Liouville Vortex And $φ^{4}$ Kink Solutions Of The Seiberg--Witten Equations

Abstract

The Seiberg--Witten equations, when dimensionally reduced to $\bf R^{2}\mit$, naturally yield the Liouville equation, whose solutions are parametrized by an arbitrary analytic function $g(z)$. The magnetic flux $Φ$ is the integral of a singular Kaehler form involving $g(z)$; for an appropriate choice of $g(z)$ , $N$ coaxial or separated vortex configurations with $Φ=\frac{2πN}{e}$ are obtained when the integral is regularized. The regularized connection in the $\bf R^{1}\mit$ case coincides with the kink solution of $φ^{4}$ theory.