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Theorem 2.9 on page 49 of Atiyah’s Geometry of Yang-Mills Fields states that there is a natural correspondence between $ U(n)$ -instantons on $ S^4$ and holomorphic vector bundles of rank $ n$ over $ \mathbb{P}^3$ with a positive real form (essentially a hermitian inner product compatible with the real structure on $ \mathbb{P}^3$ ).Read more