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.

Definition

Given a manifold and a Lie algebra valued 1-form \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 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 \mathbf{A}. In general, the Chern–Simons p-form is defined for any odd p.

Application to physics

In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms. In the gauge theory, the integral of Chern-Simons form is a global geometric invariant, and is typically gauge invariant modulo addition of an integer.