spin geometry, string geometry, fivebrane geometry …
rotation groups in low dimensions:
see also
vector bundle, 2-vector bundle, (∞,1)-vector bundle
real, complex/holomorphic, quaternionic
A spinor bundle on a smooth manifold with spin structure is a -associated bundle associated to the spin group-principal bundle lifting the tangent bundle, for 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 | principal connection | spinor | scalar field |
Lagrangian: | scalar curvature density | field strength squared | Dirac operator component density | field strength squared + potential density |
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)
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
Last revised on April 17, 2023 at 11:05:02. See the history of this page for a list of all contributions to it.