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, $\bold{A}$ over it, we can define a family of p-forms:

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

${\rm Tr} [ \bold{A} ]$ .

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

${\rm Tr} \left[ \bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}\right]$ .

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

${\rm Tr} \left[ \bold{F}\wedge\bold{F}\wedge\bold{A}-\frac{1}{2}\bold{F}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} +\frac{1}{10}\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} \right]$

where the curvature F is defined as

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

The general Chern-Simons form $ω 2 k - 1$ is defined in such a way that

$d\omega_{2k-1}={\rm Tr} \left( F^{k} \right)$ ,

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.