In theoretical physics, twistor theory was proposed by Roger Penrose in 1967[1] as a possible path[2] to quantum gravity and has evolved into a widely studied branch of theoretical and mathematical physics.
Penrose's idea was that twistor space should be the basic arena for physics from which space-time itself should emerge.
Twistor theory arose in the context of the rapidly expanding mathematical developments in Einstein's theory of general relativity in the late 1950s and in the 1960s and carries a number of influences from that period.
In particular, Roger Penrose has credited Ivor Robinson as an important early influence in the development of twistor theory, through his construction of so-called Robinson congruences.
This can be most naturally understood as the space of chiral (Weyl) spinors for 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 projective pure spinors[4][5] for the conformal group.
[6][7] 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.
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 nonlinear graviton 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
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–Higgs monopoles and instantons (see ADHM construction).
[13] An early attempt to overcome this restriction was the introduction of ambitwistors by Isenberg, Yasskin and Green,[14] and their superspace extension, super-ambitwistors, by Edward Witten.
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.
[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 and 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.
[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 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.
is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual.
corresponds to a massless particle with spin that are not localised in real space-time.
Supertwistors are a supersymmetric extension of twistors introduced by Alan Ferber in 1978.
case provides the target for Penrose's original twistor string and the
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.
[44] The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields.
Infinitesimally, these are encoded in twistor functions or cohomology classes of homogeneity −6.