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. The theory is named for Shiing-Shen Chern and James Harris Simons, co-authors of a 1974 paper entitled "Characteristic Forms and Geometric Invariants," from which the theory arose.

Definition

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 :.

The general Chern-Simons form $omega\_\{2k-1\}$ is defined in such a way that :$domega\_\{2k-1\}=\{\; m\; Tr\}\; left(\; F^\{k\}\; ight)$,

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.

ee also

*Chern-Simons theory
*Chiral anomaly
*Topological quantum field theory

References

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