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| In mathematics, '''Seiberg–Witten invariants''' are invariants of compact smooth [[4-manifold]]s introduced by {{harvtxt|Witten|1994}}, using the Seiberg–Witten theory studied by {{harvs|txt=yes|last=Seiberg|author-link=Nathan Seiberg|last2=Witten|author2-link=Edward Witten|year1=1994a|year2=1994b}} during their investigations of [[Seiberg–Witten gauge theory]].
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| Seiberg–Witten invariants are similar to [[Donaldson invariant]]s and can be used to prove similar (but sometimes slightly stronger) results about smooth 4-manifolds. They are technically much easier to work with than Donaldson invariants; for example, the [[moduli spaces of solutions of the Seiberg–Witten equations]] tend to be compact, so one avoids the hard problems involved in compactifying the moduli spaces in Donaldson theory.
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| For detailed descriptions of Seiberg–Witten invariants see {{harv|Donaldson|1996}}, {{harv|Moore|2001}}, {{harv|Morgan|1996}}, {{harv|Nicolaescu|2000}}, {{harv|Scorpan|2005|loc=Chapter 10}}. For the relation to symplectic manifolds and [[Gromov–Witten invariant]]s see {{harv|Taubes|2000}}. For the early history see {{harv|Jackson|1995}}.
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| ==Spin<sup>''c''</sup>-structures==
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| The Seiberg-Witten equations depend on the choice of a [[complex spin structure]], Spin<sup>''c''</sup>, on a 4-manifold ''M''. In 4 dimensions the group Spin<sup>''c''</sup> is
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| :(''U''(1)×Spin(4))/('''Z'''/2'''Z'''), | |
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| and there is a homomorphism from it to [[SO(4)]]. A Spin<sup>''c''</sup>-structure on ''M'' is a lift of the natural SO(4) structure on the tangent bundle (given by the [[Riemannian metric]] and orientation) to the group Spin<sup>''c''</sup>. Every smooth compact 4-manifold ''M'' has Spin<sup>''c''</sup>-structures (though most do not have [[spin structure]]s).
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| ==Seiberg–Witten equations==
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| Fix a smooth compact 4-manifold ''M'', choose a spin<sup>''c''</sup>-structure ''s'' on ''M'', and write ''W''<sup>+</sup>, ''W''<sup>−</sup> for the associated [[spinor bundle]]s, and ''L'' for the [[determinant line bundle]]. Write φ for a self-dual spinor field (a section of ''W''<sup>+</sup>) and ''A'' for a U(1) connection on ''L''.
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| The Seiberg–Witten equations for (φ,''A'') are
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| :<math>D^A\phi=0</math>
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| :<math>F^+_A=\sigma(\phi) + i\omega</math>
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| where ''D''<sup>''A''</sup> is the [[Dirac operator]] of ''A'', ''F''<sub>''A''</sub> is the curvature 2-form of ''A'', and ''F''<sub>''A''</sub><sup>+</sup> is its self-dual part, and σ is the squaring map from ''W''<sup>+</sup> to imaginary self-dual 2-forms and <math>\omega</math>
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| is a real selfdual two form, often taken to be zero or harmonic.
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| The solutions (φ,''A'') to the Seiberg–Witten equations are called '''monopoles''', as these equations are the [[field equations]] of massless [[magnetic monopoles]] on the manifold ''M''.
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| ==The moduli space of solutions==
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| The space of solutions is acted on by the gauge group, and the quotient by this action is called the '''moduli space''' of monopoles.
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| The moduli space is usually a manifold. A solution is called '''reducible''' if it is fixed by some non-trivial element of the gauge group which is equivalent to <math>\phi = 0</math>. A necessary
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| and sufficient condition for reducible solutions for a metric on ''M'' and self dual 2 forms <math>\omega</math> is that the self-dual part of the harmonic representative of the cohomology class of the determinant line bundle is equal to the harmonic part of <math>\omega/2\pi</math>. The moduli space is a manifold except at reducible monopoles. So if ''b''<sub>2</sub><sup>+</sup>(''M'')≥1 then the moduli space is a (possibly empty) manifold for generic metrics. Moreover all components have dimension
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| :<math>(c_1(s)^2-2\chi(M)-3sign(M))/4.</math>
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| The moduli space is empty for all but a finite number of spin<sup>''c''</sup> structures ''s'', and is always compact.
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| A manifold ''M'' is said to be of '''simple type''' if the moduli space is finite for all ''s''.
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| The '''simple type conjecture''' states that if ''M'' is simply connected and ''b''<sub>2</sub><sup>+</sup>(''M'')≥2 then the moduli space is finite. It is true for symplectic manifolds.
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| If ''b''<sub>2</sub><sup>+</sup>(''M'')=1 then there are examples of manifolds with moduli spaces of arbitrarily high dimension.
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| ==Seiberg–Witten invariants==
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| The Seiberg–Witten invariants are easiest to define for manifolds ''M'' of simple type. In this case the invariant is the map from spin<sup>''c''</sup> structures ''s'' to '''Z''' taking ''s'' to the number of elements of the moduli space counted with signs.
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| If the manifold ''M'' has a metric of positive scalar curvature and ''b''<sub>2</sub><sup>+</sup>(''M'')≥2 then all Seiberg–Witten invariants of ''M'' vanish.
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| If the manifold ''M'' is the connected sum of two manifolds both of which have ''b''<sub>2</sub><sup>+</sup>≥1 then all Seiberg–Witten invariants of ''M'' vanish.
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| If the manifold ''M'' is simply connected and symplectic and ''b''<sub>2</sub><sup>+</sup>(''M'')≥2 then it has a spin<sup>''c''</sup> structure ''s'' on which the Seiberg–Witten invariant is 1. In particular it cannot be split as a connected sum of manifolds with ''b''<sub>2</sub><sup>+</sup>≥1.
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| ==References==
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| *{{citation
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| |last=Donaldson|first= S. K. |authorlink=Simon Donaldson
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| |title=The Seiberg-Witten equations and 4-manifold topology.
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| |journal=Bull. Amer. Math. Soc. (N.S.) |volume=33 |year=1996|issue= 1|pages= 45–70
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| |url=http://www.ams.org/bull/1996-33-01/S0273-0979-96-00625-8/home.html|doi=10.1090/S0273-0979-96-00625-8
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| |mr=1339810}}
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| *{{citation|first=Allyn|last=Jackson|title=A revolution in mathematics|year=1995|url=http://web.archive.org/web/20100426172959/http://www.ams.org/samplings/feature-column/mathnews-revolution}}
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| *{{citation|last= Morgan|first= John W.|authorlink=John Morgan (mathematician)|title= The Seiberg-Witten equations and applications to the topology of smooth four-manifolds|series=Mathematical Notes|volume= 44|publisher= Princeton University Press|publication-place= Princeton, NJ|year= 1996|pages= viii+128| isbn= 0-691-02597-5|url=http://press.princeton.edu/titles/5866.html|mr= 1367507}}
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| *{{citation|last= Moore|first= John Douglas|title= Lectures on Seiberg-Witten invariants|edition=2nd |series= Lecture Notes in Mathematics|volume= 1629|publisher= Springer-Verlag|publication-place= Berlin|year= 2001|pages= viii+121 | isbn= 3-540-41221-2
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| |doi=10.1007/BFb0092948|mr= 1830497 }}
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| *{{springer|id=S/s120080|last=Nash|first=Ch.|title=Seiberg-Witten equations}}
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| *{{citation|last= Nicolaescu|first= Liviu I. |title=Notes on Seiberg-Witten theory|series=Graduate Studies in Mathematics|volume= 28|publisher= American Mathematical Society|publication-place= Providence, RI|year= 2000|pages= xviii+484| isbn= 0-8218-2145-8
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| |url=http://www.nd.edu/~lnicolae/swnotes.pdf|mr= 1787219}}
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| * {{citation
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| |last= Scorpan
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| |first= Alexandru
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| |year= 2005
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| |title= The wild world of 4-manifolds
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| |publisher= [[American Mathematical Society]]
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| |isbn= 978-0-8218-3749-8
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| |mr= 2136212
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| }}.
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| *{{citation
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| |last=Seiberg|first= N.|authorlink1=Nathan Seiberg|last2= Witten|first2= E. |authorlink2=Edward Witten|title=Electric-magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory
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| |journal= Nuclear Phys. B |volume= 426 |year=1994a|issue= 1|pages=19–52
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| |doi=10.1016/0550-3213(94)90124-4
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| |mr=1293681 }} {{citation
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| |title=Erratum
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| |journal= Nuclear Phys. B |volume= 430 |year=1994|issue= 2|pages=485–486|doi=10.1016/0550-3213(94)00449-8
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| |mr=1303306}}
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| *{{citation|id=|last=Seiberg|first= N.|authorlink1=Nathan Seiberg|last2= Witten|first2= E. |authorlink2=Edward Witten|title=Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD
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| |journal= Nuclear Phys. B |volume= 431 |year=1994b|issue= 3|pages=484–550
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| |doi=10.1016/0550-3213(94)90214-3|mr=1306869 }}
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| *{{citation|last= Taubes|first= Clifford Henry|authorlink=Clifford Taubes|title= Seiberg Witten and Gromov invariants for symplectic 4-manifolds|editor-first= Richard|editor-last= Wentworth|series= First International Press Lecture Series|volume= 2|publisher= International Press|publication-place=Somerville, MA|year= 2000|pages= vi+401 | isbn= 1-57146-061-6|mr= 1798809 }}
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| *{{citation|last= Witten|first= Edward |title=Monopoles and four-manifolds. |journal= Mathematical Research Letters |volume= 1 |year=1994|issue= 6|pages= 769–796|url=http://intlpress.com/site/pub/pages/journals/items/mrl/content/vols/0001/0006/00019390/index.html|mr= 1306021}}
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| {{DEFAULTSORT:Seiberg-Witten invariant}}
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| [[Category:4-manifolds]]
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| [[Category:Partial differential equations]]
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Hello! My name is Yong.
It is a little about myself: I live in France, my city of La Possession.
It's called often Eastern or cultural capital of . I've married 2 years ago.
I have 2 children - a son (Ashlee) and the daughter (Morris). We all like College football.
Here is my page :: granite countertops in Ottawa