Twistor space

<h2 id="informal-motivation">Informal motivation</h2>
In the (translated) words of <a href="/facts/Jacques_Hadamard/TgCkP3AE">Jacques Hadamard</a>: "the shortest path between two truths in the real domain passes through the complex domain." Therefore when studying four-dimensional space 
 
 
 
 
 
 R
 
 
 4
 
 
 
 
 {\displaystyle \mathbb {R} ^{4}}
 
 it might be valuable to identify it with 
 
 
 
 
 
 C
 
 
 2
 
 
 .
 
 
 {\displaystyle \mathbb {C} ^{2}.}
 
 However, since there is no canonical way of doing so, instead all <a href="/facts/Isomorphism/pj8KgkxU">isomorphisms</a> respecting orientation and metric between the two are considered. It turns out that <a href="/facts/Complex_projective_space/QmgneIzS">complex projective 3-space</a> 
 
 
 
 
 
 C
 P
 
 
 3
 
 
 
 
 {\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 
 
 
 
 
 
 R
 
 
 4
 
 
 
 
 {\displaystyle \mathbb {R} ^{4}}
 
. It turns out that <a href="/facts/Vector_bundle/cXACAa1I">vector bundles</a> with <a href="/facts/Self-dual_connection/dPN0P2ro">self-dual connections</a> on 
 
 
 
 
 
 R
 
 
 4
 
 
 
 
 {\displaystyle \mathbb {R} ^{4}}
 
(<a href="/facts/Instanton/IuXhVcRF">instantons</a>) <a href="/facts/Bijection/j4gTuTmW">correspond bijectively</a> to <a href="/facts/Holomorphic_vector_bundle/FeRrK9Bg">holomorphic vector bundles</a> on complex projective 3-space 
 
 
 
 
 
 C
 P
 
 
 3
 
 
 
 
 {\displaystyle \mathbb {CP} ^{3}}
 
.

<h2 id="formal-definition">Formal definition</h2>
For <a href="/facts/Minkowski_space/vDkfkQJn">Minkowski space</a>, denoted 
 
 
 
 
 M
 
 
 
 {\displaystyle \mathbb {M} }
 
, the solutions to the twistor equation are of the form

Ω
          
            A
          
        
        (
        x
        )
        =
        
          ω
          
            A
          
        
        −
        i
        
          x
          
            A
            
              A
              ′
            
          
        
        
          π
          
            
              A
              ′
            
          
        
      
    
    {\displaystyle \Omega ^{A}(x)=\omega ^{A}-ix^{AA'}\pi _{A'}}

where 
 
 
 
 
 ω
 
 A
 
 
 
 
 {\displaystyle \omega ^{A}}
 
 and 
 
 
 
 
 π
 
 
 A
 ′
 
 
 
 
 
 {\displaystyle \pi _{A'}}
 
 are two constant <a href="/facts/Weyl_spinor/8KhWTFaM">Weyl spinors</a> and 
 
 
 
 
 x
 
 A
 
 A
 ′
 
 
 
 =
 
 σ
 
 μ
 
 
 A
 
 A
 ′
 
 
 
 
 x
 
 μ
 
 
 
 
 {\displaystyle x^{AA'}=\sigma _{\mu }^{AA'}x^{\mu }}
 
 is a point in Minkowski space. The 
 
 
 
 
 σ
 
 μ
 
 
 =
 (
 I
 ,
 
 
 
 σ
 →
 
 
 
 )
 
 
 {\displaystyle \sigma _{\mu }=(I,{\vec {\sigma }})}
 
 are the <a href="/facts/Pauli_matrices/BdeC2wos">Pauli matrices</a>, with 
 
 
 
 A
 ,
 
 A
 
 ′
 
 
 =
 1
 ,
 2
 
 
 {\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 
 
 
 
 
 Z
 
 α
 
 
 =
 (
 
 ω
 
 A
 
 
 ,
 
 π
 
 
 A
 ′
 
 
 
 )
 
 
 {\displaystyle Z^{\alpha }=(\omega ^{A},\pi _{A'})}
 
, and with a <a href="/facts/Hermitian_form/Kp6qUA8v">hermitian form</a>

Σ
        (
        Z
        )
        =
        
          ω
          
            A
          
        
        
          
            
              
                π
                ¯
              
            
          
          
            A
          
        
        +
        
          
            
              
                ω
                ¯
              
            
          
          
            
              A
              ′
            
          
        
        
          π
          
            
              A
              ′
            
          
        
      
    
    {\displaystyle \Sigma (Z)=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}}

which is invariant under the <a href="/facts/Special_unitary_group/Qgy2orcu">group SU(2,2)</a> which is a quadruple cover of the <a href="/facts/Conformal_group/mNqA5vLP">conformal group</a> C(1,3) of compactified Minkowski spacetime.
Points in Minkowski space are related to subspaces of twistor space through the <a href="/facts/Incidence_relation/eMXGWQBw">incidence relation</a>

ω
          
            A
          
        
        =
        i
        
          x
          
            A
            
              A
              ′
            
          
        
        
          π
          
            
              A
              ′
            
          
        
        .
      
    
    {\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 
 
 
 
 
 P
 T
 
 
 
 {\displaystyle \mathbb {PT} }
 
, which is isomorphic as a complex manifold to 
 
 
 
 
 
 C
 P
 
 
 3
 
 
 
 
 {\displaystyle \mathbb {CP} ^{3}}
 
.
Given a point 
 
 
 
 x
 ∈
 M
 
 
 {\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 
 
 
 
 
 
 C
 P
 
 
 1
 
 
 
 
 {\displaystyle \mathbb {CP} ^{1}}
 
 parametrized by 
 
 
 
 
 π
 
 
 A
 ′
 
 
 
 
 
 {\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
        
        :=
        
          
            C
          
          
            4
          
        
        .
      
    
    {\displaystyle \mathbb {T} :=\mathbb {C} ^{4}.}

It has associated to it the double <a href="/facts/Fibration/h51Cx7xU">fibration</a> of <a href="/facts/Flag_manifold/rdTKdfBl">flag manifolds</a> 
 
 
 
 
 P
 
 
 ←
 
 μ
 
 
 
 F
 
 
 →
 
 ν
 
 
 
 M
 
 
 
 {\displaystyle \mathbb {P} \xleftarrow {\mu } \mathbb {F} \xrightarrow {\nu } \mathbb {M} }
 
 where 
 
 
 
 
 P
 
 
 
 {\displaystyle \mathbb {P} }
 
 is the projective twistor space

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

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

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

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

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

In the above, 
 
 
 
 
 P
 
 
 
 {\displaystyle \mathbf {P} }
 
 stands for <a href="/facts/Projective_space/7MSpA9cM">projective space</a>, 
 
 
 
 Gr
 
 
 {\displaystyle \operatorname {Gr} }
 
 a <a href="/facts/Grassmannian/QcmgLRq8">Grassmannian</a>, and 
 
 
 
 F
 
 
 {\displaystyle F}
 
 a <a href="/facts/Flag_manifold/rdTKdfBl">flag manifold</a>. The double fibration gives rise to two <a href="/facts/1%3a1_correspondence/j4gTuTmW">correspondences</a> (see also <a href="/facts/Penrose_transform/wL6vswVt">Penrose transform</a>), 
 
 
 
 c
 =
 ν
 ∘
 
 μ
 
 −
 1
 
 
 
 
 {\displaystyle c=\nu \circ \mu ^{-1}}
 
 and 
 
 
 
 
 c
 
 −
 1
 
 
 =
 μ
 ∘
 
 ν
 
 −
 1
 
 
 .
 
 
 {\displaystyle c^{-1}=\mu \circ \nu ^{-1}.}
 
 
The compactified complexified Minkowski space 
 
 
 
 
 M
 
 
 
 {\displaystyle \mathbb {M} }
 
 is embedded in 
 
 
 
 
 
 P
 
 
 5
 
 
 ≅
 
 P
 
 (
 
 ∧
 
 2
 
 
 
 T
 
 )
 
 
 {\displaystyle \mathbf {P} _{5}\cong \mathbf {P} (\wedge ^{2}\mathbb {T} )}
 
 by the <a href="/facts/Pl%25C3%25BCcker_embedding/85rFMBKE">Plücker embedding</a>; the image is the <a href="/facts/Klein_quadric/VexSnuRN">Klein quadric</a>.

<ul><li>Ward, R.S.; <a href="/facts/Raymond_O._Wells%2c_Jr./Is80Ue1q">Wells, R.O.</a> (1991). Twistor Geometry and Field Theory. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-521-42268-X.</li>
<li>Huggett, S.A.; Tod, K.P. (1994). An introduction to twistor theory. Cambridge University Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-521-45689-0.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1">Penrose, R.; MacCallum, M.A.H. (February 1973). "Twistor theory: An approach to the quantisation of fields and space-time". Physics Reports. 6 (4): 241–315. doi:10.1016/0370-1573(73)90008-2. <a href="https://dx.doi.org/10.1016/0370-1573%2873%2990008-2" target="_blank">https://dx.doi.org/10.1016/0370-1573%2873%2990008-2</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></li>
<li id="fn:2">Hodges, Andrew (2010). One to Nine: The Inner Life of Numbers. Doubleday Canada. p. 142. ISBN 978-0-385-67266-5. <a href="978-0-385-67266-5" target="_blank">978-0-385-67266-5</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></li>
</ol>

Twistor space open-in-new

Twistor space