Editing Chern–Simons form

{{Short description|Secondary characteristic classes of 3-manifolds}}
In [[mathematics]], the '''Chern–Simons forms''' are certain secondary [[characteristic class]]es.<ref>{{Cite web|title=Remarks on Chern–Simons theory|url=https://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01243-9/S0273-0979-09-01243-9.pdf|last=Freed|first=Daniel|date=January 15, 2009|access-date=April 1, 2020}}</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>{{Cite book|last1=Chern|first1=Shiing-Shen|url=https://books.google.com/books?id=uOfSa0sfJr0C&q=Characteristic+Forms+and+Geometric+Invariants&pg=PA363|title=A Mathematician and His Mathematical Work: Selected Papers of S.S. Chern|last2=Tian|first2=G.|last3=Li|first3=Peter|date=1996|publisher=World Scientific|isbn=978-981-02-2385-4|language=en}}</ref>

==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>

== Application to physics==
In 1978, [[Albert Schwarz]] formulated [[Chern–Simons theory]], early [[topological quantum field theory]], using Chern-Simons forms.<ref>{{cite journal |last=Schwartz |first=A. S. |year=1978 |title=The partition function of degenerate quadratic functional and Ray-Singer invariants |journal=Letters in Mathematical Physics |volume=2 |issue=3 |pages=247–252 |doi=10.1007/BF00406412 |bibcode=1978LMaPh...2..247S |s2cid=123231019 }}</ref>

In the [[gauge theory]], the [[Differential form|integral]] of Chern-Simons form 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==
{{Reflist}}

== Further reading ==

* {{cite journal |last1=Chern |first1=S.-S. |author-link1=Shiing-Shen Chern |last2=Simons |first2=J. |author-link2=James Harris Simons |title=Characteristic forms and geometric invariants |journal=[[Annals of Mathematics]] |series=Second Series |volume=99 |number=1 |year=1974 |pages=48–69 |jstor=1971013 |doi=10.2307/1971013 }}
* {{cite book |first=Reinhold A. |last=Bertlmann |chapter=Chern–Simons form, homotopy operator and anomaly |title=Anomalies in Quantum Field Theory |publisher=[[Clarendon Press]] |year=2001 |edition=Revised |pages=321–341 |isbn=0-19-850762-3 |chapter-url=https://books.google.com/books?id=FC_DRRUHFXEC&pg=PA321 }}

{{String theory topics |state=collapsed}}

{{DEFAULTSORT:Chern-Simons form}}
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