In the original approach,[1][2] by
Seiberg and
Witten, holomorphy and
electric-magnetic 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 two-derivative, four-
fermions low-energy 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
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
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
where are coordinates for the spinor directions of superspace.[5] Once the low energy limit is taken, the superfield is typically labelled by instead.
The so called minimal theory is given by a specific choice of ,
where is the complex coupling constant.
The minimal theory can be written on Minkowski spacetime as
with making up the chiral multiplet.
Geometry of the moduli space
For this section fix the gauge group as . A low-energy vacuum solution is an vector superfield solving the equations of motion of the low-energy 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 one-dimensional 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 electric-magnetic 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 half-plane and is the second
principal congruence subgroup, the subgroup of matrices congruent to 1 mod 2, generated by
which are the Seiberg–Witten curves. The curve becomes singular precisely when or .
Monopole condensation and confinement
The theory exhibits physical phenomena involving and linking
magnetic monopoles,
confinement, an attained
mass gap and strong-weak 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 low-energy 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
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
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 so-called -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 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
spinors generating unbroken supersymmetries. The two parameters ,
of the -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 at
can be expressed as a sum over instanton number of the products and ratios of fermionic and
bosonicdeterminants, producing the so-called
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,
Jost, Jürgen (2002). Riemannian Geometry and Geometric Analysis. Springer-Verlag.
ISBN3-540-42627-2. (See Section 7.2)
Hunter-Jones, Nicholas R. (September 2012). Seiberg–Witten Theory and Duality in N = 2 Supersymmetric Gauge Theories (Masters). Imperial College London.