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This paper describes conservation laws in general relativity (GR) dating back to the mass-energy conservation of Bondi and Sachs in the early 1960s but using 2-spinor techniques. The notion of conformal infinity is employed, and the highly original ideas of E. T. Newman are discussed in relation to twistor theory. The controversial NP constants are introduced, and their meaning is considered in a new light related to the problem of equations of motion in GR. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'. [ABSTRACT FROM AUTHOR]

The study of physical theories in various signatures has been important for uncovering structures not easily visible or definable in Lorentz signature. In split signature, global twistor constructions for conformally self-dual (SD) gravity and Yang–Mills construct solutions from twistor data that can be expressed in terms of free data without gauge freedom. This is developed for asymptotically flat SD gravity to give a fully nonlinear encoding of the asymptotic gravitational data in terms of a real homogeneous generating function h on the real twistor space. The recently discovered L w 1 + ∞ celestial symmetries, when real, act locally as passive Poisson diffeomorphisms on the real twistor space. The twistor data, h, generates an imaginary such Poisson transformation that then generates the gravitational field by shifting the real slice of the twistor space. The twistor chiral sigma models, whose correlators yield the Einstein gravity tree-level S-matrix, are reformulated as theories of holomorphic discs in twistor space whose boundaries lie on the deformed real slice determined by h. The real L w 1 + ∞ symmetries act on the corresponding formula for the S-matrix geometrically with vanishing Noether currents, but imaginary generators yield graviton vertex operators that generate gravitons in the perturbative expansion. A generating function for the all plus 1-loop amplitude, the analogous framework for Yang–Mills, possible interpretations in Lorentz signature and similar open string formulations of twistor and ambitwistor strings in 4d in split signature, are briefly discussed. [ABSTRACT FROM AUTHOR]

We consider holomorphic Poisson-BF theory on twistor space. Classically, this describes self-dual Einstein gravity on space-time, but at the quantum level it is plagued by an anomaly. The anomaly corresponds to the fact that integrability of the self-dual vacuum Einstein equations does not survive in self-dual quantum gravity. We compute the anomaly polynomials in the Poisson-BF theory, as well as in this theory coupled to a holomorphic BF theory on twistor space describing self-dual Yang–Mills. We show that all anomalies may be cancelled by further coupling to a twistor field representing a type of axion on space-time. When the twistor anomalies are cancelled, all n ≥ 4 -pt amplitudes vanish and integrability is restored. [ABSTRACT FROM AUTHOR]

In this paper, I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. At the nonrelativistic level, there is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes. This is the cosmological analogue of the Penrose effect, which can also be seen as a nonrelativistic limit of the Unruh effect. The symmetries of the associated Schrödinger equation are generated by the Newton–Hooke algebra, which arises from a nonrelativistic limit of a cosmological twistor space. [ABSTRACT FROM AUTHOR]

The existence of interacting higher-spin theories is tightly constrained by many no-go theorems. In this paper, we construct a chiral, higher-spin generalization of Yang–Mills theory in flat space which avoids these no-go theorems and has non-trivial tree-level scattering amplitudes with some higher-spin external legs. The fields and action are complex, so the theory is non-unitary and parity-violating, yet we find surprisingly compact formulae for all-multiplicity tree-level scattering amplitudes in the maximal helicity violating (MHV) sector, where the two negative helicity particles have identical but arbitrary spin. This is possible because the theory admits a perturbative expansion around its self-dual sector. Using twistor theory, we prove the classical integrability of this self-dual sector and show that it can be described on spacetime by an infinite tower of interacting massless scalar fields. We also give a twistor construction of the full theory and use it to derive the formula for the MHV amplitude. [ABSTRACT FROM AUTHOR]

The construction of amplitudes on curved space-times is a major challenge, particularly when the background has non-constant curvature. We give formulae for all tree-level graviton scattering amplitudes in curved self-dual (SD) radiative space-times; these are chiral, source-free, asymptotically flat spaces determined by free characteristic data at null infinity. Such space-times admit an elegant description in terms of twistor theory, which provides the powerful tools required to exploit their underlying integrability. The tree-level S-matrix is written in terms of an integral over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map corresponding to the helicity configuration of the external gravitons. For the MHV sector, we derive the amplitude directly from the Einstein–Hilbert action of general relativity, while other helicity configurations arise from a natural family of generating functionals and pass several consistency checks. The amplitudes in SD radiative space-times exhibit many novel features that are absent in Minkowski space, including tail effects. There remain residual integrals due to the functional degrees of freedom in the background space-time, but our formulae have many fewer such integrals than would be expected from space-time perturbation theory. In highly symmetric special cases, such as SD plane waves, the number of residual integrals can be further reduced, resulting in much simpler expressions for the scattering amplitudes. [ABSTRACT FROM AUTHOR]

We present all-multiplicity formulae, derived from first principles in the MHV sector and motivated by twistor string theory for general helicities, for the tree-level S-matrix of gluon scattering on self-dual radiative backgrounds. These backgrounds are chiral, asymptotically flat gauge fields characterised by their free radiative data, and their underlying integrability is captured by twistor theory. Tree-level gluon scattering scattering amplitudes are expressed as integrals over the moduli space of holomorphic maps from the Riemann sphere to twistor space, with the degree of the map related to the helicity configuration of the external gluons. In the MHV sector, our formula is derived from the Yang–Mills action; for general helicities the formulae are obtained using a background-coupled twistor string theory and pass several consistency tests. Unlike amplitudes on a trivial vacuum, there are residual integrals due to the functional freedom in the self-dual background, but for scattering of momentum eigenstates we are able to do many of these explicitly and even more is possible in the special case of plane wave backgrounds. In general, the number of these integrals is always less than expected from standard perturbation theory, but matches the number associated with space-time MHV rules in a self-dual background field, which we develop for self-dual plane waves. [ABSTRACT FROM AUTHOR]

On-shell kinematics for gluon scattering can be parametrized with points on the celestial sphere; in the limit where these points collide, it is known that tree-level gluon scattering amplitudes exhibit an operator product expansion (OPE)-like structure. While it is possible to obtain singular contributions to this celestial OPE, getting regular contributions from both holomorphic and anti-holomorphic sectors is more difficult. In this paper, we use twistor string theory to describe the maximal helicity violating (MHV) sector of tree-level, four-dimensional gluon scattering as an effective 2d conformal field theory on the celestial sphere. By organizing the OPE between vertex operators in this theory in terms of soft gluon descendants, we obtain all-order expressions for the celestial OPE which include all regular contributions in the collinear expansion. This gives new, all-order formulae for the collinear splitting function (in momentum space) and celestial OPE coefficients (in the conformal primary basis) of tree-level MHV gluon scattering. We obtain these results for both positive and negative helicity gluons, and for any incoming/outgoing kinematic configuration within the MHV sector. [ABSTRACT FROM AUTHOR]

The aim of this paper is to explore the conceptual and mathematical physics foundations of the work of Sir Roger Penrose. This is a transcript of an interview that took place on 15 November 2021. [ABSTRACT FROM AUTHOR]

An adaptive finite-time attitude and position coupled tracking controller for six-degree-of-freedom (6-DOF) leader-following spacecraft formation is proposed in this paper. The relative motion of the follower with respect to its desired pose is represented by twistors in a unified way without state redundancy. A fast terminal sliding surface is proposed for the twistor-based dynamic system to achieve finite-time convergence. Then an adaptive finite-time controller is developed with the unknown inertial parameters of the follower spacecraft and the upper bounds of the disturbances estimated by elaborately designed adaptive laws. The finite-time convergence of the closed-loop system is proven via Lyapunov analysis. In addition, an simple but effective strategy is proposed to eliminate the attitude unwinding phenomenon. The proposed adaptive finite-time control scheme possesses faster convergence, higher control accuracy, and more preferable control inputs. In addition, the possible numerical difficulties caused by logarithmic function of dual quaternions is avoided. Finally, simulation results demonstrate the effectiveness of the proposed algorithm. [ABSTRACT FROM AUTHOR]

The space of light rays N of a conformal Lorentz manifold (M , C) is, under some topological conditions, a manifold whose basic elements are unparametrized null geodesics. This manifold N , strongly inspired on R. Penrose's twistor theory, keeps all information of M and it could be used as a space complementing the spacetime model. In the present review, the geometry and related structures of N , such as the space of skies Σ and the contact structure H , are introduced. The causal structure of M is characterized as part of the geometry of N . A new causal boundary for spacetimes M prompted by R. Low, the L-boundary, is constructed in the case of 3–dimensional manifolds M and proposed as a model of its construction for general dimension. Its definition only depends on the geometry of N and not on the geometry of the spacetime M. The properties satisfied by the L–boundary ∂ M permit to characterize the obtained extension M ¯ = M ∪ ∂ M and this characterization is also proposed for general dimension. [ABSTRACT FROM AUTHOR]

The classical double copy relating exact solutions of biadjoint scalar, gauge and gravity theories continues to receive widespread attention. Recently, a derivation of the exact classical double copy was presented, using ideas from twistor theory, in which spacetime fields are mapped to Cech cohomology classes in twistor space. A puzzle remains, however, in how to interpret the twistor double copy, in that it relies on somehow picking special representatives of each cohomology class. In this paper, we provide two alternative formulations of the twistor double copy using the more widely-used language of Dolbeault cohomology. The first amounts to a rewriting of the Cech approach, whereas the second uses known techniques for discussing spacetime fields in Euclidean signature. The latter approach indeed allows us to identify special cohomology representatives, suggesting that further application of twistor methods in exploring the remit of the double copy may be fruitful. [ABSTRACT FROM AUTHOR]

In the framework of twistors, a saturated adaptive six-degree-of-freedom (6-DOF) control law for asteroid landing, which avoids the collision between the spacecraft and asteroid, is proposed with both the inertial parameters of the spacecraft and the bounds of the uncertainties unknown. First, the 6-DOF dynamics of the spacecraft relative to the desired coordinate frame is represented in a unified way by the application of twistors, and the collision avoidance constraint is established via separating the spacecraft and asteroid with an elaborately designed safety surface. Then, a saturated controller for the landing is proposed with the collision avoidance constraint considered by combining the backstepping method and artificial potential field technique. Further, the inertial parameters and the upper bounds of the magnitude of the uncertainties in the controller are substituted with their estimates obtained from the designed adaptive laws. Finally, the previous design is adjusted slightly to complete the adaptive control scheme. The stability of the closed-loop system is proven via Lyapunov theory, and numerical simulations are presented to demonstrate the effectiveness of the control scheme. The proposed adaptive control law can fully consider the orbit-attitude coupling and effectively enforce the collision avoidance constraint in the presence of unknowns in asteroid landing missions. [ABSTRACT FROM AUTHOR]

The search for a geometrical understanding of dualities in string theory, in particular T-duality, has led to the development of modern T-duality covariant frameworks such as Double Field Theory, whose mathematical structure can be understood in terms of generalized geometry and, more recently, para-Hermitian geometry. In this work we apply techniques associated to this doubled geometry to four-dimensional manifolds, and we show that they are particularly well-suited to the analysis of integrability in special spacetimes, especially in connection with Penrose's twistor theory and its applications to general relativity. This shows a close relationship between some of the geometrical structures in the para-Hermitian approach to double field theory and those in algebraically special solutions to the Einstein equations. Particular results include the classification of four-dimensional, possibly complex-valued, (para-)Hermitian structures in different signatures, the Lie and Courant algebroid structures of special spacetimes, and the analysis of deformations of (para-)complex structures. We also discuss a notion of "weighted algebroids" in relation to a natural gauge freedom in the framework. Finally, we analyze the connection with two- and three-dimensional (real and complex) twistor spaces, and how the former can be understood in terms of the latter, in particular in terms of twistor families. [ABSTRACT FROM AUTHOR]

We review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of M 0 , n , the moduli space of n points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle T D ∗ M 0 , n , the bundle of forms with logarithmic singularities on the divisor D as the twistor space, and K n the space of momentum invariants of n massless particles subject to momentum conservation as the analogue of space–time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain n - 3 -forms on K n , introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral n - 3 -planes in K n introduced by ABHY. [ABSTRACT FROM AUTHOR]

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions. [ABSTRACT FROM AUTHOR]

Higher Spin Gravities are scarce, but covariant actions for them are even scarcer. We construct covariant actions for contractions of Chiral Higher Spin Gravity that represent higher spin extensions of self-dual Yang-Mills and self-dual Gravity theories. The actions give examples of complete higher spin theories both in flat and (anti)-de Sitter spaces that feature gauge and gravitational interactions. The actions are based on a new description of higher spin fields, whose origin can be traced to early works on twistor theory. The new description simplifies the structure of interactions. In particular, we find a covariant form of the minimal gravitational interaction for higher spin fields both in flat and anti-de Sitter space, which resolves some of the puzzles in the literature. [ABSTRACT FROM AUTHOR]

The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings. [ABSTRACT FROM AUTHOR]

The article is dedicated to the 70th anniversary of Prof. Oleg Mushkarov – a full professor at the Institute of Mathematics and Informatics, Bulgarian Academy of Sciences (IMI-BAS) and a corresponding member of BAS. Currently, he is the Head of the Analysis, Geometry and Topology Department and Deputy Director of IMI-BAS. His main research interests are in the areas of Complex Analysis, Complex Differential Geometry, and Twistor theory. Another important aspect of Oleg’s professional involvements is his work with mathematically gifted high school students. He himself benefited from the Bulgarian system of identifying mathematical talents at an early age leading to his participation in IMO’1969, Romania. Since then, he has been involved in numerous activities dealing with nurturing and developing young mathematical talents. We present a conversation with him illustrating what his favorite problems are and why. Unlike the traditional situations in which the professor expects the students to come to a solution (possibly with his help), he presents a solution containing mistakes to a well-defined problem and “throw the gauntlet down” for them: Find the mistakes in the solution! We provide a specific example in support of his conviction that such problems are crucial in math education (since everyone, even a professional mathematician, makes errors on an everyday basis). The phases the talk in front of the white board went through were: To feel that something is wrong in his original solution. To discover where the mistake is. Finally, to find a correct solution to the problem. The very process of starting with a wrong idea, becoming aware of the error in reasoning, and finally, succeeding to correct it, which Oleg Mushkarov has kept reproducing when teaching a course on extracurricular math activities to his undergraduate students, reveals his dedication and talents to teaching in addition to his research achievements. [ABSTRACT FROM AUTHOR]

In this paper, we present new expressions for n-point NMHV tree-level gravity amplitudes. We introduce a method of factorization diagrams which is a simple graphical representation of R-invariants in Yang-Mills theory. We define the gravity analogues which we call G -invariants, and expand the NMHV gravity amplitudes in terms of these objects. We provide explicit formulas of NMHV gravity amplitudes up to eight points in terms of G -invariants, and give the general definition for any number of points. We discuss the connection to BCFW representation, special behavior under large momentum shift, the role of momentum twistors and the intricate web of spurious poles cancelation. Because of the close connection between R-invariants and the (tree-level) Amplituhedron for Yang-Mills amplitudes, we speculate that the new expansion for gravity amplitudes should correspond to the triangulation of the putative Gravituhedron geometry. [ABSTRACT FROM AUTHOR]

Superconformal geometries in spacetime dimensions D = 3, 4, 5 and 6 are discussed in terms of local supertwistor bundles over standard superspace. These natually admit superconformal connections as matrix-valued one-forms. In order to make contact with the standard superspace formalism it is shown that one can always choose gauges in which the scale parts of the connection and curvature vanish, in which case the conformal and S-supersymmetry transformations become subsumed into super-Weyl transformations. The number of component fields can be reduced to those of the minimal off-shell conformal supergravity multiplets by imposing constraints which in most cases simply consists of taking the even covariant torsion two-form to vanish. This must be supplemented by further dimension-one constraints for the maximal cases in D = 3, 4. The subject is also discussed from a minimal point of view in which only the dimension-zero torsion is introduced. Finally, we introduce a new class of supermanifolds, local super Grassmannians, which provide an alternative setting for superconformal theories. [ABSTRACT FROM AUTHOR]

We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane OP2=F4/Spin(9). By exhibiting this symmetric space as a submanifold of the Grassmannian of 10‐dimensional subspaces of the fundamental representation of F4, techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into classical Grassmannians can also be applied in this case. The originality of our approach lies on the use of the classification of nilpotent orbits in Lie algebras as described by Djoković. [ABSTRACT FROM AUTHOR]

Several methods use the Fourier transform from momentum space to twistor space to analyze scattering amplitudes in Yang–Mills theory. However, the transform has not been defined as a concrete complex integral when the twistor space is a three-dimensional complex projective space. To the best of our knowledge, this is the first study to define it as well as its inverse in terms of a concrete complex integral. In addition, our study is the first to show that the Fourier transform is an isomorphism from the zeroth Čech cohomology group to the first one. Moreover, the well-known twistor operator representations in twistor theory literature are shown to be valid for the Fourier transform and its inverse transform. Finally, we identify functions over which the application of the operators is closed. [ABSTRACT FROM AUTHOR]

Twistors can represent the spin in superstring theory and provide a theoretical explanation of the two-dimensional spherical model of elementary particles together with the surface distributions of the charges. The integrability of certain matter sectors of the strong interactions also may be traced to the action of the conformal group on collinear trajectories that occur in a twistor formulation. The amplitudes for these scattering processes in four dimensions have a representation with quantum states in a nonsupersymmetric theory defined by the low-energy limit of a solution to the N = 2 string equations with a Wick rotation of the metric with (2, 2) signature. The twistor form of an N = 2 scattering amplitude is given. [ABSTRACT FROM AUTHOR]

In the present paper we use twistor theory in order to solve two problems related to harmonic maps from surfaces to Euclidean spheres S n . First, we propose a new approach to isoperimetric eigenvalue inequalities based on energy index. Using this approach we show that for any positive k, the k-th non-zero eigenvalue of the Laplacian on the real projective plane endowed with a metric of unit area, is maximized on the sequence of metrics converging to a union of (k - 1) identical copies of round sphere and a single round projective plane. This extends the results of Li and Yau (Invent Math 69(2):269–291, 1982) for k = 1 ; Nadirashvili and Penskoi (Geom Funct Anal 28(5):1368–1393, 2018) for k = 2 ; and confirms the conjecture made in (KNPP). Second, we improve the known lower bounds for the area index of minimal two-dimensional spheres and minimal projective planes in S n . In the course of the proof we establish a twistor correspondence for Jacobi fields, which could be of independent interest for the study of moduli spaces of harmonic maps. [ABSTRACT FROM AUTHOR]

The Teukolsky equations are currently the leading approach for analysing stability of linear massless fields propagating in rotating black holes. It has recently been shown that the geometry of these equations can be understood in terms of a connection constructed from the conformal and complex structure of Petrov type D spaces. Since the study of linear massless fields by a combination of conformal, complex and spinor methods is a distinctive feature of twistor theory, and since versions of the twistor equation have recently been shown to appear in the Teukolsky equations, this raises the question of whether there are deeper twistor structures underlying this geometry. In this work we show that all these geometric structures can be understood naturally by considering a 2-dimensional twistor manifold, whereas in twistor theory the standard (projective) twistor space is 3-dimensional. [ABSTRACT FROM AUTHOR]

Jackiw –Teitelboim (JT) gravity is a 1 + 1-dimensional toy model for quantum gravity in four spacetime dimensions. In the absence of matter, JT gravity is a topological field theory and there are no local observables. The introduction of a boundary changes the situation. What was an unphysical gauge direction before turns into a physical boundary degree of freedom (a gravitational edge mode). Starting from the BF formulation of JT gravity, we develop a twistor representation for the boundary charges of JT gravity. We introduce point sources in the bulk and study the coupled gravity plus matter system at the quantum level. Eigenstates of quasi-local energy are built from the tensor product of unitary irreducible representations of. Empty patches of AdS2 are characterised by the continuous series representations, point sources are related to the discrete series representations. Physical states are constructed by fusing the representations into an invariant singlet. The singlet lies in the kernel of the constraints, which are the analogue of the Wheeler – De Witt equations in the presence of distributional matter sources. [ABSTRACT FROM AUTHOR]

We present compact analytic formulae for the one-loop amplitudes for Higgs + 4 parton scattering, 0 → ggggh, 0 → q ¯ qggh and 0 → q ¯ q q ¯ ′ q′h, mediated by a loop of massive coloured quarks. We exploit the correspondence with a theory in which a massive coloured scalar circulates in the loop to avoid a proliferation in the number of terms in the result. In addition, we use momentum twistors and high precision numerical evaluations to simplify the expressions. The analytic results in this paper, in terms of spinor products, allow construction of an efficient numerical program to calculate the amplitude. [ABSTRACT FROM AUTHOR]

It is known that the supermultiplet of beta-deformations of N = 4 supersymmetric Yang–Mills theory can be described in terms of the exterior product of two adjoint representations of the superconformal algebra. We present a supergeometrical interpretation of this fact, by evaluating the deforming operator on some special coherent states in the space of supersingletons. We also discuss generalization of this approach to other finite-dimensional deformations of the N = 4 supersymmetric Yang–Mills theory. [ABSTRACT FROM AUTHOR]

We introduce a notion of ϕ 1 -coordinated module twistor for a ϕ 1 -coordinated module of a nonlocal vertex algebra. Furthermore, we study twisted tensor products and L-R-twisted tensor products of ϕ 1 -coordinated modules of nonlocal vertex algebras and we unify these two constructions by ϕ 1 -coordinated module twistors. [ABSTRACT FROM AUTHOR]

Abstract Starting with the first-order formulation of the massless superparticle model on the A d S 5 × S 5 superbackground and presenting the momentum components tangent to A d S 5 and S 5 subspaces as bilinear combinations of the constrained S U (2) -Majorana spinors allows to bring the superparticle's Lagrangian to the form quadratic in supertwistors. The S U (2 , 2 | 4) supertwistors are assembled into a pair of S U (2) doublets, one of which has even S U (2 , 2) and odd S U (4) components, while the other has odd S U (2 , 2) and even S U (4) components. They are subject to the first-class constraints that generate the p s u (2 | 2) ⊕ u (1) gauge algebra. This justifies previously proposed group-theoretic definition of the A d S 5 × S 5 supertwistors and allows to derive the incidence relations with the (10 | 32) supercoordinates of the A d S 5 × S 5 superspace. Whenever superparticle moves within the A d S 5 subspace of the A d S 5 × S 5 space-time, twistor formulation of its Lagrangian involves just one S U (2) doublet of S U (2 , 2 | 4) supertwistors with even S U (2 , 2) and odd S U (4) components. If in addition particle's 5-momentum is null, four first-class constraints which are the s u (2) ⊕ u (1) generators single out upon quantization the states of D = 5 N = 8 gauged supergravity multiplet in the superambitwistor formulation. [ABSTRACT FROM AUTHOR]

Symmetry operators of twistor spinors and harmonic spinors can be constructed from conformal Killing-Yano (CKY) forms. Transformation operators relating twistors to harmonic spinors are found in terms of potential forms. These constructions are generalized to gauged twistor spinors and gauged harmonic spinors. The operators that transform gauged twistor spinors to gauged harmonic spinors are found. Symmetry operators of gauged harmonic spinors in terms of CKY forms are obtained. Algebraic conditions to obtain solutions of the Seiberg-Witten equations are discussed. [ABSTRACT FROM AUTHOR]

Abstract: For any integer k we construct an explicit example of a twistor space which contains a one‐parameter family of jumping rational curves, where the normal bundle changes from O ( 1 ) ⊕ O ( 1 ) to O ( k ) ⊕ O ( 2 − k ). For k > 3 the resulting anti‐self‐dual Ricci‐flat manifold is a Zariski cone in the space of holomorphic sections of O ( k ). In the case k = 2 we recover the canonical example of Hitchin's folded hyper‐Kähler manifold, where the jumping lines form a three‐parameter family. We show that in this case there exist normalisable solutions to the Schrödinger equation which extend through the fold. [ABSTRACT FROM AUTHOR]

We discuss a number of problems associated with the circle compactification of the bosonic tensile ambitwistor string with the asymmetric vacuum choice. By considering the spectrum and physical state conditions, we show that the circle radius plays a role as a tuning parameter which determines the low energy effective field theory. At the self dual point, we construct the current operators and compute OPEs between them. While the final outcome remains as yet inconclusive, several new results are obtained. Through the current algebra analysis we show that there is a subsector of the massless states where the gauge symmetry is enhanced to I S O ( 2 ) L × S U ( 2 ) R . Using the fact that the one loop partition function is not modular invariant, we show that tensile ambitwistor string partition function is exactly the same as a field theory partition function. Our result proves that despite the existence of winding modes, which is a typical characteristic of a string, the tensile ambitwistor string behaves as a point particle theory. [ABSTRACT FROM AUTHOR]

This note proves a conjecture of Sparling [Germ of a synthesis: space-time is spinorial, extra dimensions are time-like. Proc R Soc Lond Ser A Math Phys Eng Sci. 2007;463(2083):1665-1679; The Ξ-transform for conformally flat space-time. arXiv:gr-qc/0612006; 2006.], that the ultrahyperbolic wave operator and the Ξ transform on form an exact couple. [ABSTRACT FROM AUTHOR]

We propose a new world-line Lagrangian model of the D=4 massless relativistic particle with continuous spin and develop its twistorial formulation. The description uses two Penrose twistors subjected to four first class constraints. After the first quantization of the world-line twistorial model, the wave function is defined by an unconstrained function on the two-dimensional complex affine plane. We find the twistor transform that determines the space-time field of the continuous spin particle through the corresponding twistor one, which plays the role of a prepotential. It is shown that this space-time field is an exact solution of the space-time constraints defining the irreducible massless representation of the Poincaré group with continuous spin. [ABSTRACT FROM AUTHOR]

A massive rigid particle model in ( 3 + 1 ) dimensions is reformulated in terms of twistors. Beginning with a first-order Lagrangian, we establish a twistor representation of the Lagrangian for a massive particle with rigidity. The twistorial Lagrangian derived in this way remains invariant under a local U ( 1 ) × U ( 1 ) transformation of the twistor and other relevant variables. Considering this fact, we carry out a partial gauge-fixing so as to make our analysis simple and clear. We develop the canonical Hamiltonian formalism based on the gauge-fixed Lagrangian and perform the canonical quantization procedure of the Hamiltonian system. Also, we obtain an arbitrary-rank massive spinor field in ( 3 + 1 ) dimensions via the Penrose transform of a twistor function defined in the quantization procedure. Then we prove, in a twistorial fashion, that the spin quantum number of a massive particle with rigidity can take only non-negative integer values, which result is in agreement with the one shown earlier by Plyushchay. Interestingly, the mass of the spinor field is determined depending on the spin quantum number. [ABSTRACT FROM AUTHOR]

We discuss the equivalence principle in quantum mechanics in the context of Newton–Cartan geometry, and non-relativistic twistor theory. [ABSTRACT FROM AUTHOR]

Combining elements of twistor-space, phase space, and Clifford algebras, we propose a framework for the construction and quantization of certain (quadric) varieties described by Lorentz-covariant multivector coordinates. The correspondent multivectors can be parametrized by second order polynomials in the phase space. Thus the multivectors play a double role, as covariant objects in D = 2, 3, 4 Mod 8 space-time dimensions and as mechanical observables of a non-relativistic system in 2[D/2]−1 Euclidean dimensions. The latter attribute permits a dual interpretation of concepts of non-relativistic mechanics as applying to relativistic space-time geometry. Introducing the Groenewold-Moyal *-product and Wigner distributions in phase space induces Lorentz-covariant non-commutativity, and it provides the spectra of geometrical observables. We propose also new (multivector) matrix models, interpreted as descending from the interaction term of a Yang-Mills theory with minimally coupled massive fermions, in the large-N limit, which serves as a physical model containing the constructed multivector (fuzzy) geometries. We also include a section on speculative aspects on a possible cosmological effect and the origin of space-time entropy. [ABSTRACT FROM AUTHOR]

Non-conventional topology of elastic waves arises from breaking symmetry of phononic structures either intrinsically through internal resonances or extrinsically via application of external stimuli. We develop a spacetime representation based on twistor theory of an intrinsic topological elastic structure composed of a harmonic chain attached to a rigid substrate. Elastic waves in this structure obey the Klein-Gordon and Dirac equations and possesses spinorial character. We demonstrate the mapping between straight line trajectories of these elastic waves in spacetime and the twistor complex space. The twistor representation of these Dirac phonons is related to their topological and fermion-like properties. The second topological phononic structure is an extrinsic structure composed of a onedimensional elastic medium subjected to a moving superlattice. We report an analogy between the elastic behavior of this time-dependent superlattice, the scalar quantum field theory and general relativity of two types of exotic particle excitations, namely temporal Dirac phonons and temporal ghost (tachyonic) phonons. These phonons live on separate sides of a two-dimensional frequency space and are delimited by ghost lines reminiscent of the conventional light cone. Both phonon types exhibit spinorial amplitudes that can be measured by mapping the particle behavior to the band structure of elastic waves. [ABSTRACT FROM AUTHOR]

The class of hyper-CR Einstein–Weyl structures on R 3 can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Kruglikov and A. Panasyuk. Our method produces also the hyper-CR equation and can be applied to other geometric structures related to different twistor constructions. [ABSTRACT FROM AUTHOR]

The two-twistor formulation of particle mechanics in D-dimensional anti-de Sitter space for D = 4 , 5 , 7, which linearises invariance under the AdS isometry group Sp(4; $$ \mathbb{K} $$ ) for $$ \mathbb{K}=\mathbb{R},\mathbb{C},\mathrm{\mathbb{H}} $$ , is generalized to the massless N -extended 'spinning particle'. The twistor variables are gauge invariant with respect to the initial N local worldline supersymmetries; this simplifies aspects of the quantum theory such as implications of global gauge anomalies. We also give details of the two-supertwistor form of the superparticle, in particular the massive superparticle on AdS. [ABSTRACT FROM AUTHOR]

We consider the holographic duality between type-A higher-spin gravity in AdS and the free U( N) vector model. In the bulk, linearized solutions can be translated into twistor functions via the Penrose transform. We propose a holographic dual to this transform, which translates between twistor functions and CFT sources and operators. We present a twistorial expression for the partition function, which makes global higher-spin symmetry manifest, and appears to automatically include all necessary contact terms. In this picture, twistor space provides a fully nonlocal, gauge-invariant description underlying both bulk and boundary spacetime pictures. While the bulk theory is handled at the linear level, our formula for the partition function includes the effects of bulk interactions. Thus, the CFT is used to solve the bulk, with twistors as a language common to both. A key ingredient in our result is the study of ordinary spacetime symmetries within the fundamental representation of higher-spin algebra. The object that makes these 'square root' spacetime symmetries manifest becomes the kernel of our boundary/twistor transform, while the original Penrose transform is identified as a 'square root' of CPT. [ABSTRACT FROM AUTHOR]

Given a slice regular function f : Ω ⊂ H → H , with Ω ∩ R ≠ ∅ , it is possible to lift it to surfaces in the twistor space CP 3 of S 4 ≃ H ∪ { ∞ } (see Gentili et al., 2014). In this paper we show that the same result is true if one removes the hypothesis Ω ∩ R ≠ ∅ on the domain of the function f . Moreover we find that if a surface S ⊂ CP 3 contains the image of the twistor lift of a slice regular function, then S has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in CP 3 that contain the lift of a slice regular function. In addition we extend and further explore the so-called twistor transform, that is a curve in G r 2 ( C 4 ) which, given a slice regular function, returns the arrangement of lines whose lift carries on. With the explicit expression of the twistor lift and of the twistor transform of a slice regular function we exhibit the set of slice regular functions whose twistor transform describes a rational line inside G r 2 ( C 4 ) , showing the role of slice regular functions not defined on R . At the end we study the twistor lift of a particular slice regular function not defined over the reals. This example shows the effectiveness of our approach and opens some questions. [ABSTRACT FROM AUTHOR]

This paper establishes the relation between traditional results from the (Euclidean) twistor theory and chiral formulations of general relativity (GR), especially the pure connection formulation. Starting from an SU(2)-connection only, we show how to construct natural complex data on twistor space, mainly an almost Hermitian structure and a connection on some complex line bundle. Only when this almost Hermitian structure is integrable is the connection related to an anti-self-dual-Einstein metric and makes contact with the usual results. This leads to a new proof of the non-linear graviton theorem. Finally, we discuss what new strategies this "connection approach" to twistors suggests for constructing a twistor action for gravity. In Appendix A, we also review all known chiral Lagrangians for GR. [ABSTRACT FROM AUTHOR]

We review aspects of twistor theory, its aims and achievements spanning the last five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex threefold--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 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 anti-self-dual (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. [ABSTRACT FROM AUTHOR]

We show that Hopf hypersurfaces in complex projective space are constructed from half-dimensional totally complex submanifolds in complex 2-plane Grassmannian and Legendrian submanifolds in the twistor space. [ABSTRACT FROM AUTHOR]

The basic first-order differential operators of spin geometry that are Dirac operator and twistor operator are considered. Special types of spinors defined from these operators such as twistor spinors and Killing spinors are discussed. Symmetry operators of massless and massive Dirac equations are introduced and relevant symmetry operators of twistor spinors and Killing spinors are constructed from Killing–Yano (KY) and conformal Killing–Yano (CKY) forms in constant curvature and Einstein manifolds. The squaring map of spinors gives KY and CKY forms for Killing and twistor spinors, respectively. They constitute a graded Lie algebra structure in some special cases. By using the graded Lie algebra structure of KY and CKY forms, extended Killing and conformal superalgebras are constructed in constant curvature and Einstein manifolds. [ABSTRACT FROM AUTHOR]