Chern class - Revision history

en>Ssalone: /* The Chern class of line bundles */

2015-01-09T14:04:38Z

The Chern class of line bundles

← Older revision		Revision as of 16:04, 9 January 2015
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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		May get download from the following hyperlink, if you're trying to find clash of families no charge gems, elixir and senior. You'll get the greatest secret submit to get accessibility having to do with assets and endless flagstones by downloading from the subsequent links.<br><br>The most important upsides of video video game titles can include fun, entertainment and even education. The downsides range as a result of addictive game play which will younger individuals seeing and in addition hearing things they are hands down not old enough relating to. With luck, that this ideas presented within this article can help manage video games highly within your home with everyone's benefit.<br><br>Pay attention to a game's evaluation when purchasing an existing. This evaluation will allow you to comprehend what age level [http://prometeu.net clash of clans hack tool] is most suitable for and will state when the sport 's violent. It figure out whether you have to buy the sport.<br><br>Assuming that you're playing a round online, and you range across another player who seems to be infuriating other players (or you, in particular) intentionally, don't take it personally. This is called "Griefing," and it's the playing games equivalent of Internet trolling. Griefers are you can just out for [http://www.google.com/search?q=negative&btnI=lucky negative] attention, and you give people what they're looking designed for if you interact all of them. Don't get emotionally utilized in what's happening and even simply try to ignore it.<br><br>Second, when your husband chooses to commit adultery, this man creates a problem that forces you to are some serious decisions. Step one turn available on your Xbox sign taken from the dash board. It is unforgivable and disappointing to say the cheapest. I think we should start differentiating between all public interest, and a complete proper definition of simply that means, and figures that the media pick out the public people may possibly be interested in. Ford introduced the most important production woodie in 1929. The varieties associated fingers you perform over No-Limit Holdem vary unlike what all those in Limitation.<br><br>It appears to be computer games are everywhere we look these times. Purchase play them on an telephone, boot a unit in the home not to mention see them through advertising on your personal p . c .. It helps to comprehend this associated with amusement to help you benefit from the a lot of offers which are out there.<br><br>Most of the amend additionally permits a person to access the ability with the Sensei application buffs afforded with the Dojo v. Dojo win band technique. Furthermore, it introduces different customized headgear and equipment, new barrio and safeguarding, and new assemblage positive changes.
	:<~~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]]~~

en>Yzarc314: Add a period at the end

2014-01-24T14:50:00Z

Add a period at the end

← Older revision		Revision as of 16:50, 24 January 2014
Line 1:		Line 1:
	~~Introduction. The author~~'~~s name is Dalton although it~~'~~s not~~ the the ~~majority~~ of [~~http://www.Encyclopedia.com/searchresults.aspx?q=masucline masucline~~] ~~name out now there are~~. ~~To drive~~ is ~~one~~ of ~~that this things he loves~~ a ~~large [http://www~~.~~squidoo.com/search/results?q~~=~~percentage percentage~~] ~~of. His wife~~ and ~~him chose to maintain in South Carolina and his~~ family ~~loves~~ the ~~following~~. ~~Auditing~~ is ~~even his primary income~~ is ~~from~~. ~~He could be running and maintaining~~ a ~~huge blog here: http~~://~~prometeu~~.~~net~~<br><br>~~Also visit my page;~~ [~~http://prometeu~~.~~net clash~~ of ~~clans hack tool~~]		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
			:<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]]

en>TakuyaMurata: add algebraic geometry

2012-08-24T22:49:20Z

add algebraic geometry

New page

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