spin geometry, string geometry, fivebrane geometry …
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 $\rho : \mathbf{B} Spin \to$ Vect a spin representation.
A section of a spinor bundle is called a spinor (a fermion field)
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 |
$L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |
H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989)