Twistor theory

Projective twistor space ${\displaystyle \mathbb {PT} }$ is projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$, the simplest 3-dimensional compact algebraic variety. It has a physical interpretation as the space of massless particles with spin. It is the projectivisation of a 4-dimensional complex vector space, non-projective twistor space ${\displaystyle \mathbb {T} }$, 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 ${\displaystyle SO(4,2)/\mathbb {Z} _{2}}$ of Minkowski space; it is the fundamental representation of the spin 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 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. 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 ${\displaystyle \mathbb {PT} }$. 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 ${\displaystyle \mathbb {PT} }$, and the latter to certain holomorphic vector bundles over regions in ${\displaystyle \mathbb {PT} }$. 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 & 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]

Denote Minkowski 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 ${\displaystyle A=0,1;\;A'=0',1',}$ and set

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

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

${\displaystyle Z^{\alpha }{\bar {Z}}_{\alpha }=\omega ^{A}{\bar {\pi }}_{A}+{\bar {\omega }}^{A'}\pi _{A'}.}$

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

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

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

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

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• 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.

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• 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. ISBN 0-582-00466-7.
• 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. ISBN 0-582-00465-9.
• 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. ISBN 1-58488-047-3.
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• Witten, Edward (2004), "Perturbative Gauge Theory as a String Theory in Twistor Space", Communications in Mathematical Physics, 252 (1–3): 189–258, arXiv:hep-th/0312171, Bibcode:2004CMaPh.252..189W, doi:10.1007/s00220-004-1187-3, S2CID 14300396

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• Twistor newsletter archives.