In mathematics, the Chern–Simons forms are certain secondary characteristic classes. 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.

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

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