nLab
spinor bundle

Definition

A spinor bundle on a smooth manifold with spin structure is a $\rho$-associated bundle associated to the spin group-principal bundle lifting the tangent bundle, for $\rho :B\mathrm{Spin}\to$ Vect a representation of the spin group.

A section of a spinor bundle is called a spinor.

Related concepts

• spinor, spin representation

A Dirac operator acts on sections of a spinor bundle.

In physics sections of spinor bundles model matter particles: fermion. See spinors in Yang-Mills theory.

standard model of particle physics and cosmology

theory:	Einstein-	Yang-Mills-	Dirac-	Higgs
	gravity	electroweak and strong nuclear force	fermionic matter	scalar field
field content:	vielbein field $e$	principal connection $\nabla$	spinor $\psi$	scalar field $H$
Lagrangian:	scalar curvature density	field strength squared	Dirac operator component density	field strength squared + potential density
$L=$	$R(e)\mathrm{vol}(e)+$	$⟨F_{\nabla }\wedge {\star }_e{F}_{\nabla }⟩+$	$(\psi ,D_{(e,\nabla )}\psi )\mathrm{vol}(e)+$	$\nabla \overline{H}\wedge {\star }_e\nabla H+(\lambda {\mid H\mid }^4-{\mu }^2{\mid H\mid }^2)\mathrm{vol}(e)$