Chern class

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.Template:Harvs

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

$\mathrm{T}\mathrm{r}[\mathbf{A}].$

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

$\mathrm{T}\mathrm{r}[\mathbf{F}\wedge \mathbf{A}-\frac{1}{3}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}].$

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

$\mathrm{T}\mathrm{r}[\mathbf{F}\wedge \mathbf{F}\wedge \mathbf{A}-\frac{1}{2}\mathbf{F}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}+\frac{1}{10}\mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}\wedge \mathbf{A}]$

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

$d{\omega }_{2k-1}=\mathrm{T}\mathrm{r}\left(F^k\right),$

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 $\mathbf{A}$.

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

• Chern–Weil homomorphism
• Chiral anomaly
• Topological quantum field theory
• Jones polynomial

References

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