# nLab spin geometry

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

# Contents

## 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.

## References

The classical monograph on spin geometry is

Discussion with an eye towards mathematical physics:

Lecture notes:

Fundamentals of the relevant supergeometry are in

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

Last revised on September 1, 2021 at 16:00:36. See the history of this page for a list of all contributions to it.