Chern-Simons

Given a manifold and a Lie algebra valued 1-form, over it, we have:

In one dimensions, 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.

See also Topological quantum field theory and Chiral anomaly.