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| {{dablink|For applications to 4-manifolds see [[Seiberg–Witten invariant]]}}
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| {{Refimprove|date=October 2010}}
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| In [[theoretical physics]], '''Seiberg–Witten theory''' is a theory that determines an exact low-energy effective action (for massless degrees of freedom) of a N=2 supersymmetric gauge theory—namely the metric of the [[moduli space]] of vacua.
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| ==Seiberg-Witten curves==
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| In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic properties and their behavior near the singularities.In particular, in [[gauge theory]] with ''N'' = 2 [[extended supersymmetry]], the moduli space of vacua is a special Kahler manifold and its Kahler potential is constrained by above conditions.
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| In the original derivation by [[Seiberg]] and [[Witten]], they extensively used holomorphy and electric-magnetic duality
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| to constrain the prepotential, namely the metric of the moduli space of vacua.
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| Consider the example with gauge group SU(n).The classical potential is:
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| {{NumBlk|:|<math>V(x) = \frac{1}{g^2} \operatorname{Tr} [\phi , \bar{\phi} ]^2 \,</math>|{{EquationRef|1}}}}
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| This must vanish on the moduli space, so vacuum expectation value of &phi can be gauge rotated into Cartan subalgebra,
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| so it is a traceless diagonal complex matrix.
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| Because the fields φ no longer have vanishing [[Vacuum expectation value]]. Because these are now heavy due to the Higgs effect, they should be integrated out in order to find the effective N=2 Abelian gauge theory. This can be expressed in terms of a single holomorphic function F.
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| In terms of this prepotential the Lagrangian can be written in the form:
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| {{NumBlk|:| <math>\frac{1}{4\pi} \operatorname{Im} \Bigl[ \int d^4 \theta \frac{dF}{dA} \bar{A} + \int d^2 \theta \frac{1}{2} \frac{d^2 F}{dA^2} W_\alpha W^\alpha \Bigr] \,</math>|{{EquationRef|3}}}}
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| {{NumBlk|:|<math>F = \frac{i}{2\pi} \mathcal{A}^2 \operatorname{\ln}\frac{\mathcal{A}^2}{\Lambda^2} + \sum_{k=1}^\infty F_k \frac{\Lambda^{4k}}{\mathcal{A}^{4k}} \mathcal{A}^2 \,</math>|{{EquationRef|4}}}}
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| The first term is a perturbative loop calculation and the second is the [[Instanton#4d supersymmetric gauge theories|instanton]] part where k labels fixed instanton numbers.
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| From this we can get the mass of the [[Bogomol%27nyi-Prasad-Sommerfield_bound#Supersymmetry|BPS]] particles.
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| {{NumBlk|:|<math>M \approx |na+ma_D| \,</math>|{{EquationRef|5}}}}
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| {{NumBlk|:|<math> a_D = \frac{dF}{da} \,</math>|{{EquationRef|6}}}}
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| One way to interpret this is that these variables a and its dual can be expressed as [[Period_mapping#The_case_of_elliptic_curves|periods]] of a meromorphic differential on a Riemann surface called the Seiberg-Witten curve.
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| ==Seiberg-Witten prepotential via instanton counting==
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| Consider a N = 1 super Yang-Mills theory in curved 6 dimensional background.
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| After dimensional reduction on 2-torus, we obtain a 4d N = 2 super Yang-Mills theory with additional terms.
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| Turning Wilson lines to compensate holonomies of fermions on the 2-torus, we get 4d N = 2 SYM in Ω-background. Ω has 2 parameters, ε1,ε2, which go 0 in the flat limit.
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| In Ω-background, we can integrate out all the non-zero mode,so the partition function (with the boundary condition \phi →0 at x → ∞)
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| can be expressed as a sum of products and ratios of fermionic and bosonic determinants over instanton number.
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| In the limit where ε1,ε2 approach to 0, this sum is dominated by a unique saddle point.
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| On the other hand, when ε1,ε2 approach to 0,
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| {{NumBlk|:|<math> Z(a;\varepsilon_{1},\varepsilon_{2},\Lambda)=exp(-\frac{1}{\varepsilon_{1}\varepsilon_{2}}(\mathcal{F}(a;\Lambda)+O(\varepsilon_{1},\varepsilon_{2}))\,</math>|{{EquationRef|10}}}}
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| holds.
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| ==See also==
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| *[[Yang–Mills theory]]
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| *[[Argyres-Seiberg duality]]
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| *[[Gaiotto duality]]
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| ==External links==
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| *[http://xstructure.inr.ac.ru/x-bin/revtheme3.py?level=1&index1=284824&skip=0 Seiberg-Witten theory on arxiv.org]
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| *[http://arxiv.org/pdf/hep-th/9407087.pdf Electric-Magnetic Duality, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory]
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| {{DEFAULTSORT:Seiberg-Witten theory}}
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| [[Category:Supersymmetry]]
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| [[Category:Gauge theories]]
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| {{phys-stub}}
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