Summary: A trisymplectic structure on a complex \(2n\)-manifold is a three-dimensional space \(\Omega\) of closed holomorphic forms such that any element of \(\Omega\) has constant rank \(2n\), \(n\) or zero, and degenerate forms in \(\Omega\) belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyper-Kähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyper-Kähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyper-Kähler manifold \(M\) is compatible with the hyper-Kähler reduction on \(M\). As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank \(r\), charge \(c\) framed instanton bundles on \(\mathbb{CP}^3\) is a smooth trisymplectic manifold of complex dimension \(4rc\). In particular, it follows that the moduli space of rank two, charge \(c\) instanton bundles on \(\mathbb{C}\mathbb{P}^{3}\) is a smooth complex manifold dimension \(8c-3\), thus settling part of a 30-year-old conjecture.

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