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, ${\mathbf {A}}$ over it, we can define a family of p-forms:

In one dimension, the Chern–Simons 1-form is given by

${\rm {Tr}}[{\mathbf {A}}].$

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

${\rm {Tr}}\left[{\mathbf {F}}\wedge {\mathbf {A}}-{\frac {1}{3}}{\mathbf {A}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}\right].$

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

${\rm {Tr}}\left[{\mathbf {F}}\wedge {\mathbf {F}}\wedge {\mathbf {A}}-{\frac {1}{2}}{\mathbf {F}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}+{\frac {1}{10}}{\mathbf {A}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}\wedge {\mathbf {A}}\right]$

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

$d\omega _{2k-1}={\rm {Tr}}\left(F^{k}\right),$

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

• Chern, S.-S.; Simons, J (1974), "Characteristic forms and geometric invariants", Annals of Mathematics. Second Series 99 (1): 48–69, JSTOR 1971013.