We classify solutions to the Seiberg-Witten equations on X = C x sigma with finite analytic energy. The spin bundle S+ -> X splits as L+ circle plus L-. When 2 - 2g ) is a holomorphic structure on L+ and f : C -> H-0(sigma, L+, (partial derivative)over bar>) is a polynomial map. Moreover, the solution has analytic energy -4 pi(2)d center dot 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 lambda(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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