Chern-SimonsGiven 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.