Chern-Simons form
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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.
Given a manifold and a Lie algebra valued 1-form, <math>\bold{A}<math> over it, we can define a family of p-forms:
In one dimension, the Chern-Simons 1-form is given by
- <math>Tr[\bold{A}]<math>.
In three dimensions, the Chern-Simons 3-form is given by
- <math>Tr[\bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}]<math>.
In five dimensions, the Chern-Simons 5-form is given by
- <math>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}]<math>
where the curvature F is defined as
- <math>d\bold{A}+\bold{A}\wedge\bold{A}<math>.
The general Chern-Simons form <math>\omega_{2k-1}<math> is defined in such a way that <math>d\omega_{2k-1}=Tr(F^{k})<math> where the wedge product is used to define <math>F^k<math>.
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.