Intheoretical physics,twistor theory was proposed byRoger Penrose in 1967^[1] as a possible path^[2] toquantum gravity and has evolved into a widely studied branch oftheoretical andmathematical physics. Penrose's idea was thattwistor space should be the basic arena for physics from whichspace-time itself should emerge. It has led to powerful mathematical tools that have applications todifferential andintegral geometry,nonlinear differential equations andrepresentation theory, and in physics togeneral relativity,quantum field theory, and the theory ofscattering amplitudes.

Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory ofgeneral relativity in the late 1950s and in the 1960s and carries a number of influences from that period. In particular,Roger Penrose has creditedIvor Robinson as an important early influence in the development of twistor theory, through his construction of so-calledRobinson congruences.^[3]

Projectivetwistor space${\displaystyle \mathbb{P}\mathbb{T}}$ isprojective 3-space${\displaystyle {\mathbb{C}\mathbb{P}}^3}$, the simplest3-dimensionalcompact algebraic variety. It has a physical interpretation as the space ofmassless particles withspin. It is theprojectivisation of a 4-dimensionalcomplex vector space, non-projective twistor space${\displaystyle \mathbb{T}}$, with aHermitian form ofsignature (2, 2) and aholomorphicvolume form. This can be most naturally understood as the space ofchiral (Weyl)spinors for theconformal group${\displaystyle SO(4,2)/{\mathbb{Z}}_2}$ ofMinkowski space; it is thefundamental representation of thespin group${\displaystyle SU(2,2)}$ of the conformal group. This definition can be extended to arbitrary dimensions except that beyond dimension four, one defines projective twistor space to be the space of projectivepure spinors^[4][5] for the conformal group.^[6][7]

In its original form, twistor theory encodesphysical fields on Minkowski space in terms ofcomplex analytic objects on twistor space via thePenrose transform. This is especially natural formassless fields of arbitraryspin. In the first instance these are obtained viacontour integral formulae in terms of free holomorphic functions on regions in twistor space. The holomorphic twistor functions that give rise to solutions to the massless field equations can be more deeply understood asČech representatives of analyticcohomology classes on regions in${\displaystyle \mathbb{P}\mathbb{T}}$. These correspondences have been extended to certain nonlinear fields, includingself-dual gravity in Penrose'snonlineargraviton construction^[8] and self-dualYang–Mills fields in the so-called Ward construction;^[9] the former gives rise todeformations of the underlying complex structure of regions in${\displaystyle \mathbb{P}\mathbb{T}}$, and the latter to certain holomorphic vector bundles over regions in${\displaystyle \mathbb{P}\mathbb{T}}$. These constructions have had wide applications, including inter alia the theory ofintegrable systems.^[10][11][12]

The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice forYang–Mills–Higgsmonopoles andinstantons (seeADHM construction).^[13] An early attempt to overcome this restriction was the introduction ofambitwistors by Isenberg, Yasskin and Green,^[14] and theirsuperspace extension,super-ambitwistors, byEdward Witten.^[15] Ambitwistor space is the space of complexified light rays or massless particles and can be regarded as acomplexification orcotangent 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, includingfermionic and scalar fields, gave rise, in the case ofN = 1 or 2supersymmetry, to the constraint equations, while forN = 3 (or 4), the vanishing condition for supercurvature along super null lines (super ambitwistors) implied the full set offield equations, including those for the fermionic fields. This was subsequently shown to give a1-1^[clarify] 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 forinteractions beyond the self-dual sector also arose in Witten'stwistor string theory,^[20] which is a quantum theory of holomorphic maps of aRiemann surface into twistor space. This gave rise to the remarkably compact RSV (Roiban, Spradlin and Volovich) formulae for tree-levelS-matrices of Yang–Mills theories,^[21] but its gravity degrees of freedom gave rise to a version of conformalsupergravity limiting its applicability;conformal gravity is an unphysical theory containingghosts, but its interactions are combined with those of Yang–Mills theory in loop amplitudes calculated via twistor string theory.^[22]

Despite its shortcomings, twistor string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism^[23] loosely based on disconnected strings, but was given a more basic foundation in terms of a twistor action for full Yang–Mills theory in twistor space.^[24] Another key development was the introduction ofBCFW recursion.^[25] This has a natural formulation in twistor space^[26][27] that in turn led to remarkable formulations of scattering amplitudes in terms ofGrassmann integral formulae^[28][29] andpolytopes.^[30] These ideas have evolved more recently into the positiveGrassmannian^[31] andamplituhedron.

Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlyingstring theory. The extension to gravity was given by Cachazo & Skinner,^[32] and formulated as a twistor string theory formaximal supergravity by David Skinner.^[33] Analogous formulae were then found in all dimensions by Cachazo, He and Yuan for Yang–Mills theory and gravity^[34] and subsequently for a variety of other theories.^[35] They were then understood as string theories in ambitwistor space by Mason and Skinner^[36] in a general framework that includes the original twistor string and extends to give a number of new models and formulae.^[37][38][39] As string theories they have the samecritical dimensions as conventional string theory; for example thetype II supersymmetric versions are critical in ten dimensions and are equivalent to the full field theory of type II supergravities in ten dimensions (this is distinct from conventional string theories that also have a further infinite hierarchy of massive higher spin states that provide anultraviolet completion). They extend to give formulae for loop amplitudes^[40][41] and can be defined on curved backgrounds.^[42]

DenoteMinkowski 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${\displaystyle A=0,1;A^{\prime }=0^{\prime },1^{\prime },}$ and set

Non-projective twistor space${\displaystyle \mathbb{T}}$ is a four-dimensional complex vector space with coordinates denoted by${\displaystyle Z^{\alpha }=\left({\omega }^A,{\pi }_{A^{\prime }}\right)}$ where${\displaystyle {\omega }^A}$ and${\displaystyle {\pi }_{A^{\prime }}}$ are two constantWeyl spinors. The hermitian form can be expressed by defining a complex conjugation from${\displaystyle \mathbb{T}}$ to its dual${\displaystyle {\mathbb{T}}^{\ast }}$ by${\displaystyle {\overline{Z}}_{\alpha }=\left({\overline{\pi }}_A,{\overline{\omega }}^{A^{\prime }}\right)}$ so that the Hermitian form can be expressed as

${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }={\omega }^A{\overline{\pi }}_A+{\overline{\omega }}^{A^{\prime }}{\pi }_{A^{\prime }}.}$

This together with the holomorphic volume form,${\displaystyle {\epsilon }_{\alpha \beta \gamma \delta }{Z}^{\alpha }d{Z}^{\beta }\wedge d{Z}^{\gamma }\wedge d{Z}^{\delta }}$ 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

${\displaystyle {\omega }^A=i{x}^{A{A}^{\prime }}{\pi }_{A^{\prime }}.}$

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space${\displaystyle \mathbb{P}\mathbb{T},}$ which is isomorphic as a complex manifold to${\displaystyle {\mathbb{C}\mathbb{P}}^3}$. A point${\displaystyle x\in M}$ thereby determines a line${\displaystyle {\mathbb{C}\mathbb{P}}^1}$ in${\displaystyle \mathbb{P}\mathbb{T}}$ parametrised by${\displaystyle {\pi }_{A^{\prime }}.}$ 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${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }}$ vanishes, then${\displaystyle x}$ lies on a light ray, whereas if${\displaystyle Z^{\alpha }{\overline{Z}}_{\alpha }}$ 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.

Supertwistors are asupersymmetric extension of twistors introduced by Alan Ferber in 1978.^[43] Non-projective twistor space is extended byfermionic coordinates where${\displaystyle \mathcal{N}}$ is thenumber of supersymmetries so that a twistor is now given by${\displaystyle \left({\omega }^A,{\pi }_{A^{\prime }},{\eta }^i\right),i=1,\dots ,\mathcal{N}}$ 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.

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 byJ. Harnad and S. Shnider.^[4][5]

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

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.^[8] A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding ofright-handed fields. Infinitesimally, these are encoded in twistor functions orcohomology classes ofhomogeneity −6. The task of using such twistor functions in a fully nonlinear way so as to obtain aright-handed nonlinear graviton has been referred to as the (gravitational)googly problem.^[45] (The word "googly" is a term used in the game ofcricket for a ball bowled with right-handed helicity using the apparent action that would normally give rise to left-handed helicity.) The most recent proposal in this direction by Penrose in 2015 was based onnoncommutative geometry on twistor space and referred to aspalatial twistor theory.^[46] The theory is named afterBuckingham Palace, whereMichael Atiyah^[47] suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory. (The underlying twistor structure in palatial twistor theory was modeled not on the twistor space but on the non-commutativeholomorphic twistorquantum algebra.)

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

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• Mason, L. J., Hughston, L. P., and Kobak, P. K., eds (1995)Further Advances in Twistor Theory, Volume II: Integrable Systems, Conformal Geometry, and Gravitation. Pitman Research Notes in Mathematics Series 232, Longman Scientific and Technical.ISBN 0-582-00465-9.
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• Jozsa, Richard (1976), "Applications of Sheaf Cohomology in Twistor Theory."
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• Mason, L. J., "The twistor programme and twistor strings: From twistor strings to quantum gravity?"
• Sämann, Christian (2006).Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory (PhD). Universität Hannover.arXiv:hep-th/0603098.
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• Twistor newsletter archives.

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