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. See Chern and Simons (1974)

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

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 ${\displaystyle \omega _{2k-1}}$ is defined in such a way that

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