spin geometry, string geometry, fivebrane geometry …
(see also Chern-Weil theory, parameterized homotopy theory)
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 |
$L =$ | $R(e) vol(e) +$ | $\langle F_\nabla \wedge \star_e F_\nabla\rangle +$ | $(\psi , D_{(e,\nabla)} \psi) vol(e) +$ | $\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$ |
The term “spinor” is due to Paul Ehrenfest.
Élie Cartan, Theory of Spinors, Dover, first edition 1966
H. Blaine Lawson, Marie-Louise Michelsohn, chapter II, section 3 Spin geometry, Princeton University Press (1989)
Spinors in classical field theory (fermions):
Pierre Deligne, Daniel Freed, §3.4 of Classical field theory (1999) (pdf)
this is a chapter in
P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)
Radovan Dermisek qft I-8 (pdf, pdf)
Discussion relating manifolds with spinor bundles to supergeometry includes