View source for Seiberg–Witten theory

{{Short description|Theory in supersymmetric gauge theory}}
{{For|applications to 4-manifolds|Seiberg–Witten invariants}}
In [[theoretical physics]], '''Seiberg–Witten theory''' is an <math>\mathcal{N} = 2</math> [[supersymmetry|supersymmetric]] [[gauge theory]] with an exact [[renormalization|low-energy]] [[effective action]] (for massless degrees of freedom), of which the kinetic part coincides with the [[Kähler potential]] of the [[moduli space of vacua]]. Before taking the low-energy effective action, the theory is known as <math>\mathcal{N} = 2</math> '''supersymmetric Yang–Mills theory''', as the field content is a single <math>\mathcal{N} = 2</math> vector [[supermultiplet]], analogous to the field content of [[Yang–Mills theory]] being a single vector [[gauge field]] (in particle theory language) or [[principal connection|connection]] (in geometric language).

The theory was studied in detail by [[Nathan Seiberg]] and [[Edward Witten]] {{harvs|last1=Seiberg|last2=Witten|year=1994}}.

==Seiberg–Witten curves==
In general, effective [[Lagrangian (field theory)|Lagrangian]]s of supersymmetric gauge theories are largely determined by their [[holomorphic]] (really, [[meromorphic]]) properties and their behavior near the [[zeros and poles|singularities]]. In [[gauge theory]] with <math>\mathcal{N} = 2</math> [[extended supersymmetry]], the moduli space of vacua is a special [[Kähler manifold]] and its Kähler potential is constrained by above conditions.

In the original approach,<ref>{{Cite journal |arxiv = hep-th/9407087|doi = 10.1016/0550-3213(94)90124-4|title = Electric - magnetic duality, monopole condensation, and confinement in N=2 supersymmetric Yang-Mills theory|year = 1994|last1 = Seiberg|first1 = Nathan|last2 = Witten|first2 = Edward|journal = Nucl. Phys. B|volume = 426|issue = 1|pages = 19–52|bibcode = 1994NuPhB.426...19S|s2cid = 14361074}}</ref><ref>{{Cite journal |arxiv = hep-th/9408099|doi = 10.1016/0550-3213(94)90214-3|title = Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD|year = 1994|last1 = Seiberg|first1 = Nathan|last2 = Witten|first2 = Edward|journal = Nucl. Phys. B|volume = 431|issue = 3|pages = 484–550|bibcode = 1994NuPhB.431..484S|s2cid = 17584951}}</ref> by [[Nathan Seiberg|Seiberg]] and [[Edward Witten|Witten]],  holomorphy and [[Montonen–Olive duality|electric-magnetic duality]] constraints are strong enough to almost uniquely 
constrain the prepotential <math>\mathcal{F}</math> (a holomorphic function which defines the theory), and therefore the metric of the moduli space of vacua, for theories with [[SU(2)]] [[gauge group]]. 
 
More generally, consider the example with gauge group [[SU(n)]]. The classical potential is

{{NumBlk|:|<math>V(x) = \frac{1}{g^2} \operatorname{Tr} [\phi , \bar{\phi} ]^2 \,</math>|{{EquationRef|1}}}}

where <math>\phi</math> is a [[scalar field]] appearing in an expansion of superfields in the theory. The potential must vanish on the moduli space of vacua by definition, but the <math>\phi</math> need not. The [[vacuum expectation value]] of <math>\phi</math> can be gauge rotated into the [[Cartan subalgebra]], making it a [[traceless]] diagonal complex matrix <math>a</math>.

Because the fields <math>\phi</math> no longer have vanishing vacuum expectation value, other fields become massive due to the [[Higgs mechanism]] ([[spontaneous symmetry breaking]]).  They are integrated out in order to find the effective <math>\mathcal{N} = 2</math> U(1) gauge theory. Its two-derivative, four-[[fermion]]s low-energy action is given by a [[Lagrangian (field theory)|Lagrangian]] which can be expressed in terms of a single holomorphic function <math>\mathcal{F}</math> on <math>\mathcal{N} = 1</math> [[superspace]] as follows:

{{NumBlk|:| <math>\mathcal{L}_{\text{eff}}^{\mathrm{U}(1)} = \frac{1}{4\pi} \operatorname{Im} \left[ \int d^4 \theta \frac{d\mathcal{F}}{dA} \bar{A} + \int d^2 \theta \frac{1}{2} \frac{d^2 \mathcal{F}}{dA^2} W_\alpha W^\alpha \right]  \,</math>|{{EquationRef|3}}}}

where

{{NumBlk|:|<math>\mathcal{F} = \frac{i}{2\pi} \mathcal{A}^2 \operatorname{\ln}\frac{\mathcal{A}^2}{\Lambda^2} + \sum_{k=1}^\infty \mathcal{F}_k \frac{\Lambda^{4k}}{\mathcal{A}^{4k}} \mathcal{A}^2 \,</math>|{{EquationRef|4}}}}

and <math>A</math> is a [[chiral superfield]] on <math>\mathcal{N} = 1</math> superspace which fits inside the <math>\mathcal{N} = 2</math> chiral multiplet <math>\mathcal{A}</math>.

The first term is a perturbative loop calculation and the second is the [[Instanton#4d supersymmetric gauge theories|instanton]] part where <math>k</math> labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups, <math>\mathcal{F}</math> can be computed exactly using [[supersymmetric localization|localization]]<ref>{{Cite journal |arxiv = hep-th/0206161|doi = 10.4310/ATMP.2003.v7.n5.a4|title = Seiberg-Witten Prepotential from Instanton Counting|year = 2004|last1 = Nekrasov|first1 = Nikita|journal = Advances in Theoretical and Mathematical Physics|volume = 7|number = 5|pages = 831–864|s2cid = 2285041}}</ref> and the limit shape techniques.<ref>{{Cite journal |arxiv = hep-th/0306238|doi = 10.1007/0-8176-4467-9_15|title = Seiberg-Witten theory and random partitions|year = 2003|last1 = Nekrasov|first1 = Nikita|last2 = Okounkov|first2 = Andrei| journal = Prog. Math.|series = Progress in Mathematics|volume = 244|pages = 525–596|bibcode = 2003hep.th....6238N|isbn = 978-0-8176-4076-7|s2cid = 14329429}}</ref>

The Kähler potential is the kinetic part of the low energy action, and explicitly is written in terms of <math>\mathcal{F}</math> as

{{NumBlk|:|<math>K(A, \bar A) = \mathrm{Im}\left(\frac{\partial\mathcal{F}}{\partial A}\bar A\right).</math>|{{EquationRef|5}}}}

From <math>\mathcal{F}</math> we can get the mass of the [[Bogomol'nyi-Prasad-Sommerfield bound#Supersymmetry|BPS]] particles.

{{NumBlk|:|<math>M \approx |na+ma_D| \,</math>|{{EquationRef|6}}}}
{{NumBlk|:|<math> a_D = \frac{d\mathcal{F}}{da} \,</math>|{{EquationRef|7}}}}

One way to interpret this is that these variables <math>a</math> and its dual can be expressed as [[Period mapping#The case of elliptic curves|periods]] of a meromorphic [[one-form|differential]] on a [[Riemann surface]] called the Seiberg–Witten curve.

== ''N'' = 2 supersymmetric Yang–Mills theory ==
Before the low energy, or infrared, limit is taken, the action can be given in terms of a Lagrangian over <math>\mathcal{N} = 2</math> superspace with field content <math>\Psi</math>, which is a single <math>\mathcal{N} = 2</math> vector/chiral superfield in the adjoint representation of the gauge group, and a holomorphic function <math>\mathcal{F}</math> of <math>\Psi</math> called the prepotential. Then the Lagrangian is given by
<math display = block>\mathcal{L}_{SYM2} = \mathrm{Im}\mathrm{Tr}\left(\frac{1}{4\pi}\int d^2\theta d^2 \vartheta \mathcal{F}(\Psi)\right)</math>
where <math>\theta, \vartheta</math> are coordinates for the spinor directions of superspace.<ref>{{cite journal |last1=Seiberg |first1=Nathan |title=Supersymmetry and non-perturbative beta functions |journal=Physics Letters B |date=May 1988 |volume=206 |issue=1 |pages=75–80 |doi=10.1016/0370-2693(88)91265-8|bibcode=1988PhLB..206...75S }}</ref> Once the low energy limit is taken, the <math>\mathcal{N} = 2</math> superfield <math>\Psi</math> is typically labelled by <math>\mathcal{A}</math> instead.

The so called '''minimal theory''' is given by a specific choice of <math>\mathcal{F}</math>,
<math display = block>\mathcal{F}(\psi) = \frac{1}{2}\tau \Psi^2,</math>
where <math>\tau</math> is the complex coupling constant.

The minimal theory can be written on Minkowski spacetime as
<math display = block> \mathcal{L} = \frac{1}{g^2}\mathrm{Tr}\left(-\frac{1}{4}F_{\mu\nu}F^{\mu\nu} + g^2 \frac{\theta}{32\pi^2}F_{\mu\nu}*F^{\mu\nu} + (D_\mu\phi)^\dagger(D^\mu\phi) - \frac{1}{2}[\phi, \phi^\dagger]^2 - i\lambda\sigma^\mu D_\mu \bar \lambda - i \bar \psi \bar \sigma^\mu D_\mu \psi - i \sqrt{2}[\lambda, \psi]\phi^\dagger - i\sqrt{2}[\bar \lambda, \bar \psi] \phi\right)</math>
with <math>A_\mu, \lambda, \psi, \phi</math> making up the <math>\mathcal{N} = 2</math> chiral multiplet.

== Geometry of the moduli space ==
For this section fix the gauge group as <math>\mathrm{SU(2)}</math>. A low-energy vacuum solution is an <math>\mathcal{N} = 2</math> vector superfield <math>\mathcal{A}</math> solving the equations of motion of the low-energy Lagrangian, for which the scalar part <math>\phi</math> has vanishing potential, which as mentioned earlier holds if <math>[\phi, \phi^\dagger] = 0</math> (which exactly means <math>\phi</math> is a [[normal operator]], and therefore diagonalizable). The scalar <math>\phi</math> transforms in the adjoint, that is, it can be identified as an element of <math>\mathfrak{su}(2)_\mathbb{C} \cong \mathfrak{sl}(2, \mathbb{C})</math>, the [[complexification]] of <math>\mathfrak{su}(2)</math>. Thus <math>\phi</math> is traceless and diagonalizable so can be gauge rotated to (is in the [[conjugacy class]] of) a matrix of the form <math>\frac{1}{2}a\sigma_3</math> (where <math>\sigma_3</math> is the third [[Pauli matrix]]) for <math>a \in \mathbb{C}</math>. However, <math>a</math> and <math>-a</math> give conjugate matrices (corresponding to the fact the [[Weyl group]] of <math>\mathrm{SU}(2)</math> is <math>\mathbb{Z}_2</math>) so both label the same vacuum. Thus the gauge invariant quantity labelling inequivalent vacua is <math>u = a^2/2 = \mathrm{Tr}\phi^2</math>. The (classical) moduli space of vacua is a one-dimensional complex manifold (Riemann surface) parametrized by <math>u</math>, although the Kähler metric is given in terms of <math>a</math> as
<math display=block>ds^2 = \mathrm{Im}\frac{\partial^2 \mathcal{F}}{\partial a^2}dad\bar a = \mathrm{Im}da_Dd\bar a = -\frac{i}{2}(da_D d\bar a - da d\bar a_D) =: \mathrm{Im}\tau(a)dad\bar a,</math>

where <math>a_D = \frac{\partial \mathcal{F}}{\partial a}</math>. This is not invariant under an arbitrary change of coordinates, but due to symmetry in <math>a</math> and <math>a_D</math>, switching to local coordinate <math>a_D</math> gives a metric similar to the final form but with a different harmonic function replacing <math>\mathrm{Im}\tau(a)</math>. The switching of the two coordinates can be interpreted as an instance of electric-magnetic duality {{harvs|last1=Seiberg|last2=Witten|year=1994}}.

Under a minimal assumption of assuming there are only three singularities in the moduli space at <math>u = -1, +1</math> and <math>\infty</math>, with prescribed [[monodromy]] data at each point derived from quantum field theoretic arguments, the moduli space <math>\mathcal{M}</math> was found to be <math>H/\Gamma(2)</math>, where <math>H</math> is the [[Poincaré half-plane model|hyperbolic half-plane]] and <math>\Gamma(2) < \mathrm{SL}(2, \mathbb{Z})</math> is the second [[principal congruence subgroup]], the subgroup of matrices congruent to 1 mod 2, generated by
<math display = block>M_\infty = \begin{pmatrix} -1 & 2 \\ 0 & -1\end{pmatrix}, M_1 = \begin{pmatrix} 1 & 0 \\ -2 & 1\end{pmatrix}, M_{-1} = \begin{pmatrix} -1 & 2 \\ -2 & 3\end{pmatrix}.</math>
This space is a six-fold cover of the [[fundamental domain]] of the [[modular group]] and admits an explicit description as parametrizing a space of [[elliptic curve]]s <math>E_u</math> given by the vanishing of
<math display=block> y^2 = (x - 1)(x + 1)(x - u), </math>
which are the '''Seiberg–Witten curves'''. The curve becomes singular precisely when <math>u = -1, +1</math> or <math>\infty</math>.
[[File:Metric on Seiberg–Witten moduli space.jpg|none|thumb|Graph of metric function <math>\mathrm{Im}\tau(u)</math> on moduli space parametrized by <math>u</math>, with evident singularities at <math>u = -1, +1</math>. The function <math>\tau</math> is defined using the complete [[elliptic integral]] of the first kind {{harvs|last=Hunter-Jones|year=2012}}.|500x500px]]

== Monopole condensation and confinement ==
The theory exhibits physical phenomena involving and linking [[magnetic monopoles]], [[color confinement|confinement]], an attained [[mass gap]] and strong-weak duality, described in section 5.6 of {{harvs|txt|last1=Seiberg|last2=Witten|year=1994}}. The study of these physical phenomena also motivated the theory of [[Seiberg–Witten invariants]].

The low-energy action is described by the <math>\mathcal{N} = 2</math> chiral multiplet <math>\mathcal{A}</math> with gauge group <math>\mathrm{U}(1)</math>, the residual unbroken gauge from the original <math>\mathrm{SU}(2)</math> symmetry. This description is weakly coupled for large <math>u</math>, but strongly coupled for small <math>u</math>. However, at the strongly coupled point the theory admits a dual description which is weakly coupled. The dual theory has different field content, with two <math>\mathcal{N} = 1</math> chiral superfields <math>M, \tilde M</math>, and gauge field the dual photon <math>\mathcal{A}_D</math>, with a potential that gives equations of motion which are Witten's '''monopole equations''', also known as the [[Seiberg–Witten equation]]s at the critical points <math>u = \pm u_0</math> where the monopoles become massless.

In the context of Seiberg–Witten invariants, one can view [[Donaldson invariant]]s as coming from a twist of the original theory at <math>u = \infty</math> giving a [[topological field theory]]. On the other hand, Seiberg–Witten invariants come from twisting the dual theory at <math>u = \pm u_0</math>. In theory, such invariants should receive contributions from all finite <math>u</math> but in fact can be localized to the two critical points, and topological invariants can be read off from solution spaces to the monopole equations.<ref>{{cite journal |last1=Witten |first1=Edward |title=Monopoles and four-manifolds |journal=Mathematical Research Letters |date=1994 |volume=1 |issue=6 |pages=769–796 |doi=10.4310/MRL.1994.v1.n6.a13|arxiv=hep-th/9411102 |bibcode=1994MRLet...1..769W }}</ref>

==Relation to integrable systems==
The special Kähler geometry on the moduli space of vacua in Seiberg–Witten theory can be identified with the geometry of the base of complex completely [[integrable system]]. The total phase of this complex completely integrable system can be identified with the moduli space of vacua of the 4d theory compactified on a circle. The relation between Seiberg–Witten theory and integrable systems has been reviewed by [[Eric D'Hoker]] and [[Duong Hong Phong|D. H. Phong]].<ref>{{cite book|last1=D'Hoker|first1=Eric|last2=Phong|first2=D. H.|title=Theoretical Physics at the End of the Twentieth Century |date=1999-12-29|chapter=Lectures on Supersymmetric Yang-Mills Theory and Integrable Systems|pages=1–125 |doi=10.1007/978-1-4757-3671-7_1 |arxiv=hep-th/9912271|bibcode=1999hep.th...12271D|isbn=978-1-4419-2948-8 |s2cid=117202391}}</ref> See [[Hitchin system]].

==Seiberg–Witten prepotential via instanton counting==
Using supersymmetric localisation techniques, one can explicitly determine the instanton [[partition function (mathematics)|partition function]] of <math>\mathcal{N}=2</math> super Yang–Mills theory. The Seiberg–Witten prepotential can then be extracted using the localization approach<ref>{{Cite journal |arxiv = hep-th/0206161|doi = 10.4310/ATMP.2003.v7.n5.a4|title = Seiberg-Witten Prepotential from Instanton Counting|year = 2004|last1 = Nekrasov|first1 = Nikita|journal = Advances in Theoretical and Mathematical Physics|volume = 7|number = 5|pages = 831–864|s2cid = 2285041}}</ref> of [[Nikita Nekrasov]]. It arises in the flat space limit <math>\varepsilon_{1}</math>, <math>\varepsilon_{2} \to 0</math>, of the partition function of the theory subject to the so-called <math>\Omega</math>-background. 
The latter is a specific background of four dimensional <math>\mathcal{N}=2</math> [[supergravity]]. It can be engineered, formally by lifting the [[Supersymmetric gauge theory#Theories with 8 or more SUSY generators (N > 1)|super Yang–Mills theory]] to six dimensions, then compactifying on 2-[[torus]], while twisting the four dimensional spacetime around the two non-[[contractible]] cycles. In addition, one twists fermions so as to produce covariantly constant [[spinor]]s generating unbroken supersymmetries. The two parameters <math>\varepsilon_{1}</math>, <math>\varepsilon_{2}</math>
of the <math>\Omega</math>-background correspond to the angles of the spacetime rotation.

In Ω-background, all the non-zero modes can be integrated out, so the path integral with the [[boundary condition]] <math>\phi \to a </math> at <math> x \to \infty</math>  
can be expressed as a sum over instanton number of the products and ratios of fermionic and [[boson]]ic [[determinant]]s, producing the so-called [[Nikita Nekrasov|Nekrasov]] partition function.
In the limit where <math>\varepsilon_{1}</math>, <math>\varepsilon_{2}</math> approach 0, this sum is dominated by a unique [[saddle point]].  
On the other hand, when <math>\varepsilon_{1}</math>, <math>\varepsilon_{2}</math> approach 0,
{{NumBlk|:|<math> Z(a;\varepsilon_{1},\varepsilon_{2},\Lambda)=\exp \left( -\frac{1}{\varepsilon_{1}\varepsilon_{2}}\left(\mathcal{F}(a;\Lambda)+\mathcal{O}(\varepsilon_{1},\varepsilon_{2}) \right) \right)\,</math>|{{EquationRef|8}}}}
holds.

==See also==
* [[Ginzburg–Landau theory]]
*[[Donaldson theory]]
<!-- redlinks 
* [[Argyres-–Seiberg duality]]
* [[Giotto duality]]
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==References==
{{Reflist}}
*{{cite book |first=Jürgen |last=Jost |author-link=Jürgen Jost|title=Riemannian Geometry and Geometric Analysis |year=2002 |publisher=Springer-Verlag |isbn=3-540-42627-2 }} (''See Section 7.2'')
*{{cite thesis |type=Masters |last=Hunter-Jones |first=Nicholas R. |date=September 2012 |title=Seiberg–Witten Theory and Duality in N = 2 Supersymmetric Gauge Theories |publisher=Imperial College London |url=https://www.imperial.ac.uk/media/imperial-college/research-centres-and-groups/theoretical-physics/msc/dissertations/2012/Nick-Hunter-Jones-Dissertation.pdf}}

{{Supersymmetry topics}}
{{Quantum field theories}}

{{DEFAULTSORT:Seiberg-Witten theory}}
[[Category:Supersymmetric quantum field theory]]
[[Category:Gauge theories]]