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In [[mathematics]], the '''Chern–Simons forms''' are certain secondary [[characteristic class]]es. 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.{{harvs|txt=yes|last1=Chern|last2=Simons|year=1974}}


==Definition==
Given a [[manifold]] and a [[Lie algebra]] valued [[p-form|1-form]], <math>\bold{A}</math> over it, we can define a family of [[p-form]]s:


In one dimension, the '''Chern–Simons''' [[p-form|1-form]] is given by  
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:<math>{\rm Tr} [ \bold{A} ].</math>
 
In three dimensions, the '''Chern–Simons 3-form''' is given by
:<math>{\rm Tr} \left[ \bold{F}\wedge\bold{A}-\frac{1}{3}\bold{A}\wedge\bold{A}\wedge\bold{A}\right].</math>
 
In five dimensions, the '''Chern–Simons 5-form''' is given by
:<math>{\rm Tr} \left[ \bold{F}\wedge\bold{F}\wedge\bold{A}-\frac{1}{2}\bold{F}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} +\frac{1}{10}\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A}\wedge\bold{A} \right]</math>
 
where the curvature '''F''' is defined as
:<math>\bold{F} = d\bold{A}+\bold{A}\wedge\bold{A}.</math>
 
The general Chern–Simons form <math>\omega_{2k-1}</math> is defined in such a way that
:<math>d\omega_{2k-1}={\rm Tr}  \left( F^{k} \right),</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>\bold{A}</math>.
 
In general, the Chern–Simons [[p-form]] is defined for any odd ''p''. See also [[gauge theory]] for the definitions. Its [[Differential form|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==
* {{citation|last1=Chern|first1=S.-S.|authorlink1=S.-S. Chern|last2=Simons|first2=J|authorlink2=James Harris Simons|title=Characteristic forms and geometric invariants|journal=The Annals of Mathematics. Second Series|volume=99|number=1|year=1974|pages=48–69|jstor=1971013}}.
 
{{DEFAULTSORT:Chern-Simons form}}
[[Category:Homology theory]]
[[Category:Algebraic topology]]
[[Category:Differential geometry]]

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