Instantons on cylindrical manifolds and stable bundles

Abstract

Let $\Sigma$ be a smooth complex curve, and let $S$ be the product ruled surface $\Sigma \times ℂP^1$. We prove a correspondence conjectured by Donaldson between finite energy $U\left(2\right)$–instantons over $\Sigma \times S^1\times ℝ$, and rank 2 holomorphic bundles over $S$ whose restrictions to $\Sigma \times \left\{0\right\},\Sigma \times \left\{\infty \right\}$ are stable.