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In mathematics, the Chern-Simons forms are certain secondary characteristic classes. They have been found to be of interest in gauge theory, and they (especially the 3-form) define the action of Chern-Simons theory.
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
The general Chern-Simons form ω2k - 1 is defined in such a way that dω2k - 1 = Tr(Fk) where the wedge product is used to define Fk.
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.
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