Seiberg–Witten equations

On an oriented Riemannian smooth four-manifold with a Spin-c structure, let $W^{\pm }$ be the spinor bundles and $L=\det W^{+}$ their determinant line. The Seiberg–Witten equations couple a unitary connection $A$ on $L$ and a positive spinor $\psi \in \Gamma (W^{+})$:Here $D_A:\Gamma (W^{+})\to \Gamma (W^{-})$ is the Dirac operator, $F_A^{+}$ is the self-dual part of the curvature, and $q$ is the quadratic spinor map into imaginary self-dual two-forms. Numerical factors and signs in $q$ depend on the Clifford conventions; a perturbation adds a prescribed imaginary self-dual two-form.

The first equation is first-order and elliptic, while the curvature equation makes the coupled system nonlinear. Gauge transformations identify equivalent connection-spinor pairs. Solutions with $\psi =0$ are reducible and satisfy $F_A^{+}=0$ in the unperturbed system.

On closed four-manifolds, suitably perturbed moduli spaces yield the Seiberg–Witten invariant of a four-manifold. The abelian gauge equations arose from the monopole description of supersymmetric gauge theory and provide tools for distinguishing smooth four-dimensional geometry.^[ref]