# Twistor theory

Twistor theory was proposed by Roger Penrose in 1967 as a possible path to quantum gravity and has evolved into a branch of theoretical and mathematical physics. Penrose proposed that twistor space should be the basic arena for physics from which space-time itself should emerge. It leads to a powerful set of mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory and in physics to relativity and quantum field theory, in particular to scattering amplitudes.

## Overview

Mathematically, projective twistor space $\mathbb {PT}$  is a three-dimensional complex manifold, complex projective 3-space $\mathbb {CP} ^{3}$ . Physically it has the interpretation as the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space $\mathbb {T}$  with a Hermitian form of signature (2,2) and a holomorphic volume form. This can be most naturally understood as the space of chiral (Weyl) spinors for the conformal group $SO(4,2)/\mathbb {Z} _{2}$  of Minkowski space; it is the fundamental representation of the spin group $SU(2,2)$  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 projective pure spinors for the conformal group.

In its original form, twistor theory encodes physical fields on Minkowski space into 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 are more correctly understood as Cech representatives of analytic cohomology classes on regions in $\mathbb {PT}$ . These correspondences have been extended to certain nonlinear fields, including self-dual gravity in Penrose's non-linear graviton construction and self-dual Yang-Mills in Ward's construction; the former gives rise to deformations of the underlying complex structure of regions in $\mathbb {PT}$ , and the latter to certain holomorphic vector bundles over regions in $\mathbb {PT}$ . These constructions have had wide applications.

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. An early attempt to overcome this restriction was the introduction of ambitwistors by Witten and by Isenberg, Yasskin & Green. 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. These apply to general fields but the field equations are no longer so simply expressed.

Twistorial formulae for interactions beyond the self-dual sector first arose from Witten's twistor-string theory. This is a quantum theory of holomorphic maps of a Riemann surface into twistor space. It gave rise to the remarkably compact RSV (Roiban, Spradlin & Volovich) formulae for tree-level S-matrices of Yang-Mills theories, 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 in loops amplitudes calculated via twistor-string theory.

Despite its shortcomings, twistor-string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism 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. Another key development was the introduction of BCFW recursion. This has a natural formulation in twistor space that in turn led to remarkable formulations of scattering amplitudes in terms of grassmannian integral formulae and polytopes. These ideas have evolved more recently into the positive grassmannian 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, and formulated as a twistor-string theory for maximal supergravity by David Skinner. Analogous formulae were then found in all dimensions by Cachazo, He & Yuan for Yang-Mills and gravity and subsequently for a variety of other theories. They were then understood as string theories in ambitwistor space by Mason & Skinner in a general framework that includes the original twistor-string and extends to give a number of new models and formulae. 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 and can be defined on curved backgrounds.

## The twistor correspondence

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

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

Non-projective twistor space $\mathbb {T}$  is a four-dimensional complex vector space with coordinates are denoted by $Z^{\alpha }=(\omega ^{A},\pi _{A'})$  where $\omega ^{A}$  and $\pi _{A'}$  are two constant Weyl spinors. The hermitian form can be expressed by defining a complex conjugation from $\mathbb {T}$  to its dual $\mathbb {T} ^{*}$  by ${\bar {Z}}_{\alpha }=({\bar {\pi }}_{A},{\bar {\omega }}^{A'})$  so that the Hermitian form can be expressed as
$Z^{\alpha }{\bar {Z}}_{\alpha }=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}\,.$

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

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

The incidence relation is preserved under an overall re-scaling of the twistor, so usually one works in projective twistor space $\mathbb {PT}$ , which is isomorphic as a complex manifold to $\mathbb {CP} ^{3}$ . A point $x\in M$  thereby determines a line $\mathbb {CP} ^{1}$  in $\mathbb {PT}$  parametrised by $\pi _{A'}$ . A twistor $Z^{\alpha }$  is easiest understood in space-time for complex values of the coordinates where it defines a totally null two-plane that is self-dual. If $x$  is taken to be real, then if $Z^{\alpha }{\bar {Z}}_{\alpha }$  vanishes, then $x$  lies on a light ray, whereas if i$Z^{\alpha }{\bar {Z}}_{\alpha }$  is non-vanishing, there are no solutions, and indeed then $Z^{\alpha }$  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. Non-projective twistor space is extended by fermionic coordinates where ${\mathcal {N}}$  is the number of supersymmetries so that a twistor is now given by $(\omega ^{A},\pi _{A'},\eta ^{i}),i=1,\ldots ,{\mathcal {N}}$  with $\eta ^{i}$  anticommuting. The super conformal group $SU(2,2|{\mathcal {N}})$  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 ${\mathcal {N}}=4$  case provides the target for Penrose's original twistor-string and the ${\mathcal {N}}=8$  case is that for Skinner's supergravity generalisation.

### Hyper-Kähler manifolds

Hyperkähler manifolds of dimension $4k$  also admit a twistor correspondence with a twistor space of complex dimension $2k+1$ .

### Palatial twistor theory

The nonlinear graviton construction encodes only anti-self-dual, i.e., left-handed fields. A first step towards the problem of modifying twistor space so as to encode a general gravitational field is to ask to encode 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 non-linear graviton has been referred to as the (gravitational) googly problem (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 (named after Buckingham Palace, where Michael Atiyah suggested to Penrose the use of a type of "noncommutative algebra", an important component of the theory).