Twistor space

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 four-dimensional space ${\displaystyle \mathbb {R} ^{4}}$ it might be valuable to identify it with ${\displaystyle \mathbb {C} ^{2}.}$ 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 ${\displaystyle \mathbb {CP} ^{3}}$ parametrizes such isomorphisms together with complex coordinates. Thus one complex coordinate describes the identification and the other two describe a point in ${\displaystyle \mathbb {R} ^{4}}$. It turns out that vector bundles with self-dual connections on ${\displaystyle \mathbb {R} ^{4}}$(instantons) correspond bijectively to holomorphic vector bundles on complex projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$.

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. The ${\displaystyle \sigma _{\mu }=(I,{\vec {\sigma }})}$ are the Pauli matrices, with ${\displaystyle A,A^{\prime }=1,2}$ the indexes on the matrices. 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 ${\displaystyle \mathbb {PT} }$, 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

${\displaystyle \mathbb {T}$ :=\mathbb {C} ^{4}.}

It has associated to it the double fibration of flag manifolds ${\displaystyle \mathbb {P} \xleftarrow {\mu } \mathbb {F} \xrightarrow {\nu } \mathbb {M} }$ where ${\displaystyle \mathbb {P} }$ is the projective twistor space

${\displaystyle \mathbb {P} =F_{1}(\mathbb {T} )=\mathbb {CP} ^{3}=\mathbf {P} (\mathbb {C} ^{4})}$

and ${\displaystyle \mathbb {M} }$ is the compactified complexified Minkowski space

${\displaystyle \mathbb {M} =F_{2}(\mathbb {T} )=\operatorname {Gr} _{2}(\mathbb {C} ^{4})=\operatorname {Gr} _{2,4}(\mathbb {C} )}$

and the correspondence space between ${\displaystyle \mathbb {P} }$ and ${\displaystyle \mathbb {M} }$ is

${\displaystyle \mathbb {F} =F_{1,2}(\mathbb {T} )}$

In the above, ${\displaystyle \mathbf {P} }$ stands for projective space, ${\displaystyle \operatorname {Gr} }$ a Grassmannian, and ${\displaystyle F}$ a flag manifold. The double fibration gives rise to two correspondences (see also Penrose transform), ${\displaystyle c=\nu \circ \mu ^{-1}}$ and ${\displaystyle c^{-1}=\mu \circ \nu ^{-1}.}$

The compactified complexified Minkowski space ${\displaystyle \mathbb {M} }$ is embedded in ${\displaystyle \mathbf {P} _{5}\cong \mathbf {P} (\wedge ^{2}\mathbb {T} )}$ by the Plücker embedding; the image is the Klein quadric.