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

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.