Chern-Simons form

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In mathematics, the Chern-Simons forms (pronounced chen simons) 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.

[edit] 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 ω_{2k − 1} is defined in such a way that dω_{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.

[edit] See also

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