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 $\rho : \mathbf{B} Spin \to$ Vect a spin representation.

A section of a spinor bundle is called a spinor (a fermion field)

Related concepts

• Clifford bundle

• stringor bundle

• fiber bundles in physics

• rotor

• spinor, spin representation

• Dirac 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.

• fermion

  • electron, quark

  • gravitino, gluon

• Velo-Zwanziger problem

• symplectic spinor

• twistor

• stringor bundle

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
$L =$	$R(e) vol(e) +$	$\langle F_\nabla \wedge \star_e F_\nabla\rangle +$	$(\psi , D_{(e,\nabla)} \psi) vol(e) +$	$\nabla \bar H \wedge \star_e \nabla H + \left(\lambda {\vert H\vert}^4 - \mu^2 {\vert H\vert}^2 \right) vol(e)$

References

The term “spinor” is due to Paul Ehrenfest.

• Élie Cartan, Theory of Spinors, Dover, first edition 1966

• Roger Penrose, Wolfgang Rindler, Spinors and space time, in 2 vols. Cambridge Univ. Press 1984/1988.

• H. Blaine Lawson, Marie-Louise Michelsohn, chapter II, section 3 Spin geometry, Princeton University Press (1989)

• Daniel Freed, Five lectures on supersymmetry

• pdf

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 Dermisekqft I-8 (pdf, pdf)

Discussion relating manifolds with spinor bundles to supergeometry includes

• Dmitri Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Uwe Semmelmann, Killing spinors are Killing vector fields in Riemannian Supergeometry (arXiv:dg-ga/9704002)

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