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 . ${\displaystyle M}$ ${\displaystyle W^{\pm }}$

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: ${\displaystyle \phi }$ ${\displaystyle W^{+}}$ ${\displaystyle U(1)}$ ${\displaystyle A}$ ${\displaystyle L}$

${\displaystyle D^{A}\phi =0,}$

${\displaystyle F_{A}^{+}+\tau (\phi )=0.}$

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 . ${\displaystyle D^{A}}$ ${\displaystyle F_{A}}$ ${\displaystyle F_{A}^{+}:={\frac {1}{2}}(F_{A}+F_{A}^{*})}$ ${\displaystyle \tau (\phi )}$ ${\displaystyle \theta \to \langle \theta ,\phi \rangle \phi }$ ${\displaystyle W^{+}}$

Perturbed equations

For a self-dual 2-form with respect to the Riemannian metric , one considers the perturbed Seiberg-Witten equations ${\displaystyle \eta \in \Omega ^{2,+}(M,i\mathbb {R} )}$

${\displaystyle D^{A}\phi =0,}$

${\displaystyle F_{A}^{+}+\tau (\phi )+\eta =0.}$

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.