Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

Special Issue on Progress in Twistor Theory

The Guest Editors for this special issue are

Michael Eastwood (Australian National University, Australia)
Claude LeBrun (Stony Brook University, USA)
Lionel Mason (University of Oxford, England)


The motivational origins of twistor theory were initially aimed at providing a possible framework for physical theory (as formulated in December 1963 — though there were many mathematical precursors, dating back to work of Sophus Lie and Felix Klein in around 1870, or to earlier related ideas of Grassmann, Cayley, and Plücker). The intention was for a formalism that would be very specific to the 4-dimensionality of physical space-time, with its relativistic Lorentzian signature, as is directly relevant to the geometry of the physical world we perceive about us. This space-time geometry was taken to be that of Minkowskian 4-space, whereby special relativity but not gravitation is accommodated. In the “twistor correspondence” that emerged, entire light rays (null geodesics) in space-time were to be regarded as more primary elements than would actual space-time points (“events”). Various underlying physical motivations provided reasons for taking this specific correspondence seriously as a plausibly fruitful viewpoint for the description of basic physics, and some justification for this optimism was subsequently found in the surprising way in which the field equations for free massless particles of arbitrary helicity …, −2, −3/2, −1, −1/2, 0, 1/2, 1, 3/2, 2, … could all be directly encoded as holomorphic sheaf cohomology classes in twistor space.

At the time, this specific twistor correspondence presented no evident generalization that would allow it to be extended to the curved Lorentzian space-times of Einstein's general relativity. Twelve years later, however, by a roundabout route arising from attempts to address some unresolved issues of general relativity, where certain quantum considerations also contributed significantly, an unexpected connection between twistor theory and Einstein's curved-space picture of gravity actually emerged, this being what was referred to as the “non-linear graviton construction”. This brought some basic features of quantum mechanics and general relativity together in a surprising way. About a year later, Richard Ward then saw how to provide a corresponding construction for general gauge fields whereby not only the interactions of Maxwell's electromagnetism could now be expressed in twistor terms, but so also could the self-interacting Yang–Mills equations of the strong and weak nuclear forces.

Yet, these non-linear constructions were explicitly limited to the left-handed helicity (in standard conventions), which meant that the “space-times” (and “fields”) described according to these constructions were fundamentally restricted, and had to have a conformal curvature (and gauge fields) that are necessarily anti-self-dual (or self-dual, if opposite conventions are adopted). With the Lorentzian signature of the ordinary space-times of Einstein's classical theory, this restriction is huge, and the constructions directly apply, only to space-times that are conformally flat. Despite the seemingly devastating limitation that this restriction imposed on the direct physical applicability of this aspect of twistor theory, such (anti-)self-duality was found to be intriguing from a pure-mathematical perspective, because for metrics that are of positive-definite (or split) signature, much non-trivial geometry arises. Indeed, this restriction is revealed to be a fruitful one to study for is own sake, leading to many deep connections between twistor methods and integrable systems, and with such topics as Kähler and hyperKähler geometry. All these areas are elegantly illustrated in various articles in this volume.

We see from the fascinating variety of articles provided here that there is much to be gained in investigating the roles played by twistor-type correspondences in numerous different geometrical contexts. Most of these articles explore such geometrical relationships and roles for twistor-related methods of proof, finding remarkable new contexts and ingenious novelty of pure-mathematical application. Yet, the physical applications of twistor theory are themselves not left behind, and in this collection we also find a fine illustration of what is currently by far the most active area of application of twistor theory in physics, this involving powerful new procedures for the calculation of scattering amplitudes in high-energy processes.

Impressive as these modern developments indeed are, both on the pure-mathematical side and in the calculation of physical processes, the (anti-)self-dual restriction for the full (non-perturbative) applicability of twistor methods in non-linear theories has represented a fundamental barrier to twistor theory being regarded as providing a fully plausible formalism for the description of basic physics. This is most blatant with regard to gravitation, but it applies also to the full description of the other forces of Nature. As it turns out, a broader outlook has recently been under development (palatial twistor theory) that appears to provide a way of surmounting this barrier in a very natural way. It will be interesting to see how such developments may affect both the physical and the mathematical applicability of twistor theory in future years.

Roger Penrose, September 2014        

Papers in this Issue:

Twistor Topology of the Fermat Cubic
John Armstrong and Simon Salamon
SIGMA 10 (2014), 061, 12 pages   [ abs   pdf ]
Gravity in Twistor Space and its Grassmannian Formulation
Freddy Cachazo, Lionel Mason and David Skinner
SIGMA 10 (2014), 051, 28 pages   [ abs   pdf ]
Scalar Flat Kähler Metrics on Affine Bundles over CP1
Nobuhiro Honda
SIGMA 10 (2014), 046, 25 pages   [ abs   pdf ]
Twistors and Bi-Hermitian Surfaces of Non-Kähler Type
Akira Fujiki and Massimiliano Pontecorvo
SIGMA 10 (2014), 042, 13 pages   [ abs   pdf ]
Twistor Theory of the Airy Equation
Michael Cole and Maciej Dunajski
SIGMA 10 (2014), 037,  8 pages   [ abs   pdf ]
Selfdual 4-Manifolds, Projective Surfaces, and the Dunajski-West Construction
David M.J. Calderbank
SIGMA 10 (2014), 035, 18 pages   [ abs   pdf ]
Integrable Background Geometries
David M.J. Calderbank
SIGMA 10 (2014), 034, 51 pages   [ abs   pdf ]
Hyperkähler Manifolds of Curves in Twistor Spaces
Roger Bielawski
SIGMA 10 (2014), 033, 13 pages   [ abs   pdf ]

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