In its original form, twistor theory encodes
physical fields on Minkowski space in terms of
complex analytic objects on twistor space via the
Penrose transform. This is especially natural for
massless fields of arbitrary
spin. In the first instance these are obtained via
contour 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 as
Čech representatives of analytic
cohomology classes on regions in . These correspondences have been extended to certain nonlinear fields, including
self-dual gravity in Penrose's
nonlineargraviton construction[8] and self-dual
Yang–Mills fields in the so-called Ward construction;[9] the former gives rise to
deformations of the underlying complex structure of regions in , and the latter to certain holomorphic vector bundles over regions in . These constructions have had wide applications, including inter alia the theory of
integrable systems.[10][11][12]
The self-duality condition is a major limitation for incorporating the full nonlinearities of physical theories, although it does suffice for
Yang–Mills–Higgsmonopoles 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
interactions beyond the self-dual sector also arose in Witten's
twistor string theory,[20] which is a quantum theory of holomorphic maps of a
Riemann surface into twistor space. This gave rise to the remarkably compact RSV (Roiban, Spradlin & Volovich) formulae for tree-level
S-matrices of Yang–Mills theories,[21] but its gravity degrees of freedom gave rise to a version of conformal
supergravity limiting its applicability;
conformal gravity is an unphysical theory containing
ghosts, 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 of BCFW recursion.[25] This has a natural formulation in twistor space[26][27] that in turn led to remarkable formulations of scattering amplitudes in terms of
Grassmann integral formulae[28][29] and
polytopes.[30] These ideas have evolved more recently into the positive
Grassmannian[31] and
amplituhedron.
Twistor string theory was extended first by generalising the RSV Yang–Mills amplitude formula, and then by finding the underlying
string theory. The extension to gravity was given by Cachazo & Skinner,[32] and formulated as a twistor string theory for
maximal supergravity by David Skinner.[33] Analogous formulae were then found in all dimensions by Cachazo, He & 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 & 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 same
critical dimensions as conventional string theory; for example the
type 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 an
ultraviolet completion). They extend to give formulae for loop amplitudes[40][41] and can be defined on curved backgrounds.[42]
The twistor correspondence
Denote
Minkowski space by , with coordinates and Lorentzian metric signature . Introduce 2-component spinor indices and set
Non-projective twistor space is a four-dimensional complex vector space with coordinates denoted by where and are two constant
Weyl spinors. The hermitian form can be expressed by defining a complex conjugation from to its dual by so that the Hermitian form can be expressed as
This together with the holomorphic volume form, 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
The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space which is isomorphic as a complex manifold to . A point thereby determines a line in parametrised by A twistor 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 to be real, then if vanishes, then lies on a light ray, whereas if is non-vanishing, there are no solutions, and indeed then 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 is the
number of supersymmetries so that a twistor is now given by with anticommuting. The super conformal group 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 case provides the target for Penrose's original twistor string and the 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 also admit a twistor correspondence with a twistor space of complex dimension .[44]
Palatial twistor theory
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 of
right-handed fields. Infinitesimally, these are encoded in twistor functions or
cohomology classes of
homogeneity −6. The task of using such twistor functions in a fully nonlinear way so as to obtain a right-handed nonlinear graviton has been referred to as the (gravitational) googly problem.[45] (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.[46] The theory is named after
Buckingham Palace, where
Michael 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-commutative
holomorphic twistor
quantum algebra.)
^Penrose, Roger (1987). "On the Origins of Twistor Theory". In Rindler, Wolfgang; Trautman, Andrzej (eds.). Gravitation and Geometry, a Volume in Honour of Ivor Robinson. Bibliopolis.
ISBN88-7088-142-3.
^
abHarnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions". Journal of Mathematical Physics. 33 (9): 3197–3208.
doi:
10.1063/1.529538.
^Ward, R. S. (1990). Twistor geometry and field theory. Wells, R. O. (Raymond O'Neil), 1940-. Cambridge [England]: Cambridge University Press.
ISBN978-0521422680.
OCLC17260289.
^Mason, Lionel J; Woodhouse, Nicholas M J (1996). Integrability, self-duality, and twistor theory. Oxford: Clarendon Press.
ISBN9780198534983.
OCLC34545252.
^Dunajski, Maciej (2010). Solitons, instantons, and twistors. Oxford: Oxford University Press.
ISBN9780198570622.
OCLC507435856.
^Harnad, J.; Légaré, M.; Hurtubise, J.; Shnider, S. (1985). "Constraint equations and field equations in supersymmetric N = 3 Yang-Mills theory". Nuclear Physics B. 256: 609–620.
doi:
10.1016/0550-3213(85)90410-9.
^Harnad, J.; Hurtubise, J.; Shnider, S. (1989). "Supersymmetric Yang-Mills equations and supertwistors". Annals of Physics. 193 (1): 40–79.
doi:
10.1016/0003-4916(89)90351-5.
^Arkani-Hamed, Nima; Bourjaily, Jacob L.; Cachazo, Freddy; Goncharov, Alexander B.; Postnikov, Alexander; Trnka, Jaroslav (2012-12-21). "Scattering Amplitudes and the Positive Grassmannian".
arXiv:1212.5605 [
hep-th].
Roger Penrose and
Wolfgang Rindler (1984), Spinors and Space-Time; vol. 1, Two-Spinor Calculus and Relativitic Fields, Cambridge University Press, Cambridge.
Roger Penrose and Wolfgang Rindler (1986), Spinors and Space-Time; vol. 2, Spinor and Twistor Methods in Space-Time Geometry, Cambridge University Press, Cambridge.
Hughston, L. P. (1979) Twistors and Particles. Springer Lecture Notes in Physics 97, Springer-Verlag.
ISBN978-3-540-09244-5.
Hughston, L. P. and Ward, R. S., eds (1979) Advances in Twistor Theory. Pitman.
ISBN0-273-08448-8.
Mason, L. J. and Hughston, L. P., eds (1990) Further Advances in Twistor Theory, Volume I: The Penrose Transform and its Applications. Pitman Research Notes in Mathematics Series 231, Longman Scientific and Technical.
ISBN0-582-00466-7.
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.
ISBN0-582-00465-9.
Mason, L. J., Hughston, L. P., Kobak, P. K., and Pulverer, K., eds (2001) Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces. Research Notes in Mathematics 424, Chapman and Hall/CRC.
ISBN1-58488-047-3.
Sämann, Christian (2006). Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory (PhD). Universit ̈at Hannover.
arXiv:hep-th/0603098.