Editing Chern–Simons form (section)

==Definition==
Given a [[manifold]] and a [[Lie algebra]] valued [[Multilinear form|1-form]] <math>\mathbf{A}</math> over it, we can define a family of [[Multilinear form|''p''-forms]]:<ref>{{Cite web|title=Chern-Simons form in nLab|url=https://ncatlab.org/nlab/show/Chern-Simons+form|website=ncatlab.org|access-date=May 1, 2020}}</ref>

In one dimension, the '''Chern–Simons''' [[Multilinear form|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 ''F<sup>k</sup>''. 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 [[Multilinear form|''p''-form]] is defined for any odd ''p''.<ref>{{Cite web|title=Introduction To Chern-Simons Theories|url=http://www.physics.rutgers.edu/~gmoore/TASI-ChernSimons-StudentNotes.pdf|last=Moore|first=Greg|date=June 7, 2019|website=University of Texas|access-date=June 7, 2019}}</ref>