Twistor theory

In

Projective twistor space ${\displaystyle \mathbb {PT} }$ is projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$, the simplest

${\displaystyle \mathbb {T} }$

${\displaystyle SO(4,2)/\mathbb {Z} _{2}}$ of

${\displaystyle SU(2,2)}$

In its original form, twistor theory encodes

${\displaystyle \mathbb {PT} }$

of the underlying complex structure of regions in ${\displaystyle \mathbb {PT} }$, and the latter to certain holomorphic vector bundles over regions in ${\displaystyle \mathbb {PT} }$. These constructions have had wide applications, including inter alia the theory of

The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for Yang–Mills–Higgs monopoles and instantons (see ADHM construction).^[13] An early attempt to overcome this restriction was the introduction of ambitwistors by Isenberg, Yasskin & Green,^[14] and their superspace extension, super-ambitwistors, by Edward Witten.^[15] Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as a complexification or cotangent bundle of the original twistor description. By extending the ambitwistor correspondence to suitably defined formal neighborhoods, Isenberg, Yasskin and Green^[14] showed the equivalence between the vanishing of the curvature along such extended null lines and the full Yang-Mills field equations.^[14] Witten^[15] showed that a further extension, within the framework of super Yang-Mills theory, including fermionic and scalar fields, gave rise, in the case of N=1 or 2 supersymmetry, to the constraint equations, while for N=3 (or 4), the vanishing condition for supercurvature along super null lines (super ambitwistors) implied the full set of field equations, including those for the fermionic fields. This was subsequently shown to give a 1-1 equivalence between the null curvature constraint equations and the supersymmetric Yang-Mills field equations.^[16][17] Through dimensional reduction, it may also be deduced from the analogous super-ambitwistor correspondence for 10-dimensional, N=1 super Yang Mills theory.^[18][19]

Twistorial formulae for

Denote Minkowski space by ${\displaystyle M}$, with coordinates ${\displaystyle x^{a}=(t,x,y,z)}$ and Lorentzian metric ${\displaystyle \eta _{ab}}$ signature ${\displaystyle (1,3)}$. Introduce 2-component spinor indices ${\displaystyle A=0,1;\;A'=0',1',}$ and set

${\displaystyle x^{AA'}={\frac {1}{\sqrt {2}}}{\begin{pmatrix}t-z&x+iy\\x-iy&t+z\end{pmatrix}}.}$

Non-projective twistor space ${\displaystyle \mathbb {T} }$ is a four-dimensional complex vector space with coordinates denoted by ${\displaystyle Z^{\alpha }=\left(\omega ^{A},\,\pi _{A'}\right)}$ where ${\displaystyle \omega ^{A}}$ and ${\displaystyle \pi _{A'}}$ are two constant

${\displaystyle \mathbb {T} }$

${\displaystyle \mathbb {T} ^{*}}$

${\displaystyle {\bar {Z}}_{\alpha }=\left({\bar {\pi }}_{A},\,{\bar {\omega }}^{A'}\right)}$

${\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha }=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}.}$

This together with the holomorphic volume form, ${\displaystyle \varepsilon _{\alpha \beta \gamma \delta }Z^{\alpha }dZ^{\beta }\wedge dZ^{\gamma }\wedge dZ^{\delta }}$ is invariant under the group SU(2,2), 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

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

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space ${\displaystyle \mathbb {PT} ,}$ which is isomorphic as a complex manifold to ${\displaystyle \mathbb {CP} ^{3}}$. A point ${\displaystyle x\in M}$ thereby determines a line ${\displaystyle \mathbb {CP} ^{1}}$ in ${\displaystyle \mathbb {PT} }$ parametrised by ${\displaystyle \pi _{A'}.}$ A twistor ${\displaystyle Z^{\alpha }}$ is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. Take ${\displaystyle x}$ to be real, then if ${\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha }}$ vanishes, then ${\displaystyle x}$ lies on a light ray, whereas if ${\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha }}$ is non-vanishing, there are no solutions, and indeed then ${\displaystyle Z^{\alpha }}$ corresponds to a massless particle with spin that are not localised in real space-time.

Variations

Supertwistors

Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978.^[43] Non-projective twistor space is extended by fermionic coordinates where ${\displaystyle {\mathcal {N}}}$ is the number of supersymmetries so that a twistor is now given by ${\displaystyle \left(\omega ^{A},\,\pi _{A'},\,\eta ^{i}\right),i=1,\ldots ,{\mathcal {N}}}$ with ${\displaystyle \eta ^{i}}$ anticommuting. The super conformal group ${\displaystyle SU(2,2|{\mathcal {N}})}$ naturally acts on this space and a supersymmetric version of the Penrose transform takes cohomology classes on supertwistor space to massless supersymmetric multiplets on super Minkowski space. The ${\displaystyle {\mathcal {N}}=4}$ case provides the target for Penrose's original twistor string and the ${\displaystyle {\mathcal {N}}=8}$ case is that for Skinner's supergravity generalisation.

Higher dimensional generalization of the Klein correspondence

A higher dimensional generalization of the Klein correspondence underlying twistor theory, applicable to isotropic subspaces of conformally compactified (complexified) Minkowski space and its super-space extensions, was developed by J. Harnad and S. Shnider.^[4][5]

Hyperkähler manifolds

Hyperkähler manifolds of dimension ${\displaystyle 4k}$ also admit a twistor correspondence with a twistor space of complex dimension ${\displaystyle 2k+1}$.^[44]

Palatial twistor theory

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.

• Background independence
• Complex spacetime
• History of loop quantum gravity
• Robinson congruences
• Spin network

Notes