Twistor theory

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. It has led to powerful mathematical tools that have applications to differential and integral geometry, nonlinear differential equations and representation theory, and in physics to general relativity, quantum field theory, and the theory of scattering amplitudes.

Contents

• Overview
• The twistor correspondence
• Variations
• Supertwistors
• Higher dimensional generalization of the Klein correspondence
• Hyperkähler manifolds
• Palatial twistor theory
• See also
• Notes
• References
• Further reading
• External links

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. ^[3]

Overview

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 and 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^{[ clarify ]} 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 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. ^[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 and 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 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. ^[42]

The twistor correspondence

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.

Variations

See also

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

Notes

Related Research Articles

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References

• Roger Penrose (2004), The Road to Reality , Alfred A. Knopf, ch. 33, pp. 958–1009.
• 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.

Further reading

• Atiyah, Michael; Dunajski, Maciej; Mason, Lionel J. (2017). "Twistor theory at fifty: from contour integrals to twistor strings". Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences. 473 (2206): 20170530. arXiv: 1704.07464 . Bibcode:2017RSPSA.47370530A. doi: 10.1098/rspa.2017.0530 . PMC   5666237 . PMID   29118667. S2CID   5735524.
• 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, S2CID   123628735
• 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 (2–3): 581–611. Bibcode:1999CSF....10..581P. doi:10.1016/S0960-0779(98)00333-6.
• 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

External links

• Penrose, Roger (1999), "Einstein's Equation and Twistor Theory: Recent Developments"
• Penrose, Roger; Hadrovich, Fedja. "Twistor Theory."
• Hadrovich, Fedja, "Twistor Primer."
• Penrose, Roger. "On the Origins of Twistor Theory."
• Jozsa, Richard (1976), "Applications of Sheaf Cohomology in Twistor Theory."
• Dunajski, Maciej (2009). "Twistor Theory and Differential Equations". J. Phys. A: Math. Theor. 42 (40): 404004. arXiv: 0902.0274 . Bibcode:2009JPhA...42N4004D. doi:10.1088/1751-8113/42/40/404004. S2CID   62774126.
• Andrew Hodges, Summary of recent developments.
• Huggett, Stephen (2005), "The Elements of 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 (PhD). Universit ̈at Hannover. arXiv: hep-th/0603098 .
• Sparling, George (1999), "On Time Asymmetry."
• Spradlin, Marcus (2012). "Progress and Prospects in Twistor String Theory" (PDF). hdl:11299/130081.
• MathWorld: Twistors.
• Universe Review: "Twistor Theory."
• Twistor newsletter archives.