Template:Short description In mathematics, the Chern–Simons forms are certain secondary characteristic classes.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref> 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.<ref>Template:Cite book</ref>
DefinitionEdit
Given a manifold and a Lie algebra valued 1-form <math>\mathbf{A}</math> over it, we can define a family of p-forms:<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
In one dimension, the Chern–Simons 1-form is given by
- <math>\operatorname{Tr} [ \mathbf{A} ].</math>
In three dimensions, the Chern–Simons 3-form is given by
- <math>\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].</math>
In five dimensions, the Chern–Simons 5-form is given by
- <math>
\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} </math>
where the curvature F is defined as
- <math>\mathbf{F} = d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}.</math>
The general Chern–Simons form <math>\omega_{2k-1}</math> is defined in such a way that
- <math>d\omega_{2k-1}= \operatorname{Tr}(F^k),</math>
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 <math>\mathbf{A}</math>.
In general, the Chern–Simons p-form is defined for any odd p.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>
Application to physicsEdit
In 1978, Albert Schwarz formulated Chern–Simons theory, early topological quantum field theory, using Chern-Simons forms.<ref>Template:Cite journal</ref>
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.