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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.  
 
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}}.
 
==Spin<sup>''c''</sup>-structures==
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
 
:(''U''(1)&times;Spin(4))/('''Z'''/2'''Z'''),  
 
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).
 
==Seiberg–Witten equations==
Fix a smooth compact 4-manifold ''M'', choose a spin<sup>''c''</sup>-structure ''s'' on ''M'', and write ''W''<sup>+</sup>, ''W''<sup>&minus;</sup> for the associated [[spinor bundle]]s, and ''L'' for the [[determinant line bundle]]. Write &phi; for a self-dual spinor field (a section of ''W''<sup>+</sup>) and ''A'' for a U(1) connection on ''L''.  
The Seiberg–Witten equations for (&phi;,''A'') are
:<math>D^A\phi=0</math>
:<math>F^+_A=\sigma(\phi) + i\omega</math>
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 &sigma; is the squaring map from ''W''<sup>+</sup> to imaginary self-dual 2-forms and <math>\omega</math>
is a real selfdual two form, often taken to be zero or harmonic.  
 
The solutions (&phi;,''A'') to the Seiberg–Witten equations are called '''monopoles''', as these equations are the [[field equations]] of massless [[magnetic monopoles]] on the manifold ''M''.
 
==The moduli space of solutions==
The space of solutions is acted on by the gauge group, and the quotient by this action is called the '''moduli space''' of monopoles.  
 
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
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'')&ge;1 then the moduli space is a (possibly empty) manifold for generic metrics. Moreover all components have dimension
:<math>(c_1(s)^2-2\chi(M)-3sign(M))/4.</math>
 
The moduli space is empty for all but a finite number of spin<sup>''c''</sup> structures ''s'', and is always compact.
 
A manifold ''M'' is said to be of '''simple type''' if the moduli space is finite for all ''s''.  
The '''simple type conjecture''' states that if ''M'' is simply connected and ''b''<sub>2</sub><sup>+</sup>(''M'')&ge;2 then the moduli space is finite. It is true for symplectic manifolds.  
If ''b''<sub>2</sub><sup>+</sup>(''M'')=1 then there are examples of manifolds with moduli spaces of arbitrarily high dimension.
 
==Seiberg–Witten invariants==
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.
 
If the manifold ''M'' has a metric of positive scalar curvature and ''b''<sub>2</sub><sup>+</sup>(''M'')&ge;2 then all Seiberg–Witten invariants of ''M'' vanish.
 
If the manifold ''M'' is the connected sum of two manifolds both of which have ''b''<sub>2</sub><sup>+</sup>&ge;1 then all Seiberg–Witten invariants of ''M'' vanish.
 
If the manifold ''M'' is simply connected and symplectic and ''b''<sub>2</sub><sup>+</sup>(''M'')&ge;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>&ge;1.
 
==References==
*{{citation
|last=Donaldson|first= S. K. |authorlink=Simon Donaldson
|title=The Seiberg-Witten equations and 4-manifold topology. 
|journal=Bull. Amer. Math. Soc. (N.S.)  |volume=33  |year=1996|issue= 1|pages= 45–70
|url=http://www.ams.org/bull/1996-33-01/S0273-0979-96-00625-8/home.html|doi=10.1090/S0273-0979-96-00625-8
|mr=1339810}}
*{{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}}
*{{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}}
*{{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
|doi=10.1007/BFb0092948|mr= 1830497 }}
*{{springer|id=S/s120080|last=Nash|first=Ch.|title=Seiberg-Witten equations}}
*{{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
|url=http://www.nd.edu/~lnicolae/swnotes.pdf|mr= 1787219}}
* {{citation
|last= Scorpan
|first= Alexandru
|year= 2005
|title= The wild world of 4-manifolds
|publisher= [[American Mathematical Society]]
|isbn= 978-0-8218-3749-8
|mr= 2136212
}}.
*{{citation
|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
|journal= Nuclear Phys. B |volume= 426  |year=1994a|issue= 1|pages=19–52
|doi=10.1016/0550-3213(94)90124-4
|mr=1293681 }} {{citation
|title=Erratum
|journal= Nuclear Phys. B |volume= 430  |year=1994|issue= 2|pages=485–486|doi=10.1016/0550-3213(94)00449-8
|mr=1303306}}
*{{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
|journal= Nuclear Phys. B |volume= 431  |year=1994b|issue= 3|pages=484–550
|doi=10.1016/0550-3213(94)90214-3|mr=1306869 }}
*{{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 }}
*{{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}}
 
{{DEFAULTSORT:Seiberg-Witten invariant}}
[[Category:4-manifolds]]
[[Category:Partial differential equations]]

Revision as of 19:59, 26 February 2014

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