= Penrose-Ward correspondence
{c}
A <holomorphic vector bundle> on an appropriate twistor domain, trivial on its space-time incidence lines, determines a complex anti-self-dual gauge <gauge connection>. Split its patching <matrix> on each line and apply the tangent vector fields $\pi^{A'}\partial_{AA'}$. They annihilate the unsplit twistor patching <matrix>, so the two logarithmic <derivatives> of the factors agree and define a global degree-one <function> of $\pi$. It is consequently $\pi^{A'}A_{AA'}$. The inverse splitting <bundle frame> is parallel along each <alpha-plane>; commuting these parallel equations proves that the primed <gauge curvature> <two-component spinor> vanishes. Reality conditions recover unitary Euclidean <gauge fields>; Euclidean <instanton> bundles need triviality on the real <twistor lines>, not on every possible complex line.
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