arXiv research

Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.

The paper examines curvature properties of twistor spaces.

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to di…

Examines quantum mechanics equivalence with Newtonian geometry.

New examples of structures found using twistor space.

Classifies all toric Kahler surfaces with twistor 2-forms.

Study calibrated geometry in hyperkähler cones and their related spaces.

Study the geometry of twistor spaces with rotating circle action.

In this paper, following the constructions of N. R. O'Brian, J. H. Rawnsley and I. Vaisman, we define four almost Hermitian structures (up to conjugation) on the twistor space of a Hermitian surface by using canonical connections, including the Lichnerowicz connection and the Chern connection. We also study the relatio…

New sigma models compute graviton scattering amplitudes from quaternionic geometry.

Study of algebraic curves and surfaces in flag manifold using twistor geometry.

We introduce the symplectic twistor operator $T_s$ in symplectic spin geometry, as a symplectic analogue of the twistor operator in Riemannian spin geometry. We focus on the real dimension 2 and compute the space of its solutions on ${\mathbb R}^2$. Our analysis is based on the techniques of metaplectic Howe duality.

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…

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general K…

We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimens…

We show that given a conformal structure whose holonomy representation fixes a totally lightlike subspace of arbitrary dimension, there is always a local metric in the conformal class off a singular set which is Ricci-isotropic and gives rise to a parallel, totally lightlike distribution on the tangent bundle. This nat…

Study of CR twistor model $Q^{2,2}$ and its sections.

This work applies Double Field Theory to four-dimensional manifolds, revealing connections to integrability and twistor theory.

The resolved conifold geometry is linked to a special Kähler manifold and an instanton-corrected hyperkähler manifold.

We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.

The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces…

In this paper, we construct tools from the holomorphic twistor spaces that we introduced in \cite{Gindi1} to derive results about the complex geometries of their base manifolds. In particular, we develop a new approach to studying generalized Kahler manifolds that leads to insights into their real and holomorphic Poiss…

We introduce in this paper normal twistor equations for differential forms and study their solutions, the so-called normal conformal Killing forms. The twistor equations arise naturally from the canonical normal Cartan connection of conformal geometry. Reductions of its holonomy are related to solutions of the normal t…

We explicitly describe all SO(7)-invariant almost quaternion-Hermitian structures on the twistor space of the six sphere and determine the types of their intrinsic torsion.

The aim of our article is the study of solution space of the symplectic twistor operator $T_s$ in symplectic spin geometry on standard symplectic space $({\mathbb R}^{2n},ω)$, which is the symplectic analogue of the twistor operator in (pseudo)Riemannian spin geometry. In particular, we observe a substantial difference…

Study of complex structures on product twistor spaces for 4D manifolds.

Study BGG operators on homogeneous conformal geometries.

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry à la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the…

We deal here with the geometry of the twistor fibration $\mathcal{Z} \to \bb{S}^3_1$ over the De Sitter 3-space. The total space $\mathcal{Z}$ is a five dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on $\mathcal{Z}$ we study the harmonic map equation …

Study classifies certain Einstein 4-manifolds with twistorial properties.

We continue to study twistor spaces on the connected sum of four complex projective planes, whose anticanonical map is of degree two over the image. In particular, we determine the defining equation of the branch divisor of the anticanonical map in an explicit form. Together with previous two articles (arXiv:1009.3153 …

New construction of self-dual black holes using quadrics.

Simplified proof of spin^c structures using twistor spaces.

We exploit techniques from classical (real and complex) algebraic geometry for the study of the standard twistor fibration $π:\mathbb{CP}^{3}\to S^{4}$. We prove three results about the topology of the twistor discriminant locus of an algebraic surface in $\mathbb{CP}^{3}$. First of all we prove that, with the exceptio…

This is an expanded version of a series of lectures delivered at the 25th Winter School ``Geometry and Physics'' in Srni. After a short introduction to Cartan geometries and parabolic geometries, we give a detailed description of the equivalence between parabolic geometries and underlying geometric structures. The seco…

In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.

Generic distributions on 5- and 6-manifolds give rise to conformal structures that were discovered by P. Nurowski resp. R. Bryant. We describe both as Fefferman-type constructions and show that for orientable distributions one obtains conformal spin structures. The resulting conformal spin geometries are then character…

Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symme…

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 releva…

It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed ove…

We use the CR geometry of the standard hyperquadric in complex projective three-space to give a detailed twistor description of conformal foliations in Euclidean three-space.

In 1995 D. Joyce explicitly constructed a series of self-dual metrics with torus action on the connected sums of complex projective planes. In this paper we explicitly construct the twistor spaces of some of Joyce's self-dual metrics. Starting from a fiber space whose fibers are compact singular toric surfaces, we appl…

Study invariant operators and vanishing theorems in CR geometry.

The paper explores curvature constraints on Kodaira dimension for specific almost Hermitian manifolds.

For a semisimple Lie group $G$ with parabolic subgroups $Q\subset P\subset G$, we associate to a parabolic geometry of type $(G,P)$ on a smooth manifold $N$ the correspondence space $\Cal CN$, which is the total space of a fiber bundle over $N$ with fiber a generalized flag manifold, and construct a canonical parabolic…

This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space-time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the 6-dimensional case in which twisto…

New insights connect strong coupling SYM amplitudes to hyperkähler geometry.

The article resolves complex structures in transport twistor spaces, proving a Newlander-Nirenberg theorem.