= Seiberg–Witten equations
{c}
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 $\psi\in\Gamma(W^+)$:
$$
D_A\psi=0,\qquad F_A^+=q(\psi).
$$
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.<Monopoles and Four-Manifolds by Edward Witten>[Section2, monopole equations and their moduli spaces]{ref}
Back to article page
