SeibergWitten

Seiberg-Witten theory refers to a framework of exact results for four-dimensional N=2 supersymmetric gauge theories, formulated by Nathan Seiberg and Edward Witten in 1994. It provides the exact low-energy effective action on the Coulomb branch, where the nonabelian gauge group is generically broken to a abelian subgroup and the physics is captured by a holomorphic prepotential F.

In the canonical SU(2) case, the low-energy theory is described by a complex parameter u that coordinates the moduli space of vacua and by a pair of special functions a(u) and aD(u) obtained from a Seiberg-Witten curve and a Seiberg-Witten differential. The effective gauge coupling is encoded in the complexified parameter τ = daD/da, and all low-energy couplings, masses, and the BPS spectrum can be read from the periods of the differential over the curve. Concretely, the theory is organized around an elliptic curve (the Seiberg-Witten curve) together with a meromorphic differential; the electric and magnetic variables are given by period integrals a(u) = ∮A λSW and aD(u) = ∮B λSW.

Key physical insights include the existence of singularities on the Coulomb branch where certain states become massless, such as monopoles or dyons, signaling electric-magnetic duality. Near these points, a dual description in terms of light magnetic degrees of freedom emerges, which explains phenomena akin to confinement via monopole condensation in a supersymmetric context. The Seiberg-Witten solution extends to larger gauge groups and theories with matter, revealing a rich structure of dualities and nonperturbative effects.

Beyond physics, the framework inspired mathematical developments, notably Seiberg-Witten invariants, and influenced the study of four-manifolds alongside Donaldson theory, linking quantum field theory to geometry.