In differential geometry, given a

spin structure In differential geometry, a spin structure on an orientable Riemannian manifold allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. Spin structures have wide applications to mathematical ...

on an

n

-dimensional orientable Riemannian manifold

(M, g),\,

one defines the spinor bundle to be the

complex vector bundle In mathematics, a complex vector bundle is a vector bundle whose fibers are complex vector spaces. Any complex vector bundle can be viewed as a real vector bundle through the restriction of scalars. Conversely, any real vector bundle ''E'' can be ...

\pi_\colon\to M\,

associated to the corresponding

principal bundle In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...

\pi_\colon\to M\,

of spin frames over

M

and the

spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equ ...

of its structure group

(n)\,

on the space of

spinor In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...

\Delta_n.

. A section of the spinor bundle

\,

is called a spinor field.

Formal definition

Let

(,F_)

be a

on a Riemannian manifold

(M, g),\,

that is, an equivariant lift of the oriented orthonormal frame bundle

\mathrm F_(M)\to M

with respect to the double covering

\rho\colon (n)\to (n)

of the special orthogonal group by the spin group. The spinor bundle

\,

is defined to be the

=\times_\Delta_n\,

associated to the

via the

\kappa\colon (n)\to (\Delta_n),\,

where

()\,

denotes the

group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...

unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the co ...

s acting on a Hilbert space

.\,

It is worth noting that the spin representation

\kappa

is a faithful and

unitary representation In mathematics, a unitary representation of a group ''G'' is a linear representation π of ''G'' on a complex Hilbert space ''V'' such that π(''g'') is a unitary operator for every ''g'' ∈ ''G''. The general theory is well-developed in case ''G ...

of the group

(n).

pages 20 and 24

Formal definition

See also

Notes

Further reading