Twistor theory at fifty: from contour integrals to twistor strings

Abstract

We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- the twistor space. After giving an elementary construction of this space we demonstrate how solutions to linear and nonlinear equations of mathematical physics: anti-self-duality (ASD) equations on Yang--Mills, or conformal curvature can be encoded into twistor cohomology. These twistor correspondences yield explicit examples of Yang--Mills, and gravitational instantons which we review. They also underlie the twistor approach to integrability: the solitonic systems arise as symmetry reductions of ASD Yang--Mills equations, and Einstein--Weyl dispersionless systems are reductions of ASD conformal equations. We then review the holomorphic string theories in twistor and ambitwistor spaces, and explain how these theories give rise to remarkable new formulae for the computation of quantum scattering amplitudes. Finally we discuss the Newtonian limit of twistor theory, and its possible role in Penrose's proposal for a role of gravity in quantum collapse of a wave function.

Minor corrections, and additional references. Final version, to appear in the Proceedings of the Royal Society A. 49 pages, 6 Figures. Dedicated to Roger Penrose and Nick Woodhouse at 85 and 67 years

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• The Complete Planar S-matrix of N=4 SYM as a Wilson Loop in Twistor Space
• Ambitwistor strings in 4-dimensions
• From Twistor Actions to MHV Diagrams
• From twistors to twisted geometries
• Gravity from Rational Curves
• Two-Loop Scattering Amplitudes from the Riemann Sphere
• Perturbative gravity at null infinity
• Ambitwistor strings at null infinity and subleading soft limits
• A Worldsheet Theory for Supergravity
• All Tree-Level MHV Form Factors in $\mathcal{N}=4$ SYM from Twistor Space
• Quantisation of twistor theory by cocycle twist
• The Inverse Spectral Transform for the Dunajski hierarchy and some of its reductions, I: Cauchy problem and longtime behavior of solutions
• Twistor/ambitwistor strings and null-superstrings in spacetime of D=4,10 and 11 dimensions
• Twisted geometries, twistors and conformal transformations
• Supertwistor formulation for higher dimensional superstrings
• Anyons in Geometric Models of Matter
• Quaternion-Kaehler four-manifolds and Przanowski's function

Cited by in corpus (15)

• Scattering Amplitudes from Intersection Theory
• A Twistorial Foundation for the Classical Double Copy
• Tractor geometry of asymptotically flat spacetimes
• $L_{\infty }$-Algebras, the BV Formalism, and Classical Fields
• The character map in (twisted differential) non-abelian cohomology
• Quaternion algebra on 4D superfluid quantum space-time. Gravitomagnetism
• Twistor description of spinning particles in AdS
• Dilaton from Tractor and Matter Field from Twistor
• A note on Penrose's Spin-Geometry Theorem and the geometry of `empirical quantum angles'
• Gravitation and General Relativity at King's College London
• Self-adjoint Elements in the Pseudo-unitary Group $\mathbf{U}\left(p,p\right)$
• An octahedron of complex null rays, and conformal symmetry breaking
• Gravitational Scattering in the High-Energy Limit
• Twistor Theory of Dancing Paths
• Integrability of anti-self-dual vacuum Einstein equations with nonzero cosmological constant: an infinite hierarchy of nonlocal conservation laws