Spin^c structure

In spin geometry, a spin^c structure (or complex spin structure) is a generalization of a spin structure. In mathematics, these are used to describe spinor bundles and spinors, which in physics are used to describe spin, an intrinsic angular momentum of particles after which they have been named. Since spin^c structures also exist under weakened conditions, which might not allow spin structures, they provide a suitable alternative for such situations. Orientable manifolds with a spin^c structure are called spin^c manifolds. C stands for the complex numbers, which are denoted ${\displaystyle \mathbb {C} }$ and appear in the definition of the underlying spin^c group.

In four dimensions, a spin^c structure defines two complex plane bundles, which can be used to describe negative and positive chirality of spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifolds.