Chern-Simons form

In mathematics, the Chern-Simons forms (pronounced chen simons ) 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.

Given a manifold and a Lie algebra valued p-form, old{A} over it, we can define a family of p-forms:

In one dimension, the Chern-Simons p-form is given by :Tr[old{A}].

In three dimensions, the Chern-Simons 3-form is given by :Tr[old{F}wedgeold{A}-frac{1}{3}old{A}wedgeold{A}wedgeold{A}].

In five dimensions, the Chern-Simons 5-form is given by :Tr[old{F}wedgeold{F}wedgeold{A}-frac{1}{2}old{F}wedgeold{A}wedgeold{A}wedgeold{A} +frac{1}{10}old{A}wedgeold{A}wedgeold{A}wedgeold{A}wedgeold{A}]

where the curvature F is defined as :dold{A}+old{A}wedgeold{A}.

The general Chern-Simons form omega_{2k-1} is defined in such a way that domega_{2k-1}=Tr(F^{k}) where the wedge product is used to define F^k.

See gauge theory for more details.

In general, the Chern-Simons p-form is defined for any odd p. See gauge theory for the definitions. Its Differential form over a p dimensional manifold is a homotopy invariant. This value is called the Chern class.

See also Topological quantum field theory and Chiral anomaly.