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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''.
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| == ADHM data ==
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| The ADHM construction uses the following data:
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| * complex [[vector space]]s ''V'' and ''W'' of dimension ''k'' and ''N'',
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| * ''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'',
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| * 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>
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| * a [[complex number|complex]] moment map <math>\displaystyle\mu_c = [B_1,B_2]+IJ.</math>
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| Then the ADHM construction claims that, given certain regularity conditions,
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| * 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,
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| * 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
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| * The [[metric (mathematics)|metric]] on the [[moduli space]] of instantons is that inherited from the flat metric on ''B'', ''I'' and ''J''.
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| ==Generalizations==
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| ===Noncommutative instantons===
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| 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.
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| ===Vortices===
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| 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.
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| == The construction formula ==
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| Let ''x'' be the 4-dimensional [[Euclidean space|Euclidean]] [[spacetime]] coordinates written in [[quaternion]]ic notation
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| <math>x_{ij}=\begin{pmatrix}z_2&z_1\\-\bar{z_1}&\bar{z_2}\end{pmatrix}.</math>
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| Consider the 2''k'' × (''N'' + 2''k'') matrix
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| :<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>
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| Then the conditions <math>\displaystyle\mu_r=\mu_c=0</math> are equivalent to the factorization condition
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| :<math>\Delta\Delta^\dagger=\begin{pmatrix}f^{-1}&0\\0&f^{-1}\end{pmatrix}</math> where ''f''(''x'') is a ''k'' × ''k'' [[Hermitian matrix]].
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| Then a hermitian [[projection (linear algebra)|projection]] operator ''P'' can be constructed as
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| :<math>P=\Delta^\dagger\begin{pmatrix}f&0\\0&f\end{pmatrix}\Delta.</math> | |
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| 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.
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| A regularity condition on the rank of Δ guaranteed the completeness condition
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| :<math>P+UU^\dagger=1. \, </math>
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| The anti-selfdual [[connection (mathematics)|connection]] is then constructed from ''U'' by the formula
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| :<math>A_m=U^\dagger \partial_m U.</math>
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| ==See also==
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| *[[Monad (linear algebra)]]
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| ==References==
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| *{{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}}
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| *{{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 }}
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| *[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.cmp/1103922679 On the Construction of Monopoles]
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| [[Category:Quantum field theory]]
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| [[Category:Differential geometry]]
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| [[Category:Quantum chromodynamics]]
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