# 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. See Chern and Simons (1974)

## Definition

Given a manifold and a Lie algebra valued 1-form, ${\displaystyle {\mathbf {A}}}$ over it, we can define a family of p-forms:

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

${\displaystyle {\rm {Tr}}[{\mathbf {A} }].}$

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

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

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

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

where the curvature F is defined as

${\displaystyle {\mathbf {F} }=d{\mathbf {A} }+{\mathbf {A} }\wedge {\mathbf {A} }.}$

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

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

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 ${\displaystyle {\mathbf {A}}}$.

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.