Editing Twistor theory (section)

==Variations<!--'History of twistor theory' redirects here-->==
===Supertwistors===

Supertwistors are a [[supersymmetry|supersymmetric]] extension of twistors introduced by Alan Ferber in 1978.<ref name="Fer">{{Citation|bibcode=1978NuPhB.132...55F|doi = 10.1016/0550-3213(78)90257-2|title=Supertwistors and conformal supersymmetry|year=1978|last1=Ferber|first1=A.|journal=Nuclear Physics B|volume=132|issue = 1|pages=55–64|postscript=. }}</ref> Non-projective twistor space is extended by [[fermion]]ic coordinates where <math>\mathcal{N}</math> is the [[extended supersymmetry|number of supersymmetries]] so that a twistor is now given by <math>\left(\omega^A,\, \pi_{A'},\, \eta^i\right), i = 1, \ldots, \mathcal{N}</math> with <math>\eta^i</math> anticommuting. The super conformal group <math>SU(2,2|\mathcal{N})</math> 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 <math>\mathcal{N} = 4</math> case provides the target for Penrose's original twistor string and the <math>\mathcal{N} = 8</math> case is that for Skinner's supergravity generalisation.

=== Higher dimensional generalization of the Klein correspondence ===

A higher dimensional generalization of the [[Klein quadric | Klein correspondence]] underlying twistor theory, applicable to isotropic subspaces of conformally compactified (complexified) Minkowski space and its super-space extensions, was developed by [[John Harnad |J. Harnad]] and S. Shnider.<ref name ="HS1"/><ref name = "HS2"/>

=== Hyperkähler manifolds ===
[[Hyperkähler manifold]]s of dimension <math>4k</math> also admit a twistor correspondence with a twistor space of complex dimension <math>2k+1</math>.<ref>{{cite journal|last4=Roček | first4=M. | last3=Lindström | first3=U. | last2=Karlhede | first2=A. | last1=Hitchin | first1=N. J. | title=Hyper-Kähler metrics and supersymmetry | url=https://projecteuclid.org/download/pdf_1/euclid.cmp/1104116624 |mr=877637 | year=1987 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=108 | issue=4 | pages=535–589 | doi=10.1007/BF01214418| bibcode=1987CMaPh.108..535H | s2cid=120041594 }}</ref>

===Palatial twistor theory<!--'Googly problem' and 'Palatial twistor theory' redirect here-->===
The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.<ref name="Penrose1976"/> A first step towards the problem of modifying twistor space so as to encode a general gravitational field is the encoding of [[Chirality (physics)|right-handed]] fields. Infinitesimally, these are encoded in twistor functions or [[cohomology]] classes of [[Homogeneous function|homogeneity]] −6. The task of using such twistor functions in a fully nonlinear way so as to obtain a ''[[Helicity (particle physics)|right-handed]]'' nonlinear graviton has been referred to as the ('''gravitational''') '''googly problem'''<!--boldface per WP:R#PLA-->.<ref name="Penrose1000">Penrose 2004, p. 1000.</ref> (The word "[[googly]]" is a term used in the game of [[cricket]] 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 on [[noncommutative geometry]] on twistor space and referred to as '''palatial twistor theory'''<!--boldface per WP:R#PLA-->.<ref>{{Cite journal|doi=10.1098/rsta.2014.0237|title=Palatial twistor theory and the twistor googly problem|year=2015|last1=Penrose|first1=Roger|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=373|issue=2047|pmid=26124255|s2cid=13038470|page=20140237|bibcode=2015RSPTA.37340237P |doi-access=free}}</ref> The theory is named after [[Buckingham Palace]], where [[Michael Atiyah]]<ref>[https://www.quantamagazine.org/20160303-michael-atiyahs-mathematical-dreams/ "Michael Atiyah's Imaginative State of Mind"] – ''[[Quanta Magazine]]''</ref> 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-commutative [[holomorphic]] twistor [[Quantum group|quantum algebra]].)