# 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)

## Definition

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 *F ^{k}*. 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.

## See also

## References

- Chern, S.-S.; Simons, J. (1974). "Characteristic forms and geometric invariants".
*Annals of Mathematics*. Second Series.**99**(1): 48–69. doi:10.2307/1971013. JSTOR 1971013. - Bertlmann, Reinhold A. (2001). "Chern–Simons form, homotopy operator and anomaly".
*Anomalies in Quantum Field Theory*(Revised ed.). Clarendon Press. pp. 321–341. ISBN 0-19-850762-3.