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

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 $2-2g\le c_1(S^{+})[\Sigma ]<0$, 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 $-4{\pi }^{2}d\cdot c_1(S^{+})[\Sigma ]$ 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.

53 pages