Chern-Simons form

Given a manifold and a Lie algebra valued 1-form, over it, we can define a family of p-forms:

In one dimension, the Chern-Simons 1-form is given by

In three dimensions, the Chern-Simons 3-form is given by

In five dimensions, the Chern-Simons 5-form is given by

where the curvature F is defined as

See gauge theory for more details.

In general, the Chern-Simons p-form is defined for any odd p. See gauge theory for the definitions. Its integral over a p dimensional manifold is a homotopy invariant. This value is called the Chern number.