On finite energy monopoles on $\mathbb{C}\times \Sigma$

TL;DR

This paper classifies finite energy solutions to the Seiberg-Witten equations on the product of the complex plane and a Riemann surface, revealing their structure and energy characteristics.

Contribution

It provides a classification theorem for solutions with finite energy, relating them to holomorphic data and polynomial maps, and estimates their decay rates.

Findings

Moduli space characterized by holomorphic structures and polynomial maps.

Solutions have energy proportional to the degree of the polynomial map.

All solutions are reducible when the first Chern class vanishes.

Abstract

Let $X=\mathbb{C}\times\Sigma$ be the product of the complex plane and a compact Riemann surface. We establish a classification theorem of solutions to the Seiberg-Witten equation on $X$ with finite analytic energy. The spin bundle $S^+\to X$ splits as $L^+\oplus L^-$. When $2-2g\leq c_1(S^+)[\Sigma]<0$, the moduli space is in bijection with the moduli space of pairs $((L^+,\bar{\partial}), f)$ where $(L^+,\bar{\partial})$ is a holomorphic structure on $L^+$ and $f: \mathbb{C}\to H^0(\Sigma, L^+,\bar{\partial})$ is a polynomial map. Moreover, the solution has analytic energy $-4\pi^2d\cdot c_1(S^+)[\Sigma]$ if $f$ has degree $d$. When $c_1(S^+)=0$, all solutions are reducible and the moduli space is the space of flat connections on $\bigwedge^2 S^+$. We also estimate the decay rate of these solutions at infinity.