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en>Yobot: WP:CHECKWIKI error fixes + other fixes, removed orphan tag using AWB (10065)

2014-03-28T20:45:52Z

WP:CHECKWIKI error fixes + other fixes, removed orphan tag using AWB (10065)

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	~~In mathematical physics, the '''ADHM construction''' or '''monad construction''~~' is ~~the construction of all [[instanton]]s using methods of linear algebra by [[Michael Atiyah]], [[Vladimir Drinfeld]], [[Nigel Hitchin]], [[Yuri I~~. ~~Manin]] in their paper ''Construction of Instantons''~~.		The writer's name is Christy. Credit authorising is how he tends to make money. My spouse doesn't like it the way I do but what I really like doing is caving but I don't have the time lately. My wife and I live in Mississippi but now I'm contemplating other options.<br><br>Look at my web blog: accurate [http://netwk.hannam.ac.kr/xe/data_2/85669 free psychic readings] [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 free online tarot card readings] ([http://Www.Herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies http://Www.Herandkingscounty.com/])

	~~== ADHM data ==~~
	~~The ADHM construction uses the following data:~~

	* complex [[vector space]]s ''V'' and ''W'' of dimension ''k'' and ''N'',
	* ''k'' × ''k'' complex matrices ''B''<sub>1</sub>, ''B''<sub>2</sub>, a ''k'' × ''N'' complex matrix ''I'' and a ''N'' × ''k'' complex matrix ''J'',
	* a [[real number\|real]] [[moment map]] <math>\mu_r = [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+II^\dagger-J^\dagger J,</math>
	* a [[complex number\|complex]] moment map <math>\displaystyle\mu_c = [B_1,B_2]+IJ.</math>

	~~Then~~ the ~~ADHM construction claims that, given certain regularity conditions,~~
	* Given ''B''<sub>1</sub>, ''B''<sub>2</sub>, ''I~~'', ''J'' such that <math>\mu_r=\mu_c=0</math>, an anti-self-dual [[instanton]] in a [[special unitary group\|SU(''N'')]] [[gauge theory]] with [[instanton]] number ''k'' can be constructed,~~
	* All anti-self-dual [[instantons]] can be obtained in this way and are in one-to-one correspondence with solutions up to a U(''k'') rotation which acts on each ''B'' in the [[adjoint representation]] and on ''I~~'' and ''J'' via the [[fundamental representation\|fundamental]] and antifundamental representations~~
	* The [[metric (mathematics)\|metric]] on the [[moduli space]] of instantons is ~~that inherited from the flat metric on ''B'', ''~~I'~~' and ''J''.~~

	~~==Generalizations==~~
	~~===Noncommutative instantons===~~
	In a [[noncommutative geometry\|noncommutative]] gauge theory, the ADHM construction is identical but the moment map <math>\vec\mu </math> is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the [[identity matrix]]. In this case instantons exist even when the ~~gauge group is U(1)~~. ~~The noncommutative instantons were discovered by [[Nikita Nekrasov]]~~ and ~~[[Albert Schwarz]]~~ in ~~1998~~.

	~~===Vortices===~~
	~~Setting ''B''~~<~~sub~~>2<~~/sub~~> and ''J'' to zero, one obtains the classical moduli space of nonabelian [[Abrikosov vortex\|vortices]] in a [[supersymmetry\|supersymmetric]] gauge theory with an equal number of colors and flavors, as was demonstrated in [http://~~www~~.~~slac~~.~~stanford~~.~~edu/spires~~/~~find~~/~~hep/www?eprint=hep-th~~/~~0306150 Vortices, instantons and branes~~]~~. The generalization to greater numbers of flavors appeared in~~ [http://~~www~~.~~slac.stanford~~.~~edu/spires/find/hep/www~~?~~eprint~~=hep-th/0602170 Solitons in the Higgs phase: The Moduli matrix approach]. In both cases the [[Fayet-Iliopoulos term]], which determines a [[squark]] [[top quark condensate\|condensate]], plays the role of the noncommutativity parameter in the real moment map.

	~~== The construction formula ==~~
	~~Let ''x'' be the 4-dimensional [[Euclidean space\|Euclidean]] [[spacetime]] coordinates written in [[quaternion]]ic notation~~
	~~<math>x_{ij}=\begin{pmatrix}z_2&z_1\\-\bar{z_1}~~&~~\bar{z_2}\end{pmatrix}.</math>~~

	~~Consider the 2''k'' × (''N'' + 2''k'') matrix~~

	~~:<math>\Delta~~= ~~\begin{pmatrix}I~~&~~B_2+z_2&B_1+z_1\\J^\dagger&-B_1^\dagger-\bar{z_1}&B_2^\dagger+\bar{z_2}\end{pmatrix}.</math>~~

	~~Then the conditions <math>\displaystyle\mu_r~~=~~\mu_c=0</math> are equivalent to the factorization condition~~
	~~:<math>\Delta\Delta^\dagger=\begin{pmatrix}f^{-1}&0\\0&f^{-1}\end{pmatrix}</math> where ''f''~~(~~''x'') is a ''k'' × ''k''~~ [~~[Hermitian matrix]].~~

	~~Then a hermitian [[projection (linear algebra)\|projection]] operator ''P'' can be constructed as~~
	:~~<math>P=\Delta^\dagger\begin{pmatrix}f&0\\0&f\end{pmatrix}\Delta.<~~/~~math>~~

	The [[nullspace]] of Δ(''x'') is of ''N'' dimension for generic ''x''. The basis vector for this null-space can be assembled into an (''N'' + 2''k'') × ''N'' matrix ''U''(''x'') with orthonormalization condition ''U''<sup>†</~~sup>''U'' = 1~~.

	~~A regularity condition on the rank of Δ guaranteed the completeness condition~~
	~~:<math>P+UU^\dagger=1~~. ~~\, <~~/~~math>~~

	~~The anti-selfdual [[connection (mathematics)\|connection]] is then constructed from ''U'' by the formula~~
	~~:<math>A_m=U^\dagger \partial_m U.<~~/~~math>~~

	~~==See also==~~

	*[[Monad (linear algebra)]]

	~~==References==~~

	*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-~~link=Michael Atiyah \| title=Geometry of Yang~~-~~Mills fields \| publisher=Scuola Normale Superiore Pisa, Pisa \| id={{MathSciNet \| id = 554924}} \| year=1979}}~~
	*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-~~link=Michael Atiyah \| last3=Hitchin \| first3=N. J. \| last2=Drinfeld \| first2=V. G. \| last4=Manin \| first4=Yuri Ivanovich \| author4~~-~~link=Yuri Ivanovich Manin \| title=Construction of instantons \| url=http://dx.doi.org/10.1016/0375-9601(78)90141-X \| doi=10.1016/0375~~-~~9601(78)90141~~-~~X \| id={{MR\|598562}} \| year=1978 \| journal=Physics Letters A \| issn=0375~~-~~9601 \| volume=65 \| issue=3 \| pages=185–187\|bibcode = 1978PhLA...65..185A }}~~
	*[http://~~projecteuclid~~.~~org/DPubS/Repository/1~~.0/~~Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103922679 On the Construction of Monopoles]~~

	~~[[Category:Quantum field theory]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Quantum chromodynamics]~~]

en>Soham: Cleanup

2013-10-21T07:12:48Z

Cleanup

← Older revision		Revision as of 09:12, 21 October 2013
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	The ~~individual who wrote~~ the ~~article is called Jayson Hirano~~ and ~~he totally digs~~ that ~~title. My spouse doesn~~'~~t like it~~ the ~~way~~ I ~~do but what~~ I ~~really like doing~~ is ~~caving~~ but ~~I don~~'~~t have~~ the ~~time lately~~. ~~Her family members life~~ in ~~Ohio but her~~ [http://~~formalarmour~~.~~com~~/~~index~~.~~php?do~~=/~~profile~~-~~26947~~/~~info~~/ are ~~psychics real] husband wants them~~ to ~~transfer. Invoicing~~ is ~~what I do for~~ a ~~living but I~~'~~ve usually wanted my personal business~~.<br><br>[~~http:~~/~~/www~~.~~aseandate~~.~~com~~/~~index~~.~~php?m~~=~~member_profile&p~~=~~profile&id~~=~~13352970 cheap psychic readings~~] ~~Have a look at my homepage; tarot readings~~, [http://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997~~ http://~~mybrandcp~~.~~com~~/xe/~~board_XmDx25~~/~~107997~~],		In mathematical physics, the '''ADHM construction''' or '''monad construction''' is the construction of all [[instanton]]s using methods of linear algebra by [[Michael Atiyah]], [[Vladimir Drinfeld]], [[Nigel Hitchin]], [[Yuri I. Manin]] in their paper ''Construction of Instantons''.

			== ADHM data ==
			The ADHM construction uses the following data:

			* complex [[vector space]]s ''V'' and ''W'' of dimension ''k'' and ''N'',
			* ''k'' × ''k'' complex matrices ''B''<sub>1</sub>, ''B''<sub>2</sub>, a ''k'' × ''N'' complex matrix ''I'' and a ''N'' × ''k'' complex matrix ''J'',
			* a [[real number\|real]] [[moment map]] <math>\mu_r = [B_1,B_1^\dagger]+[B_2,B_2^\dagger]+II^\dagger-J^\dagger J,</math>
			* a [[complex number\|complex]] moment map <math>\displaystyle\mu_c = [B_1,B_2]+IJ.</math>

			Then the ADHM construction claims that, given certain regularity conditions,
			* Given ''B''<sub>1</sub>, ''B''<sub>2</sub>, ''I'', ''J'' such that <math>\mu_r=\mu_c=0</math>, an anti-self-dual [[instanton]] in a [[special unitary group\|SU(''N'')]] [[gauge theory]] with [[instanton]] number ''k'' can be constructed,
			* All anti-self-dual [[instantons]] can be obtained in this way and are in one-to-one correspondence with solutions up to a U(''k'') rotation which acts on each ''B'' in the [[adjoint representation]] and on ''I'' and ''J'' via the [[fundamental representation\|fundamental]] and antifundamental representations
			* The [[metric (mathematics)\|metric]] on the [[moduli space]] of instantons is that inherited from the flat metric on ''B'', ''I'' and ''J''.

			==Generalizations==
			===Noncommutative instantons===
			In a [[noncommutative geometry\|noncommutative]] gauge theory, the ADHM construction is identical but the moment map <math>\vec\mu </math> is set equal to the self-dual projection of the noncommutativity matrix of the spacetime times the [[identity matrix]]. In this case instantons exist even when the gauge group is U(1). The noncommutative instantons were discovered by [[Nikita Nekrasov]] and [[Albert Schwarz]] in 1998.

			===Vortices===
			Setting ''B''<sub>2</sub> and ''J'' to zero, one obtains the classical moduli space of nonabelian [[Abrikosov vortex\|vortices]] in a [[supersymmetry\|supersymmetric]] gauge theory with an equal number of colors and flavors, as was demonstrated in [http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/0306150 Vortices, instantons and branes]. The generalization to greater numbers of flavors appeared in [http://www.slac.stanford.edu/spires/find/hep/www?eprint=hep-th/0602170 Solitons in the Higgs phase: The Moduli matrix approach]. In both cases the [[Fayet-Iliopoulos term]], which determines a [[squark]] [[top quark condensate\|condensate]], plays the role of the noncommutativity parameter in the real moment map.

			== The construction formula ==
			Let ''x'' be the 4-dimensional [[Euclidean space\|Euclidean]] [[spacetime]] coordinates written in [[quaternion]]ic notation
			<math>x_{ij}=\begin{pmatrix}z_2&z_1\\-\bar{z_1}&\bar{z_2}\end{pmatrix}.</math>

			Consider the 2''k'' × (''N'' + 2''k'') matrix

			:<math>\Delta= \begin{pmatrix}I&B_2+z_2&B_1+z_1\\J^\dagger&-B_1^\dagger-\bar{z_1}&B_2^\dagger+\bar{z_2}\end{pmatrix}.</math>

			Then the conditions <math>\displaystyle\mu_r=\mu_c=0</math> are equivalent to the factorization condition
			:<math>\Delta\Delta^\dagger=\begin{pmatrix}f^{-1}&0\\0&f^{-1}\end{pmatrix}</math> where ''f''(''x'') is a ''k'' × ''k'' [[Hermitian matrix]].

			Then a hermitian [[projection (linear algebra)\|projection]] operator ''P'' can be constructed as
			:<math>P=\Delta^\dagger\begin{pmatrix}f&0\\0&f\end{pmatrix}\Delta.</math>

			The [[nullspace]] of Δ(''x'') is of ''N'' dimension for generic ''x''. The basis vector for this null-space can be assembled into an (''N'' + 2''k'') × ''N'' matrix ''U''(''x'') with orthonormalization condition ''U''<sup>†</sup>''U'' = 1.

			A regularity condition on the rank of Δ guaranteed the completeness condition
			:<math>P+UU^\dagger=1. \, </math>

			The anti-selfdual [[connection (mathematics)\|connection]] is then constructed from ''U'' by the formula
			:<math>A_m=U^\dagger \partial_m U.</math>

			==See also==

			*[[Monad (linear algebra)]]

			==References==

			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| title=Geometry of Yang-Mills fields \| publisher=Scuola Normale Superiore Pisa, Pisa \| id={{MathSciNet \| id = 554924}} \| year=1979}}
			*{{Citation \| last1=Atiyah \| first1=Michael Francis \| author1-link=Michael Atiyah \| last3=Hitchin \| first3=N. J. \| last2=Drinfeld \| first2=V. G. \| last4=Manin \| first4=Yuri Ivanovich \| author4-link=Yuri Ivanovich Manin \| title=Construction of instantons \| url=http://dx.doi.org/10.1016/0375-9601(78)90141-X \| doi=10.1016/0375-9601(78)90141-X \| id={{MR\|598562}} \| year=1978 \| journal=Physics Letters A \| issn=0375-9601 \| volume=65 \| issue=3 \| pages=185–187\|bibcode = 1978PhLA...65..185A }}
			*[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103922679 On the Construction of Monopoles]

			[[Category:Quantum field theory]]
			[[Category:Differential geometry]]
			[[Category:Quantum chromodynamics]]

en>Whouk: change link

2012-05-31T12:28:58Z

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The individual who wrote the article is called Jayson Hirano and he totally digs that title. My spouse doesn't like it the way I do but what I really like doing is caving but I don't have the time lately. Her family members life in Ohio but her [http://formalarmour.com/index.php?do=/profile-26947/info/ are psychics real] husband wants them to transfer. Invoicing is what I do for a living but I've usually wanted my personal business.<br><br>[http://www.aseandate.com/index.php?m=member_profile&p=profile&id=13352970 cheap psychic readings] Have a look at my homepage; tarot readings, [http://mybrandcp.com/xe/board_XmDx25/107997 http://mybrandcp.com/xe/board_XmDx25/107997],