nLab spin geometry

Idea

The differential geometry of manifolds with spin structure is called spin geometry. It studies spin group-principal bundles, spin representations, the associated spinor bundles, and the Dirac operators acting on spaces of sections of these bundles, hence also their index theory and generally K-theory.

In physics

The relevance of spin geometry in physics rests on the fact that in quantum mechanics and quantum field theory in general and in the standard model of particle physics in particular, fermions such as the electron are mathematically modeled as sections of spin-bundles. The very term spin originates in the fact that the quanta of these fields behave to some extent as if they had an intrinsic angular momentum, as if they were spinning about an axis as a classical top.

Spin geometry also plays a central role in supersymmetric quantum field theory such as supergravity.

Related concepts

• spin group. spin representation, spinor, spin

• Fierz identity

• geometric algebra

References

A classical monograph:

• H. Blaine Lawson, Marie-Louise Michelsohn, Spin geometry, Princeton University Press (1989) [ISBN 9780691085425]

Discussion with an eye towards mathematical physics:

• Gerd Rudolph, Matthias Schmidt, Chapter 5 of: Differential Geometry and Mathematical Physics Part II. Fibre Bundles, Topology and Gauge Fields, Springer (2017) [doi:10.1007/978-94-024-0959-8]

Lecture notes:

• José Figueroa-O'Farrill, PG course on Spin Geometry, 2010

Fundamentals of the relevant supergeometry are in

• Pierre Deligne, Daniel Freed, Supersolutions (arXiv:hep-th/9901094)

  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)

Discussion in the context of conformal geometry is in

• Pierre Anglès, section 2.2.1.2 of Conformal Groups in Geometry and Spin Structures, Progress in Mathematical Physics 2008