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We find all hyper-Kähler 4-manifolds admitting conformal Kähler structures with respect to either orientation, and we show that these structures can be expressed as a combination of twistor elementary states (and possibly a self-dual dyon) in locally flat spaces. The complex structures of different flat pieces are not compatible however, reflecting that the global geometry is not a linear superposition. For either orientation, the space must be Gibbons–Hawking (thus excluding the Atiyah–Hitchin metric), and, if the orientations are opposite, it must also be toric and have an irreducible Killing tensor. We also show that the only hyper-Kähler 4-metric with a non-constant Killing–Yano tensor is the half-flat Taub–NUT instanton.

We give a simple interpretation of the adapted complex structure of Lempert–Szöke and Guillemin–Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant hypercomplex structure on a neighbourhood of X in TTX, where X is a real-analytic manifold equipped with a linear connection. We show that the Nahm equations arise naturally in this context: for a connection with zero curvature and arbitrary torsion, the real sections of the twistor space can be obtained by solving Nahm's equations in the Lie algebra of certain vector fields. Finally, we show that, if we start with a metric connection, then our construction yields an SO(3)-invariant hyperkähler metric.

Lorentz invariant supersymmetric deformations of superspaces based on Moyal star product parametrized by Majorana spinor λ_a and Ramond Grassmannian vector in the spinor realization⁴² are proposed. The map of supergravity background into composite supercoordinates: valid up to the second order corrections in deformation parameter h and transforming the background dependent Lorentz noninvariant (anti)commutators of supercoordinates into their invariant Moyal brackets is revealed. We found one of the deformations to depend on the axial vector and to vanish for the θ components with the same chiralities. The deformations in the (super)twistor picture are discussed.

Tree-level gluon scattering amplitudes in Yang-Mills theory frequently display simple mathematical structure which is completely obscure in the calculation of Feynman diagrams. We describe a novel way of calculating these amplitudes, motivated by a conjectured relation to twistor space, in which the problem of summing Feynman diagrams is replaced by the problem of solving a certain set of algebraic equations.

We give a short discussion/review of the recent developments expressing the perturbative scattering amplitudes in Yang-Mills theory, specifically for the theory, in terms of holomorphic curves in a supersymmetric twistor space. Holomorphic curves, which are maps of CP¹ to the supertwistor space, can also be interpreted as the lowest Landau level wave functions; this point of view is also briefly explained.

We extend the Shirafuji model for massless particles with primary space–time coordinates and composite four-momenta to a model for massive particles with spin and electric charge. The primary variables in the model are the space–time four-vector, four scalars describing spin and charge degrees of freedom as well as a pair of Weyl spinors. The geometric description proposed in this paper provides an intermediate step between the free purely twistorial model in two-twistor space in which both space–time and four-momenta vectors are composite, and the standard particle model, where both space–time and four-momenta vectors are elementary. We quantize the model and find explicitly the first-quantized wave functions describing relativistic particles with mass, spin and electric charge. The space–time coordinates in the model are not commutative; this leads to a wave function that depends only on one covariant projection of the space–time four-vector (covariantized time coordinate) defining plane wave solutions.

The action for a massless particle in 4D Minkowski space–time has a worldline-time reversing symmetry corresponding to CPT invariance of the quantum theory. The analogous symmetry of the ${𝒩}$-extended superparticle is shown to be anomalous when ${𝒩}$ is odd; in the supertwistor formalism this is because a CPT-violating worldline-Chern–Simons term is needed to preserve the chiral $U(1)$ gauge invariance. This accords with the fact that no massless ${𝒩}=1$ super-Poincaré irrep is CPT-self-conjugate. There is a CPT self-conjugate supermultiplet when ${𝒩}$ is even, but it has $2^{{𝒩}+1}$ states when $\frac{1}{2}{𝒩}$ is odd (e.g. the ${𝒩}=2$ hypermultiplet) in contrast to just $2^{{𝒩}}$ when $\frac{1}{2}{𝒩}$ is even (e.g. the ${𝒩}=4$ Maxwell supermultiplet). This is shown to follow from a Kramers degeneracy of the superparticle state space when $\frac{1}{2}{𝒩}$ is odd.

For many years now it has become conventional ([E. Witten, Phys. Today 49 (1996) 24; D. Gross, Einstein and the quest for a unified theory, in Einstein for the 21st Century: His Legacy in Science, Art, and Modern Culture, eds. P. L. Galison, G. Holton and S. S. Schweber (Princeton University Press, 2008), pp. 287–297; N. Arkani-Hamed, Spacetime is doomed (2010)]) for theorists to argue that “spacetime is doomed”, with the difficulties in finding a quantum theory of gravity implying the necessity of basing a fundamental theory on something quite different than usual notions of spacetime geometry. But what is this spacetime geometry that is doomed? In this paper, we will explore how our understanding of 4-dimensional geometry has evolved since Einstein, leading to new ideas about such geometry which may not be doomed at all.

The waveform of a binary black hole coalescence appears to be both simple and universal. In this essay we argue that the dynamics should admit a separation into “fast and slow” degrees of freedom, such that the latter are described by an integrable system of equations, accounting for the simplicity and universality of the waveform. Given that Painlevé transcendents are a smoking gun of integrable structures, we propose the Painlevé-II transcendent as the key structural element threading a hierarchy of asymptotic models aiming at capturing different (effective) layers in the dynamics. Ward’s conjecture relating integrable and (anti-)self-dual solutions can provide the avenue to encode background binary black hole data in (nonlocal) twistor structures.

An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, two-component spinor calculus, conformal gravity, α-planes in Minkowski space-time, α-surfaces and twistor geometry, anti-self-dual space-times and Penrose transform, spin-3/2 potentials, heaven spaces and heavenly equations.

This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far-reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to non-commutative symplectic symmetric spaces.

Some properties of the Clifford algebras and are presented, and three isomorphisms between the Dirac–Clifford algebra and are exhibited, in order to construct conformal maps and twistors, using the paravector model of spacetime. The isomorphism between the twistor space inner product isometry group SU(2,2) and the group $pin₊(2,4) is also investigated, in the light of a suitable isomorphism between and . After reviewing the conformal spacetime structure, conformal maps are described in Minkowski spacetime as the twisted adjoint representation of $pin₊(2,4), acting on paravectors. Twistors are then presented via the paravector model of Clifford algebras and related to conformal maps in the Clifford algebra over the Lorentzian ℝ^4,1 spacetime. We construct twistors in Minkowski spacetime as algebraic spinors associated with the Dirac–Clifford algebra using one lower spacetime dimension than standard Clifford algebra formulations, since for this purpose, the Clifford algebra over ℝ^4,1 is also used to describe conformal maps, instead of ℝ^2,4. Our formalism sheds some new light on the use of the paravector model and generalizations.

In this contribution I would like to review some topics in which Jerzy Lukierski has been one of the main players and contributed over decades with many interesting and important results. We shall consider various ways in which relativistic particles can be endowed with spin degrees of freedom with the use of twistor-like variables and supersymmetrization, which upon quantization lead either to higher-spin fields (in D > 3) or to anions (in D = 3).

In this paper, a generic counterexample to the strong cosmic censor conjecture is exhibited. More precisely — taking into account that the conjecture lacks any precise formulation yet — first we make sense of what one would mean by a "generic counterexample" by introducing the mathematically unambigous and logically stronger concept of a "robust counterexample". Then making use of Penrose' nonlinear graviton construction (i.e. twistor theory) and a Wick rotation trick we construct a smooth Ricci-flat but not flat Lorentzian metric on the largest member of the Gompf — Taubes uncountable radial family of large exotic ℝ⁴'s. We observe that this solution of the Lorentzian vacuum Einstein's equations with vanishing cosmological constant provides us with a sort of counterexample which is weaker than a "robust counterexample" but still reasonable to consider as a "generic counterexample". It is interesting that this kind of counterexample exists only in four dimensions.

Superspace is considered as space of parameters of the supercoherent states defining the basis for oscillator-like unitary irreducible representations of the generalized superconformal group $SU(2m,2n|2N)$ in the field of quaternions $ℍ$. The specific construction contains naturally the supertwistor one of the previous work by Litov and Pervushin [1] and it is shown that in the case of extended supersymmetry such an approach leads to the separation of a class of superspaces and its groups of motion. We briefly discuss this particular extension to the domain of quaternionic superspaces as nonlinear realization of some kind of the affine and the superconformal groups with the final end to include also the gravitational field [6] (this last possibility to include gravitation, can be realized on the basis of Ref. 12 where the coset $\frac{Sp(8)}{SL(4R)}\sim$$\frac{SU(2,2)}{SL(2C)}$ was used in the non supersymmetric case). It is shown that this quaternionic construction avoid some unconsistencies appearing at the level of the generators of the superalgebras (for specific values of $p$ and $q$; $p+q=N$) in the twistor one.

We show that (specifically scaled) equations of shear-free null geodesic congruences on the Minkowski space-time possess intrinsic self-dual, restricted gauge and algebraic structures. The complex eikonal, Weyl 2-spinor, $SL(2,ℂ)$ Yang–Mills and complex Maxwell fields, the latter produced by integer-valued electric charges (“elementary” for the Kerr-like congruences), can all be explicitly associated with any shear-free null geodesic congruence. Using twistor variables, we derive the general solution of the equations of the shear-free null geodesic congruence (as a modification of the Kerr theorem) and analyze the corresponding “particle-like” field distributions, with bounded singularities of the associated physical fields. These can be obtained in a straightforward algebraic way and exhibit nontrivial collective dynamics simulating physical interactions.

In 2003, Witten proposed a topological string theory in twistor space that is dual to a weakly coupled gauge theory. This has lead to new developments in computing gauge theory scattering amplitudes. In these proceedings, we review two new methods, the MHV vertices construction and the on-shell recursion relations.

We discuss recent developments in p-adic geometry, ranging from foundational results such as the degeneration of the Hodge-to-de Rham spectral sequence for “compact p-adic manifolds” over new period maps on moduli spaces of abelian varieties to applications to the local and global Langlands conjectures, and the construction of “universal” p-adic cohomology theories. We finish with some speculations on how a theory that combines all primes p, including the archimedean prime, might look like.

The exact Kerr-Schild (KS) solutions for electromagnetic excitations of black-holes have the form of singular beams supported on twistor lines. The beams have very strong back reaction on metric and horizon and produce a fluctuating KS geometry. Its holographic structure forms a pre-quantum spacetime taking intermediate position between the Classical and Quantum gravity.

Contradiction between Quantum theory and Gravity contains two principal points: i/uncompatibility with the quantum statement on the pointlike electron and ii/distortion of the plane waves and the related Fourier transform in curved spaces. The general point of view on the priority of Quantum theory doesn't lead to progress, and we suggest to listen attentively to Gravity's view, which suggests for item i/ singular pp-waves and twistorial Fourier transform, and for item ii/ the Kerr-Newman (KN) electron model based on the twistorial structure of the Kerr theorem. Both suggestions are related with the holographic twistorial structure of the Kerr-Schild (KS) geometry, indicating the twistorial KS way to Quantum Gravity.