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 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.
See also Topological quantum field theory and Chiral anomaly.