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 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

\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 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 spin structure on a

(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 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. The spinor bundle

\,

is defined to be the complex vector bundle

=\times_\Delta_n\,

associated to the spin structure

via the spin representation

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

where

()\,

denotes the group of

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