Twistor theory

Mathematically, projective twistor space ${\displaystyle \mathbb {PT} }$ is a three-dimensional complex manifold, complex projective 3-space ${\displaystyle \mathbb {CP} ^{3}}$. It has the physical interpretation of 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 for the conformal group.[3][4]

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 ${\displaystyle \mathbb {PT} }$. These correspondences have been extended to certain nonlinear fields, including self-dual gravity in Penrose's non-linear graviton construction[5] and self-dual Yang-Mills in Ward's construction;[6] 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.[7][8][9]

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.[10] An early attempt to overcome this restriction was the introduction of ambitwistors by Witten[11] and by Isenberg, Yasskin & Green.[12] 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.[13] 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,[14] 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.[15]

Despite its shortcomings, twistor-string theory led to rapid developments in the study of scattering amplitudes. One was the so-called MHV formalism[16] 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.[17] Another key development was the introduction of BCFW recursion.[18] This has a natural formulation in twistor space[19][20] that in turn led to remarkable formulations of scattering amplitudes in terms of grassmannian integral formulae[21][22] and polytopes.[23] These ideas have evolved more recently into the positive grassmannian[24] 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,[25] and formulated as a twistor-string theory for maximal supergravity by David Skinner.[26] Analogous formulae were then found in all dimensions by Cachazo, He & Yuan for Yang-Mills and gravity[27] and subsequently for a variety of other theories.[28] They were then understood as string theories in ambitwistor space by Mason & Skinner[29] in a general framework that includes the original twistor-string and extends to give a number of new models and formulae.[30][31][32] 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[33][34] and can be defined on curved backgrounds.[35]

Denote Minkowski space by ${\displaystyle \mathbb {M} }$, 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 ${\displaystyle Z^{\alpha }=(\omega ^{A},\pi _{A'})}$ 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 ${\displaystyle {\bar {Z}}_{\alpha }=({\bar {\pi }}_{A},{\bar {\omega }}^{A'})}$ 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
• Penrose transform
• Spin network
• Twistor strings

• Roger Penrose, The Road to Reality, Alfred A. Knopf, 2004.

• Atiyah, M., Dunajski, M., and Mason, L. J. (2017). "Twistor theory at fifty: from contour integrals to twistor strings". Proc. R. Soc. A. 473 (2206): 20170530. doi:10.1098/rspa.2017.0530. ISSN 1364-5021.
• Baird, P., "An Introduction to Twistors."
• Huggett, S. and Tod, K. P. (1994). An Introduction to Twistor Theory, second edition. Cambridge University Press. ISBN 9780521456890. OCLC 831625586.
• Hughston, L. P. (1979) Twistors and Particles. Springer Lecture Notes in Physics 97, Springer-Verlag. ISBN 978-3-540-09244-5.
• Hughston, L. P. and Ward, R. S., eds (1979) Advances in Twistor Theory. Pitman. ISBN 0-273-08448-8.
• 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. 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.
• Penrose, Roger (1967), "Twistor Algebra", Journal of Mathematical Physics, 8 (2): 345–366, Bibcode:1967JMP.....8..345P, doi:10.1063/1.1705200, MR 0216828, archived from the original on 2013-01-12
• Penrose, Roger (1968), "Twistor Quantisation and Curved Space-time", International Journal of Theoretical Physics, 1 (1): 61–99, Bibcode:1968IJTP....1...61P, doi:10.1007/BF00668831
• Penrose, Roger (1969), "Solutions of the Zero‐Rest‐Mass Equations", Journal of Mathematical Physics, 10 (1): 38–39, Bibcode:1969JMP....10...38P, doi:10.1063/1.1664756, archived from the original on 2013-01-12
• Penrose, Roger (1977), "The Twistor Programme", Reports on Mathematical Physics, 12 (1): 65–76, Bibcode:1977RpMP...12...65P, doi:10.1016/0034-4877(77)90047-7, MR 0465032
• Penrose, Roger (1999) "The Central Programme of Twistor Theory," Chaos, Solitons and Fractals 10: 581–611.
• 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

• Penrose, Roger (1999), "Einstein's Equation and Twistor Theory: Recent Developments"
• Penrose, Roger; Hadrovich, Fedja. "Twistor Theory."
• Dunajski, Maciej, "Twistor Theory and Differential Equations."
• Hadrovich, Fedja, "Twistor Primer."
• Andrew Hodges, "Twistor Theory and the Twistor Programme." Includes many links.
• Huggett, Stephen (2005), "The Elements of Twistor Theory."
• Richard Jozsa (1976), "Applications of Sheaf Cohomology in Twistor Theory."
• Mason, L. J., "The twistor programme and twistor strings:From twistor strings to quantum gravity?"
• Sämann, Christian (2006), "Aspects of Twistor Geometry and Supersymmetric Field Theories within Superstring Theory."
• Sparling, George (1999), "On Time Asymmetry."
• Spradlin, Marcus (2006), "Progress and Prospects In Twistor String Theory."
• MathWorld: Twistors.
• Universe Review: "Twistor Theory."
• Twistor newsletter archives.