# nLab spinor bundle

Contents

## Spin geometry

spin geometry

Dynkin labelsp. orth. groupspin grouppin groupsemi-spin group
SO(2)Spin(2)Pin(2)
B1SO(3)Spin(3)Pin(3)
D2SO(4)Spin(4)Pin(4)
B2SO(5)Spin(5)Pin(5)
D3SO(6)Spin(6)
B3SO(7)Spin(7)
D4SO(8)Spin(8)SO(8)
B4SO(9)Spin(9)
D5SO(10)Spin(10)
B5SO(11)Spin(11)
D6SO(12)Spin(12)
$\vdots$$\vdots$
D8SO(16)Spin(16)SemiSpin(16)
$\vdots$$\vdots$
D16SO(32)Spin(32)SemiSpin(32)

string geometry

## Ninebrane geometry

#### Bundles

bundles

fiber bundles in physics

# Contents

## 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 $\rho : \mathbf{B} Spin \to$ 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
gravityelectroweak and strong nuclear forcefermionic matterscalar field
field content:vielbein field $e$principal connection $\nabla$spinor $\psi$scalar field $H$
Lagrangian:scalar curvature densityfield strength squaredDirac operator component densityfield 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)$

## References

The term “spinor” is due to Paul Ehrenfest.

Spinors in classical field theory (fermions):

Discussion relating manifolds with spinor bundles to supergeometry includes

## History

• 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 December 18, 2017 at 00:41:30. See the history of this page for a list of all contributions to it.