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Let X = X¯ − D be a smooth quasi-projective curve. We previously constructed a Deligne–Hitchin moduli space with Hecke gauge groupoid for connections of rank 2. We extend this construction to the case of any rank r, although still keeping a genericity hypothesis. The formal neighborhood of a preferred section corresponding to a tame harmonic bundle is governed by a mixed twistor structure. [ABSTRACT FROM AUTHOR]

In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that C P 3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold. [ABSTRACT FROM AUTHOR]

Let X be a quasi-projective curve, compactified to (Y, D) with X = Y - D . We construct a Deligne–Hitchin twistor space out of moduli spaces of framed λ -connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the λ -connection. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the construction of Deligne–Hitchin twistor space by gluing two certain Hodge moduli spaces. We investigate some properties of such generalized Deligne–Hitchin twistor space as a complex analytic manifold. More precisely, we show it admits holomorphic sections whose normal bundle contains a semistable subbundle with positive degree and whose energy is semi-negative, and it carries a balanced metric. Moreover, we also study the automorphism groups of the Hodge moduli spaces and the generalized Deligne–Hitchin twistor spaces. [ABSTRACT FROM AUTHOR]

Let E be an oriented vector bundle of rank 4 over a torus T 2 with a metric h and E ^ the twistor space associated with E. We will study the horizontality in E ^ with respect to the connection ∇ ^ induced by an h-connection ∇ . We will describe the subset of the fiber E ^ a of E ^ at a point a ∈ T 2 given by horizontality along normal polygonal curves in R 2 for an initial value at a in the case where the subset is finite. We will see that if E ^ has a partially horizontal section Ω , then there exists an h-connection ∇ ′ related to ∇ such that h, ∇ ′ and Ω give a Kähler structure of E. We will make analogous discussions for an oriented vector bundle of rank 4 over T 2 with a neutral metric. [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]

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]

We propose a generalization of Verbitsky's global Torelli theorem in the framework of compact Kähler irreducible holomorphically symplectic orbifolds by adapting Huybrechts' proof [17]. As intermediate step needed, we also provide a generalization of the twistor space and the projectivity criterion based on works of Campana [8] and Huybrechts [21] respectively. [ABSTRACT FROM AUTHOR]

In this paper, we extend the notion of Bekenstein–Hawking entropy for a cover of a site. We deduce a new class of discrete dynamical system on a site and we introduce the Bekenstein–Hawking entropy for each member of it. We present an upper bound for the Bekenstein–Hawking entropy of the iterations of a dynamical system. We define a conjugate relation on the set of dynamical systems on a site and we prove that the Bekenstein–Hawking entropy preserves under this relation. We also prove that the twistor correspondence preserves the Bekenstein–Hawking entropy. [ABSTRACT FROM AUTHOR]

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold. [ABSTRACT FROM AUTHOR]

In this short note, we review the well-known result that there is no orthogonal complex structure on S 6 with respect to the round metric. [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]

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]

We propose a new construction of compact non-Kähler Calabi–Yau manifolds with balanced metrics and study the Strominger system on them. In particular, we obtain explicit solutions to the Strominger system with degeneracies on Σ g × T 4 , where Σ g is an immersed minimal surface of genus g ≥ 3 in flat T 3 . [ABSTRACT FROM AUTHOR]

A study of tensors on the quaternionic projective plane HP 2 arising from a stable 3-form on C 6 and an associated action of SU (3) is related to the existence of a holomorphic rank 3 vector bundle over CP 5 discovered by Horrocks. It also leads to the construction of SU (3) invariant Spin (7) structures on HP 2 , which are characterised in terms of associated 4-forms. [ABSTRACT FROM AUTHOR]

Let M be a hyperkähler manifold, and η a closed, positive (1, 1)-form with rk η < dim M . We associate to η a family of complex structures on M , called a degenerate twistor family, and parametrized by a complex line. When η is a pullback of a Kähler form under a Lagrangian fibration L , all the fibers of degenerate twistor family also admit a Lagrangian fibration, with the fibers isomorphic to that of L . Degenerate twistor families can be obtained by taking limits of twistor families, as one of the Kähler forms in the hyperkähler triple goes to η . [ABSTRACT FROM AUTHOR]

A trisymplecti cstructure on a complex $2n$-manifold is a three-dimensional space ${\rm\Omega}$ of closed holomorphic forms such that any element of ${\rm\Omega}$ has constant rank $2n$, $n$ or zero, and degenerate forms in ${\rm\Omega}$ belong to a non-degenerate quadric hypersurface. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of regular rational curves in the twistor space of a hyperkähler manifold, and define a trisymplectic reduction of a trisymplectic manifold, which is a complexified form of a hyperkähler reduction. We prove that the trisymplectic reduction in the space of regular rational curves on the twistor space of a hyperkähler manifold $M$ is compatible with the hyperkähler reduction on $M$. As an application of these geometric ideas, we consider the ADHM construction of instantons and show that the moduli space of rank $r$, charge $c$ framed instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth trisymplectic manifold of complex dimension $4rc$. In particular, it follows that the moduli space of rank two, charge $c$ instanton bundles on $\mathbb{C}\mathbb{P}^{3}$ is a smooth complex manifold dimension $8c-3$, thus settling part of a 30-year-old conjecture. [ABSTRACT FROM AUTHOR]

In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2, 1)-bundle over a paraquaternionic Kähler manifold. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we prove that the CR manifold 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. [Copyright &y& Elsevier]

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a -manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy -manifold. We interpret -instanton bundles as CR-holomorphic bundles on its twistor space. [Copyright &y& Elsevier]

We discuss the harmonic spheres conjecture that the space of harmonic maps of the Riemann sphere into the loop space of a compact Lie group G are related to the moduli space of Yang-Mills G-fields on the four-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D2 V = 0 on a five-dimensional manifold $${\mathbf{Y}}$$. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in { V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space $${\mathcal{Z}}$$ has $${ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}$$, projecting thus over $${\mathbb C\mathbb P_2}$$ with twistor lines mapping onto plane conics. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts. [Copyright &y& Elsevier]

We construct complete noncompact Riemannian metrics with G 2-holonomy on noncompact orbifolds that are ℝ3-bundles with the twistor space [InlineMediaObject not available: see fulltext.] as a spherical fiber. [ABSTRACT FROM AUTHOR]

Abstract: We investigate the behavior of derivatives of the fundamental solution of a parabolic equation for the square of Dirac operator on a twistor space when the metric is blown up in the base space direction. Such a blowing up operation is expected to be an effective method for extracting some intrinsic values from various geometric invariants, most of whose cores consist of some derivatives of the fundamental solution. [Copyright &y& Elsevier]

Abstract: We study surfaces whose twistor lifts are harmonic sections, and characterize these surfaces in terms of their second fundamental forms. As a corollary, under certain assumptions for the curvature tensor, we prove that the twistor lift is a harmonic section if and only if the mean curvature vector field is a holomorphic section of the normal bundle. For surfaces in four-dimensional Euclidean space, a lower bound for the vertical energy of the twistor lifts is given. Moreover, under a certain assumption involving the mean curvature vector field, we characterize a surface in four-dimensional Euclidean space in such a way that the twistor lift is a harmonic section, and its vertical energy density is constant. [Copyright &y& Elsevier]

In this brief review, we summarize the new development on the correspondence between noncommutative (NC) field theory and gravity, shortly referred to as the NCFT/Gravity correspondence. We elucidate why a gauge theory in NC spacetime should be a theory of gravity. A basic reason for the NCFT/Gravity correspondence is that the Λ-symmetry (or B-field transformations) in NC spacetime can be considered as a par with diffeomorphisms, which results from the Darboux theorem. This fact leads to a striking picture about gravity: Gravity can emerge from a gauge theory in NC spacetime. Gravity is then a collective phenomenon emerging from gauge fields living in fuzzy spacetime. [ABSTRACT FROM AUTHOR]

Abstract: We give a proof that the sphere does not admit an integrable orthogonal complex structure using simple differential geometric methods. This appears as a corollary of a general analogous result concerning pseudo-spheres. We study the twistor space of a pseudo-Riemannian manifold in both the holomorphic and pseudo-Riemannian directions. In particular, we construct the twistor space of a pseudo-sphere as a known pseudo-Kähler symmetric space. This leads to the explicit, unexpected computation of the exterior derivative of the Kähler form on the base manifold. [Copyright &y& Elsevier]

Abstract: We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is , then the manifold is locally isomorphic to the twistor space of a quaternionic Kähler manifold with positive scalar curvature. If the manifold is complete, then it is globally isomorphic to such a twistor space. [Copyright &y& Elsevier]

A chiral superfield strength in ${\mathcal N} = 2$ conformal supergravity at linearized level is obtained by acting two superspace derivatives on ${\mathcal N} = 4$ chiral superfield strength which can be described in terms of ${\mathcal N} = 4$ twistor superfields. By decomposing SU(4)R representation of ${\mathcal N} = 4$ twistor superfields into the SU(2)R representation with an invariant U(1)R charge, the surviving ${\mathcal N} = 2$ twistor superfields contain the physical states of ${\mathcal N} = 2$ conformal supergravity. These ${\mathcal N} = 2$ twistor superfields are functions of homogeneous coordinates of weighted complex projective space WCP3|4 where the two weighted fermionic coordinates have weight -1 and 3. [ABSTRACT FROM AUTHOR]

Abstract: A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov–Gauduchon generalization of the Goldberg–Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon''s twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira–Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira–Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed. [Copyright &y& Elsevier]

We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces C P 3 and F 1 , 2 (C 3) with respect to both the Kähler structure as well as the nearly Kähler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds. [ABSTRACT FROM AUTHOR]

Donaldson-Friedman constructed anti-self-dual classes on K 3 # 3 CP 2 ‾ using twistor space. We show that some of these conformal classes have almost-Kähler representatives. [ABSTRACT FROM AUTHOR]

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we consider surfaces in self-dual Einstein manifolds whose twistor lifts are harmonic sections. In particular, we state the stability of the twistor lifts as harmonic sections and determine such surfaces of genus zero. This paper is a short survey of our previous results in Hasegawa (2007, 2009, 2011) . [Copyright &y& Elsevier]