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,

\bold{A}

over it, we can define a family of p-forms: In one dimension, the Chern-Simons 1-form is given by

Tr \bold{A}

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

Tr \bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}

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

Tr +\frac{1}{10}\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}

where the curvature F is defined as

d\bold{A}+\bold{A}\wedge\bold{A}

The general Chern-Simons form

\omega_{2k-1}

is defined in such a way that

d\omega_{2k-1}=Tr(F^{k})

where the wedge product is used to define

F^k

. 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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