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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 $\mathbb{P}^3$). A key result used to associate a holomorphic bundle to an instanton is Theorem 1.2 on page 46 which asserts that on an hermitian vector bundle $E$ equipped with a metric connection $\nabla$ over a complex manifold there is a unique holomorphic structure on E such that $\nabla$ is the Chern connection of $E$. This theorem is used on the lifted bundle with lifted metric and connection (the lift is given by pullback with the twistor fibration map).

Atiyah says that Theorem 2.9 "can easily be generalized to the orthogonal and symplectic case". I can't see how to go from instanton to holomorphic bundle in the orthogonal case if Theorem 1.2 doesn't apply (the lifted bundle will have a quadratic form, not a hermitian form).

Thoughts?

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  • $\begingroup$ This is explained in more detail here (remark 3 on p.332). In the $O(n)$ case you can take the complexification of the standard representation on $R^n$. This gives an hermitian vector bundle of rank $n$ on $S^4$, to which you can apply Theorem 2.9. $\endgroup$ – Gil Bor Feb 8 '18 at 19:50

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