Twistor space

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In mathematics, twistor space is four-dimensional complex space T := C⁴. It has associated to it the double fibration of flag manifolds P ←_μ F _ν→ M, where

projective twistor space

P := F₁(T) = P₃(C) = P(C⁴)

compactified complexified Minkowski space

M := F₂(T) = G₂(C⁴) = G_2,4(C)

the correspondence space between P and M

F := F_1,2(T)

In the above P stands for projective space, G a Grassmannian, and F a flag manifold. The double fibration gives rise to two correspondences, c := ν . μ⁻¹ and c⁻¹ := μ . ν⁻¹.

M is embedded in P₅ ~=~ P(Λ²T) by the Plücker embedding and the image is the Klein quadric.

[edit] Rationale

In the (translated) words of Jacques Hadamard: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying R⁴ it might be valuable to identify it with C². However, since there is no canonical way of doing so, instead all isomorphisms respecting orientation and metric between the two are considered. It turns out that complex projective 3-space P₃(C) parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in R⁴. It turns out that vector bundles with self-dual connections on R⁴(instantons) correspond bijectively to holomorphic bundles on complex projective 3-space P₃(C).