Chern–Simons form

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
 * $$\operatorname{Tr} [ \mathbf{A} ].$$

In three dimensions, the Chern–Simons 3-form is given by
 * $$\operatorname{Tr} \left[ \mathbf{F} \wedge \mathbf{A}-\frac{1}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \right] = \operatorname{Tr} \left[ d\mathbf{A} \wedge \mathbf{A} + \frac{2}{3} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\right].$$

In five dimensions, the Chern–Simons 5-form is given by

\begin{align} & \operatorname{Tr} \left[ \mathbf{F}\wedge\mathbf{F} \wedge \mathbf{A}-\frac{1}{2} \mathbf{F} \wedge\mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} +\frac{1}{10} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge\mathbf{A} \right] \\[6pt] = {} & \operatorname{Tr} \left[ d\mathbf{A}\wedge d\mathbf{A} \wedge \mathbf{A} + \frac{3}{2} d\mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A} +\frac{3}{5} \mathbf{A} \wedge \mathbf{A} \wedge \mathbf{A}\wedge\mathbf{A}\wedge\mathbf{A} \right] \end{align} $$

where the curvature F is defined as
 * $$\mathbf{F} = d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.$$

The general Chern–Simons form $$\omega_{2k-1}$$ is defined in such a way that
 * $$d\omega_{2k-1}= \operatorname{Tr}(F^k),$$

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.