Twistor space

In mathematics, twistor space is the complex vector space of solutions of the twistor equation ${\displaystyle \nabla _{A'}^{(A}\Omega ^{B)}=0}$. It was described in the 1960s by Roger Penrose and MacCallum.[1] According to Andrew Hodges, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.[2]

For Minkowski space, denoted ${\displaystyle \mathbb {M} }$, the solutions to the twistor equation are of the form

${\displaystyle \Omega ^{A}(x)=\omega ^{A}-ix^{AA'}\pi _{A'}}$

where ${\displaystyle \omega ^{A}}$ and ${\displaystyle \pi _{A'}}$ are two constant Weyl spinors and ${\displaystyle x^{AA'}=\sigma _{\mu }^{AA'}x^{\mu }}$ is a point in Minkowski space. This twistor space is a four-dimensional complex vector space, whose points are denoted by ${\displaystyle Z^{\alpha }=(\omega ^{A},\pi _{A'})}$, and with a hermitian form

${\displaystyle \Sigma (Z)=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}}$

which is invariant under the group SU(2,2) which is a quadruple cover of the conformal group C(1,3) of compactified Minkowski spacetime.

Points in Minkowski space are related to subspaces of twistor space through the incidence relation

${\displaystyle \omega ^{A}=ix^{AA'}\pi _{A'}.}$

This incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space, denoted PT, which is isomorphic as a complex manifold to ${\displaystyle \mathbb {CP} ^{3}}$.

Given a point ${\displaystyle x\in M}$ it is related to a line in projective twistor space where we can see the incidence relation as giving the linear embedding of a ${\displaystyle \mathbb {CP} ^{1}}$ parametrized by ${\displaystyle \pi _{A'}}$.

The geometric relation between projective twistor space and complexified compactified Minkowski space is the same as the relation between lines and two-planes in twistor space; more precisely, twistor space is

T := C4. It has associated to it the double fibration of flag manifolds Pμ F νM, 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 ~=~ P2T) by the Plücker embedding and the image is the Klein quadric.

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).