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The term “spinor” is due to Paul Ehrenfest, see the historical references at spin.
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 October 22, 2023 at 09:27:38. See the history of this page for a list of all contributions to it.