Seiberg-Witten equation
The Seiberg-Witten equations come from the Seiberg-Witten theory in theoretical physics . Their solutions are called monopolies . In mathematics is the moduli space of its solutions for the construction Seiberg-Witten invariants used.
Equations
Let be a compact , differentiable manifold with a Riemannian metric and a spin c structure with associated spinor bundles .
The Seiberg-Witten equations are equations for a "self-dual spinor" (i.e., a cut of.. ) And a - related to the determinant bundle . They are:
In this case, referred to the Dirac operator of the connection, the curvature shape of the link, their self-dual component, and the non-marking portion of the endomorphism of .
Perturbed equations
For a self-dual 2-form with respect to the Riemannian metric , one considers the perturbed Seiberg-Witten equations
literature
- N. Seiberg, E. Witten: Electric-Magnetic Duality, Monopole Condensation, and Confinement in N = 2 Supersymmetric Yang-Mills Theory , Nuclear Physics B, Volume 426, Issue 1, September 5, 1994, pages 19-52.