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.