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, 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 is defined in such a way that
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 .
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.