Seiberg–Witten theory
In theoretical physics, Seiberg–Witten theory is an supersymmetric gauge theory with an exact lowenergy 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 lowenergy effective action, the theory is known as supersymmetric Yang–Mills theory, as the field content is a single vector supermultiplet, analogous to the field content of Yang–Mills theory being a single vector gauge field (in particle theory language) or connection (in geometric language).
The theory was studied in detail by Nathan Seiberg and Edward Witten (Seiberg & Witten 1994).
Seiberg–Witten curves[edit]
In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities. In gauge theory with 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,^{[1]}^{[2]} by Seiberg and Witten, holomorphy and electricmagnetic duality constraints are strong enough to almost uniquely constrain the prepotential (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

(1)
where 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 need not. The vacuum expectation value of can be gauge rotated into the Cartan subalgebra, making it a traceless diagonal complex matrix .
Because the fields 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 U(1) gauge theory. Its twoderivative, fourfermions lowenergy action is given by a Lagrangian which can be expressed in terms of a single holomorphic function on superspace as follows:

(3)
where

(4)
and is a chiral superfield on superspace which fits inside the chiral multiplet .
The first term is a perturbative loop calculation and the second is the instanton part where labels fixed instanton numbers. In theories whose gauge groups are products of unitary groups, can be computed exactly using localization^{[3]} and the limit shape techniques.^{[4]}
The Kähler potential is the kinetic part of the low energy action, and explicitly is written in terms of as

(5)
From we can get the mass of the BPS particles.

(6)

(7)
One way to interpret this is that these variables and its dual can be expressed as periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve.
N = 2 supersymmetric Yang–Mills theory[edit]
Before the low energy, or infrared, limit is taken, the action can be given in terms of a Lagrangian over superspace with field content , which is a single vector/chiral superfield in the adjoint representation of the gauge group, and a holomorphic function of called the prepotential. Then the Lagrangian is given by
The so called minimal theory is given by a specific choice of ,
The minimal theory can be written on Minkowski spacetime as
Geometry of the moduli space[edit]
For this section fix the gauge group as . A lowenergy vacuum solution is an vector superfield solving the equations of motion of the lowenergy Lagrangian, for which the scalar part has vanishing potential, which as mentioned earlier holds if (which exactly means is a normal operator, and therefore diagonalizable). The scalar transforms in the adjoint, that is, it can be identified as an element of , the complexification of . Thus is traceless and diagonalizable so can be gauge rotated to (is in the conjugacy class of) a matrix of the form (where is the third Pauli matrix) for . However, and give conjugate matrices (corresponding to the fact the Weyl group of is ) so both label the same vacuum. Thus the gauge invariant quantity labelling inequivalent vacua is . The (classical) moduli space of vacua is a onedimensional complex manifold (Riemann surface) parametrized by , although the Kähler metric is given in terms of as
where . This is not invariant under an arbitrary change of coordinates, but due to symmetry in and , switching to local coordinate gives a metric similar to the final form but with a different harmonic function replacing . The switching of the two coordinates can be interpreted as an instance of electricmagnetic duality (Seiberg & Witten 1994).
Under a minimal assumption of assuming there are only three singularities in the moduli space at and , with prescribed monodromy data at each point derived from quantum field theoretic arguments, the moduli space was found to be , where is the hyperbolic halfplane and is the second principal congruence subgroup, the subgroup of matrices congruent to 1 mod 2, generated by
Monopole condensation and confinement[edit]
The theory exhibits physical phenomena involving and linking magnetic monopoles, confinement, an attained mass gap and strongweak duality, described in section 5.6 of Seiberg and Witten (1994). The study of these physical phenomena also motivated the theory of Seiberg–Witten invariants.
The lowenergy action is described by the chiral multiplet with gauge group , the residual unbroken gauge from the original symmetry. This description is weakly coupled for large , but strongly coupled for small . 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 chiral superfields , and gauge field the dual photon , with a potential that gives equations of motion which are Witten's monopole equations, also known as the Seiberg–Witten equations at the critical points where the monopoles become massless.
In the context of Seiberg–Witten invariants, one can view Donaldson invariants as coming from a twist of the original theory at giving a topological field theory. On the other hand, Seiberg–Witten invariants come from twisting the dual theory at . In theory, such invariants should receive contributions from all finite 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.^{[6]}
Relation to integrable systems[edit]
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 D. H. Phong.^{[7]} See Hitchin system.
Seiberg–Witten prepotential via instanton counting[edit]
Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of super Yang–Mills theory. The Seiberg–Witten prepotential can then be extracted using the localization approach^{[8]} of Nikita Nekrasov. It arises in the flat space limit , , of the partition function of the theory subject to the socalled background. The latter is a specific background of four dimensional supergravity. It can be engineered, formally by lifting the super Yang–Mills theory to six dimensions, then compactifying on 2torus, while twisting the four dimensional spacetime around the two noncontractible cycles. In addition, one twists fermions so as to produce covariantly constant spinors generating unbroken supersymmetries. The two parameters , of the background correspond to the angles of the spacetime rotation.
In Ωbackground, all the nonzero modes can be integrated out, so the path integral with the boundary condition at can be expressed as a sum over instanton number of the products and ratios of fermionic and bosonic determinants, producing the socalled Nekrasov partition function. In the limit where , approach 0, this sum is dominated by a unique saddle point. On the other hand, when , approach 0,

(8)
holds.
See also[edit]
References[edit]
 ^ Seiberg, Nathan; Witten, Edward (1994). "Electric  magnetic duality, monopole condensation, and confinement in N=2 supersymmetric YangMills theory". Nucl. Phys. B. 426 (1): 19–52. arXiv:hepth/9407087. Bibcode:1994NuPhB.426...19S. doi:10.1016/05503213(94)901244. S2CID 14361074.
 ^ Seiberg, Nathan; Witten, Edward (1994). "Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD". Nucl. Phys. B. 431 (3): 484–550. arXiv:hepth/9408099. Bibcode:1994NuPhB.431..484S. doi:10.1016/05503213(94)902143. S2CID 17584951.
 ^ Nekrasov, Nikita (2004). "SeibergWitten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hepth/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041.
 ^ Nekrasov, Nikita; Okounkov, Andrei (2003). "SeibergWitten theory and random partitions". Prog. Math. Progress in Mathematics. 244: 525–596. arXiv:hepth/0306238. Bibcode:2003hep.th....6238N. doi:10.1007/0817644679_15. ISBN 9780817640767. S2CID 14329429.
 ^ Seiberg, Nathan (May 1988). "Supersymmetry and nonperturbative beta functions". Physics Letters B. 206 (1): 75–80. doi:10.1016/03702693(88)912658.
 ^ Witten, Edward (1994). "Monopoles and fourmanifolds". Mathematical Research Letters. 1 (6): 769–796. doi:10.4310/MRL.1994.v1.n6.a13.
 ^ D'Hoker, Eric; Phong, D. H. (19991229). "Lectures on Supersymmetric YangMills Theory and Integrable Systems". Theoretical Physics at the End of the Twentieth Century. pp. 1–125. arXiv:hepth/9912271. Bibcode:1999hep.th...12271D. doi:10.1007/9781475736717_1. ISBN 9781441929488. S2CID 117202391.
 ^ Nekrasov, Nikita (2004). "SeibergWitten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hepth/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041.
 Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. SpringerVerlag. ISBN 3540426272. (See Section 7.2)
 HunterJones, Nicholas R. (September 2012). Seiberg–Witten Theory and Duality in N = 2 Supersymmetric Gauge Theories (Masters). Imperial College London.