Chern-Simons 3-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.
See also Topological quantum field theory and Chiral anomaly.
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