On finite energy monopoles on $C \times \Sigma$

We classify solutions to the Seiberg–Witten equations on $X=\mathbb{C} \times \Sigma$ with finite analytic energy. The spin bundle $S^{+} \to X$ splits as $L^{+} \oplus L^{-}$. When $2 - 2g \leq c_1(S^{+}) [\Sigma] \lt 0$, the moduli space is identified 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 it is the space of flat connections on $\bigwedge^2 S^{+}$.

Solutions will have either exponential decay or power law decay according to the polynomial map $f$. We give a complete criterion for these cases.

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Received 29 December 2018

Accepted 13 August 2019

Published 29 November 2022