Chern–Simons form

Definition

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

${\displaystyle \mathrm{Tr}[\mathbf{A}].}$

In three dimensions, the Chern–Simons 3-form is given by

${\displaystyle \mathrm{Tr}\left[\mathbf{F}\wedge \mathbf{A}-\frac{1}{3}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right]=\mathrm{Tr}\left[d\mathbf{A}\wedge \mathbf{A}+\frac{2}{3}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right].}$

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

${\displaystyle \begin{array}{cc}& \mathrm{Tr}\left[\mathbf{F}\wedge \mathbf{F}\wedge \mathbf{A}-\frac{1}{2}\mathbf{F}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}+\frac{1}{10}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right]\\ =& \mathrm{Tr}\left[d\mathbf{A}\wedge d\mathbf{A}\wedge \mathbf{A}+\frac{3}{2}d\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}+\frac{3}{5}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\right]\end{array}}$

where the curvature F is defined as

${\displaystyle \mathbf{F}=d\mathbf{A}+\mathbf{A}\wedge \mathbf{A}.}$

The general Chern–Simons form ${\displaystyle {\omega }_{2k-1}}$ is defined in such a way that

${\displaystyle d{\omega }_{2k-1}=\mathrm{Tr}(F^k),}$

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 ${\displaystyle \mathbf{A}}$.

In general, the Chern–Simons p-form is defined for any odd p.