nLab spinor bundle



Spin geometry





A spinor bundle on a smooth manifold with spin structure is a ρ\rho-associated bundle associated to the spin group-principal bundle lifting the tangent bundle, for ρ:BSpin\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

gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field eeprincipal connection \nablaspinor ψ\psiscalar field HH
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield strength squared + potential density
L=L = R(e)vol(e)+R(e) vol(e) + F eF +\langle F_\nabla \wedge \star_e F_\nabla\rangle + (ψ,D (e,)ψ)vol(e)+ (\psi , D_{(e,\nabla)} \psi) vol(e) + H¯ eH+(λ|H| 4μ 2|H| 2)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The term “spinor” is due to Paul Ehrenfest, see the historical references at spin.

Spinors in classical field theory (fermions):

Discussion relating manifolds with spinor bundles to supergeometry includes


  • B. L.Van derWaerden, Exclusion principle and spin, in Theoretical Physics in the Twentieth Century: A Memorial Volume to Wolfgang Pauli, ed. M. Fierz and V. F. Weisskopf, New York: Interscience, 1960

Last revised on October 22, 2023 at 09:27:38. See the history of this page for a list of all contributions to it.