In theoretical physics, Seiberg–Witten theory is an supersymmetric gauge theory with an exact 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 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 curves
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, 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
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:
where
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 and the limit shape techniques.
The Kähler potential is the kinetic part of the low energy action, and explicitly is written in terms of as
From we can get the mass of the BPS particles.
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. 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