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) = G2,4(C)
- the correspondence space between P and M
- F := F1,2(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).
[edit] See also
[edit] References
- Ward, R.S. and Wells, Raymond O. Jr., Twistor Geometry and Field Theory, Cambridge University Press (1991). ISBN 0-521-42268-X.
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