Existence of Solutions to the Seiberg-Witten Vortex Equations with Exponential Decay on the Plane

TL;DR

This paper demonstrates the existence of both exponentially decayed and polynomial growth solutions in the moduli space of a Hitchin-type reduction of Seiberg-Witten equations on the plane, extending previous vortex results.

Contribution

It establishes the presence of diverse solution types in the moduli space of reduced Seiberg-Witten equations, expanding understanding of their solution structure.

Findings

Existence of exponentially decayed solutions

Existence of polynomial growth solutions

Extension of vortex solution results to Seiberg-Witten equations

Abstract

Clifford Taubes showed that the moduli space of the variational equation of the Yang-Mills-Higgs functional on the plane is non-empty, and its elements correspond to "vortices". Inspired by this result, in this paper, we show that the moduli space of the Hitchin-type dimensional reduction of the Seiberg-Witten equations on the plane contains both exponentially decayed solutions and polynomial growth solutions.