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. 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.Chern and Simons (1974)

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

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

The general Chern–Simons form $\omega_{2k-1}$ is defined in such a way that

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

where the wedge product is used to define Fk. The right-hand side of this equation is proportional to the k-th Chern character of the connection $\bold{A}$.

In general, the Chern–Simons p-form is defined for any odd p. See also gauge theory for the definitions. Its integral over a p-dimensional manifold is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.