the group
S
p
i
n
(
n
)
.
