differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

, 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 mathemati ...

on an

n

-dimensional orientable

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

(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 pr ...

\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 equ ...

\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 equi ...

of its

structure group In mathematics, and particularly topology, a fiber bundle ( ''Commonwealth English'': fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...

(n)\,

on the space of

spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...

\Delta_n

. A section of the spinor bundle

\,

is called a spinor field.

Formal definition

Let

(,F_)

be a

on a

(M, g),\,

that is, an

equivariant In mathematics, equivariance is a form of symmetry for functions from one space with symmetry to another (such as symmetric spaces). A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, ...

lift of the oriented

orthonormal frame bundle In mathematics, a frame bundle is a principal fiber bundle F(E) associated with any vector bundle ''E''. The fiber of F(E) over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E_x''. The general linear group acts naturally on ...

\mathrm F_(M)\to M

with respect to the double covering

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

of the

special orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...

by the

spin group In mathematics the spin group, denoted Spin(''n''), page 15 is a Lie group whose underlying manifold is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathbb_2 \to \o ...

. 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 iden ...

unitary operator In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Non-trivial examples include rotations, reflections, and the Fourier operator. Unitary operators generalize unitar ...

s acting on a

Hilbert space In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The ...

.\,

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 the ca ...

of the group

(n).

pages 20 and 24

Formal definition

See also

Notes

Further reading