# Chern-Simons form

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.

Given a manifold and a Lie algebra valued 1-form, [itex]\bold{A}[itex] over it, we can define a family of p-forms:

In one dimension, the Chern-Simons 1-form is given by

[itex]Tr[\bold{A}][itex].

In three dimensions, the Chern-Simons 3-form is given by

[itex]Tr[\bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}][itex].

In five dimensions, the Chern-Simons 5-form is given by

[itex]Tr[\bold{F}\wedge\bold{F}\wedge\bold{A}-\frac{1}{2}\bold{F}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} +\frac{1}{10}\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}][itex]

where the curvature F is defined as

[itex]d\bold{A}+\bold{A}\wedge\bold{A}[itex].

The general Chern-Simons form [itex]\omega_{2k-1}[itex] is defined in such a way that [itex]d\omega_{2k-1}=Tr(F^{k})[itex] where the wedge product is used to define [itex]F^k[itex].

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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