Seiberg–Witten theory
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.
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).
In general, effective Lagrangians of supersymmetric gauge theories are largely determined by their holomorphic (really, meromorphic) properties and their behavior near the singularities.
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
The potential must vanish on the moduli space of vacua by definition, but the
can be gauge rotated into the Cartan subalgebra, making it a traceless diagonal complex matrix
no longer have vanishing vacuum expectation value, other fields become massive due to the Higgs mechanism (spontaneous symmetry breaking).
Its two-derivative, four-fermions low-energy action is given by a Lagrangian which can be expressed in terms of a single holomorphic function
The first term is a perturbative loop calculation and the second is the instanton part where
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
and its dual can be expressed as periods of a meromorphic differential on a Riemann surface called the Seiberg–Witten curve.
Before the low energy, or infrared, limit is taken, the action can be given in terms of a Lagrangian over
vector/chiral superfield in the adjoint representation of the gauge group, and a holomorphic function
solving the equations of motion of the low-energy Lagrangian, for which the scalar part
is traceless and diagonalizable so can be gauge rotated to (is in the conjugacy class of) a matrix of the form
give conjugate matrices (corresponding to the fact the Weyl group of
The (classical) moduli space of vacua is a one-dimensional complex manifold (Riemann surface) parametrized by
This is not invariant under an arbitrary change of coordinates, but due to symmetry in
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).
, with prescribed monodromy data at each point derived from quantum field theoretic arguments, the moduli space
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).
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.
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.
Using supersymmetric localisation techniques, one can explicitly determine the instanton partition function of
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.
can be expressed as a sum over instanton number of the products and ratios of fermionic and bosonic determinants, producing the so-called Nekrasov partition function.
