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

TL;DR

This paper explores solutions to the Seiberg--Witten equations that reduce to Liouville and kink solutions, revealing connections between vortex configurations, the Liouville equation, and $^{4}$ theory.

Contribution

It demonstrates how the Seiberg--Witten equations yield Liouville and kink solutions, linking vortex configurations with classical field theory solutions.

Findings

Vortex solutions parametrized by an analytic function g(z)

Regularized magnetic flux yields quantized vortex configurations

Connection to kink solutions in $^{4}$ theory

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 $\Phi$ 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 $\Phi=\frac{2\pi N}{e}$ are obtained when the integral is regularized. The regularized connection in the $\bf R^{1}\mit$ case coincides with the kink solution of $\varphi^{4}$ theory.