323 results on '"Twistor space"'
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2. Hopf hypersurfaces in complex projective space and half-dimensional totally complex submanifolds in complex 2-plane Grassmannians II.
- Author
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Kimura, Makoto
- Subjects
- *
HOPF algebras , *HYPERSURFACES , *PROJECTIVE spaces , *SUBMANIFOLDS , *GRASSMANN manifolds , *TWISTOR theory - Abstract
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]
- Published
- 2017
- Full Text
- View/download PDF
3. Generalized almost even-Clifford manifolds and their twistor spaces
- Author
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Luis Fernando Hernández-Moguel and Rafael Herrera
- Subjects
Physics ,Pure mathematics ,010102 general mathematics ,twistor space ,01 natural sciences ,even-clifford structure ,15a66 ,generalized complex structure ,Twistor theory ,53c15 ,Computer Science::Emerging Technologies ,32l25 ,53c28 ,53d18 ,0103 physical sciences ,Generalized complex structure ,QA1-939 ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics - Abstract
Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.
- Published
- 2021
4. Product twistor spaces and Weyl geometry
- Author
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Johann Davidov
- Subjects
Mathematics - Differential Geometry ,Physics ,Applied Mathematics ,General Mathematics ,Geometry ,Twistor theory ,Differential Geometry (math.DG) ,Product (mathematics) ,Bundle ,FOS: Mathematics ,Torsion (algebra) ,Twistor space ,Mathematics::Differential Geometry ,Connection (algebraic framework) ,Mathematics::Symplectic Geometry ,Metric connection - Abstract
Motivated by generalized geometry (\`a 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-symmetric torsion. These structures are defined by means of the connection $D$ and four (K\"ahler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry and this is used to supply examples satisfying the conditions., Comment: to appear in PAMS
- Published
- 2020
5. Twistor constructions for higher-spin extensions of (self-dual) Yang-Mills
- Author
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Tung Tran
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Spacetime ,Generalization ,FOS: Physical sciences ,Inverse ,Yang–Mills existence and mass gap ,QC770-798 ,Higher Spin Gravity ,Action (physics) ,Twistor theory ,General Relativity and Quantum Cosmology ,High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,High Energy Physics - Theory (hep-th) ,Topological Field Theories ,Nuclear and particle physics. Atomic energy. Radioactivity ,Penrose transform ,Twistor space ,Integrable Field Theories ,Mathematical physics - Abstract
We present the inverse Penrose transform (the map from spacetime to twistor space) for self-dual Yang-Mills (SDYM) and its higher-spin extensions on a flat background. The twistor action for the higher-spin extension of SDYM (HS-SDYM) is of BF-type. By considering a deformation away from the self-dual sector of HS-SDYM, we discover a new action that describes a higher-spin extension of Yang-Mills theory (HS-YM). The twistor action for HS-YM is a straightforward generalization of the Yang-Mills one., Comment: 25 pages. v2: corrections, changed title, improved presentation
- Published
- 2021
6. Twistor Actions for Integrable Systems
- Author
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Robert F. Penna
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Chern-Simons Theories ,Integrable system ,Sigma model ,Spacetime ,General relativity ,Supergravity ,FOS: Physical sciences ,QC770-798 ,Mathematical Physics (math-ph) ,Twistor theory ,General Relativity and Quantum Cosmology ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,Nuclear and particle physics. Atomic energy. Radioactivity ,Differential and Algebraic Geometry ,Dilaton ,Twistor space ,Integrable Field Theories ,Mathematical Physics ,Sigma Models ,Mathematical physics - Abstract
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., 22 pages
- Published
- 2021
7. The Weyl double copy from twistor space
- Author
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Silvia Nagy, Chris D. White, and Erick Chacón
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,Current (mathematics) ,Formalism (philosophy) ,Scalar (mathematics) ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,QC770-798 ,01 natural sciences ,General Relativity and Quantum Cosmology ,Twistor theory ,Theoretical physics ,High Energy Physics - Phenomenology (hep-ph) ,Nuclear and particle physics. Atomic energy. Radioactivity ,0103 physical sciences ,Differential and Algebraic Geometry ,Scattering Amplitudes ,010306 general physics ,Physics ,Spacetime ,010308 nuclear & particles physics ,Gauge (firearms) ,Scattering amplitude ,High Energy Physics - Phenomenology ,High Energy Physics - Theory (hep-th) ,Twistor space ,Classical Theories of Gravity - Abstract
The Weyl double copy is a procedure for relating exact solutions in biadjoint scalar, gauge and gravity theories, and relates fields in spacetime directly. Where this procedure comes from, and how general it is, have until recently remained mysterious. In this paper, we show how the current form and scope of the Weyl double copy can be derived from a certain procedure in twistor space. The new formalism shows that the Weyl double copy is more general than previously thought, applying in particular to gravity solutions with arbitrary Petrov types. We comment on how to obtain anti-self-dual as well as self-dual fields, and clarify some conceptual issues in the twistor approach., 38 pages, 1 figure
- Published
- 2021
8. Integrable Complex Structures on Twistor Spaces
- Author
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Steven Gindi
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Integrable system ,General Mathematics ,010102 general mathematics ,Fibered knot ,01 natural sciences ,Twistor theory ,Differential Geometry (math.DG) ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics - Abstract
We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three compatible complex structures on the twistor space that have equal torsions., Comment: New results on Hermitian properties of twistor spaces
- Published
- 2019
9. Twistor space of a generalized quaternionic manifold
- Author
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Guillaume Deschamps
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Hypercomplex number ,Mathematics - Complex Variables ,General Mathematics ,Bismut connection ,Manifold ,Connection (mathematics) ,Twistor theory ,Differential Geometry (math.DG) ,53D18, 53C28, 51P05, 32Q15, 32Q60 ,Generalized complex structure ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Complex Variables (math.CV) ,Quaternion ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
We first make a little survey of the twistor theory for hypercomplex, generalized hypercomplex, quaternionic or generalized quaternionic manifolds. This last theory was iniated by Pantilie, who shows that any generalized almost quaternionic manifold equipped with an appropriate connection admit a twistor space with an almost generalized complex structure. The aim of this article is to give an integrability criterion for this generalized almost complex structure and to give some examples especially in the case of generalized hyperk\"ahler manifolds using the generalized Bismut connection, introduced by Gualtieri., Comment: 19 pages
- Published
- 2021
10. Twistor spaces on foliated manifolds
- Author
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Robert Wolak and Rouzbeh Mohseni
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Twistor theory ,High Energy Physics::Theory ,Differential Geometry (math.DG) ,Normal bundle ,FOS: Mathematics ,Foliation (geology) ,Twistor space ,Mathematics::Differential Geometry ,53C12, 53C28 ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
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.
- Published
- 2021
11. Complex multiplication in twistor spaces
- Author
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Daniel Huybrechts
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Complex multiplication ,0102 computer and information sciences ,01 natural sciences ,K3 surface ,Twistor theory ,Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,010201 computation theory & mathematics ,FOS: Mathematics ,Twistor space ,Transcendental number ,Mathematics::Differential Geometry ,0101 mathematics ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarized K3 surface with complex multiplication, all algebraic fibres of its twistor space away from the equator have complex multiplication as well., Comment: 21 pages. Revision takes into account various insightful comments of two anonymous referees. In particular the equation of the CM extension for the twistor fibre (Corollary 3.10) has been corrected and from Section 3.2 the assumption (\ell'.\ell')>0 has been added. To appear in IMRN
- Published
- 2021
- Full Text
- View/download PDF
12. Energy of sections of the Deligne–Hitchin twistor space
- Author
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Markus Roeser, Florian Beck, and Sebastian Heller
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Twistor methods in differential geometry ,General Mathematics ,Holomorphic function ,Computer Science::Digital Libraries ,01 natural sciences ,Twistor theory ,Mathematics::Algebraic Geometry ,Line bundle ,0103 physical sciences ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,ddc:510 ,Relationships between algebraic curves and integrable systems ,Mathematics::Symplectic Geometry ,Hyper-Kähler and quaternionic Kähler geometry ,Mathematics ,Energy functional ,Meromorphic function ,Mathematics::Complex Variables ,Vector bundles on curves and their moduli ,010102 general mathematics ,Differential geometric aspects of harmonic maps ,Dewey Decimal Classification::500 | Naturwissenschaften::510 | Mathematik ,Moduli space ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Computer Science::Mathematical Software ,Twistor space ,010307 mathematical physics ,Mathematics::Differential Geometry - Abstract
We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages
- Published
- 2021
- Full Text
- View/download PDF
13. Degenerate twistor spaces for hyperkähler manifolds.
- Author
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Verbitsky, Misha
- Subjects
- *
TWISTOR theory , *LAGRANGIAN functions , *SUBMANIFOLDS , *MATHEMATICAL complexes , *TOPOLOGICAL spaces - Abstract
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]
- Published
- 2015
- Full Text
- View/download PDF
14. Traintrack Calabi-Yaus from Twistor Geometry
- Author
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Cristian Vergu and Matthias Volk
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,010308 nuclear & particles physics ,FOS: Physical sciences ,Geometry ,Branched surface ,01 natural sciences ,Supersymmetric Gauge Theory ,K3 surface ,Twistor theory ,Momentum ,Mathematics::Algebraic Geometry ,Intersection ,High Energy Physics - Theory (hep-th) ,K3 ,Supersymmetric gauge theory ,Genus (mathematics) ,0103 physical sciences ,lcsh:QC770-798 ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,Twistor space ,Scattering Amplitudes ,010306 general physics - Abstract
We describe the geometry of the leading singularity locus of the traintrack integral family directly in momentum twistor space. For the two-loop case, known as the elliptic double box, the leading singularity locus is a genus one curve, which we obtain as an intersection of two quadrics in $\mathbb{P}^{3}$. At three loops, we obtain a K3 surface which arises as a branched surface over two genus-one curves in $\mathbb{P}^{1} \times \mathbb{P}^{1}$. We present an analysis of its properties. We also discuss the geometry at higher loops and the supersymmetrization of the construction., Comment: 23 pages, 5 figures
- Published
- 2020
- Full Text
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15. Jumps, folds and singularities of Kodaira moduli spaces
- Author
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Paul Tod, James Gundry, and Maciej Dunajski
- Subjects
Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Space (mathematics) ,01 natural sciences ,Manifold ,Moduli space ,Twistor theory ,Normal bundle ,Cone (topology) ,0103 physical sciences ,Twistor space ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - 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-Kahler manifold, where the jumping lines form a three-parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.
- Published
- 2018
16. Generalized metrics and generalized twistor spaces
- Author
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Johann Davidov
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53D18, 53C28 ,General Mathematics ,010102 general mathematics ,Space (mathematics) ,01 natural sciences ,Manifold ,Twistor theory ,Differential Geometry (math.DG) ,Bundle ,0103 physical sciences ,Metric (mathematics) ,FOS: Mathematics ,Tangent space ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry \`a 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 manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms., Comment: typos corrected, minor changes, to appear in Math. Z
- Published
- 2018
17. A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces
- Author
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Reinier Storm
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Complex Variables ,Circle bundle ,010102 general mathematics ,Structure (category theory) ,Fibration ,Surface (topology) ,Submanifold ,01 natural sciences ,Hermitian matrix ,Twistor theory ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,53C28, 53C42 ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
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 Kahler structure as well as the nearly Kahler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.
- Published
- 2019
18. Twistor interpretation of slice regular functions
- Author
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Amedeo Altavilla
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,010102 general mathematics ,53C28, 30G35, 53C55, 14J26 ,General Physics and Astronomy ,01 natural sciences ,Twistor theory ,Lift (mathematics) ,Differential Geometry (math.DG) ,0103 physical sciences ,Algebraic surface ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Complex Variables (math.CV) ,0101 mathematics ,Arrangement of lines ,Mathematical Physics ,Mathematics - Abstract
Given a slice regular function $f:\Omega\subset\mathbb{H}\to \mathbb{H}$, with $\Omega\cap\mathbb{R}\neq \emptyset$, it is possible to lift it to a surface in the twistor space $\mathbb{CP}^{3}$ of $\mathbb{S}^4\simeq \mathbb{H}\cup \{\infty\}$ (see~\cite{gensalsto}). In this paper we show that the same result is true if one removes the hypothesis $\Omega\cap\mathbb{R}\neq \emptyset$ on the domain of the function $f$. Moreover we find that if a surface $\mathcal{S}\subset\mathbb{CP}^{3}$ contains the image of the twistor lift of a slice regular function, then $\mathcal{S}$ has to be ruled by lines. Starting from these results we find all the projective classes of algebraic surfaces up to degree 3 in $\mathbb{CP}^{3}$ 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 $\mathbb{G}r_2(\mathbb{C}^4)$ 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 $\mathbb{G}r_2(\mathbb{C}^4)$, showing the role of slice regular functions not defined on $\mathbb{R}$. 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., Comment: 29 pages
- Published
- 2018
19. An inclusive immersion into a quaternionic manifold and its invariants
- Author
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Kazuyuki Hasegawa
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,01 natural sciences ,Twistor theory ,53C26 ,53C28 ,Quaternionic representation ,0103 physical sciences ,Immersion (mathematics) ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Algebraic curve ,0101 mathematics ,Invariant (mathematics) ,Quaternionic projective space ,Mathematics - Abstract
金沢大学人間社会研究域学校教育系 / Institute of Human and Social science, Teacher Education, We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary. © 2017, Springer-Verlag Berlin Heidelberg., Embargo Period 12 months
- Published
- 2017
20. On the twistor space of a quaternionic contact manifold
- Author
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Alt, Jesse
- Subjects
- *
TWISTOR theory , *QUATERNIONS , *CONTACT manifolds , *MANIFOLDS (Mathematics) , *TOPOLOGICAL spaces , *GEOMETRIC analysis , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2011
- Full Text
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21. A CR twistor space of a -manifold
- Author
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Verbitsky, Misha
- Subjects
- *
TWISTOR theory , *MANIFOLDS (Mathematics) , *CALIBRATION , *VECTOR spaces , *INSTANTONS , *DIFFERENTIAL geometry - Abstract
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]
- Published
- 2011
- Full Text
- View/download PDF
22. Twistor spaces and the general adiabatic expansions
- Author
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Nagase, Masayoshi
- Subjects
- *
TWISTOR theory , *GEOMETRIC congruences , *FIELD theory (Physics) , *MATHEMATICAL analysis - Abstract
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]
- Published
- 2007
- Full Text
- View/download PDF
23. The Coulomb Branch of 3d $${\mathcal{N}= 4}$$ N = 4 Theories
- Author
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Davide Gaiotto, Tudor Dimofte, and Mathew Bullimore
- Subjects
Physics ,010308 nuclear & particles physics ,High Energy Physics::Lattice ,010102 general mathematics ,Quiver ,Magnetic monopole ,Statistical and Nonlinear Physics ,01 natural sciences ,Moduli space ,Twistor theory ,Theoretical physics ,0103 physical sciences ,Coulomb ,Nahm equations ,Twistor space ,Gauge theory ,0101 mathematics ,Mathematical Physics - Abstract
We propose a construction for the quantum-corrected Coulomb branch of a general 3d gauge theory withN = 4 supersymmetry, in terms of local coordinates associated with an abelianized theory. In a xed complex structure, the holomorphic functions on the Coulomb branch are given by expectation values of chiral monopole operators. We construct the chiral ring of such operators, using equivariant integration over BPS moduli spaces. We also quantize the chiral ring, which corresponds to placing the 3d theory in a 2d Omega background. Then, by unifying all complex structures in a twistor space, we encode the full hyperkahler metric on the Coulomb branch. We verify our proposals in a multitude of examples, including SQCD and linear quiver gauge theories, whose Coulomb branches have alternative descriptions as solutions to Bogomolnyi and/or Nahm equations.
- Published
- 2017
24. Limits of Riemannian 4-manifolds and the symplectic geometry of their twistor spaces
- Author
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Joel Fine
- Subjects
Physics ,Instanton ,General Mathematics ,010102 general mathematics ,Holomorphic function ,Curvature ,01 natural sciences ,Twistor theory ,0103 physical sciences ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Ricci curvature ,Scalar curvature ,Symplectic geometry ,Mathematical physics - Abstract
The twistor space of a Riemannian 4-manifold carries two almost complex structures, $J_+$ and $J_-$, and a natural closed 2-form $\omega$. This article studies limits of manifolds for which $\omega$ tames either $J_+$ or $J_-$. This amounts to a curvature inequality involving self-dual Weyl curvature and Ricci curvature, and which is satisfied, for example, by all anti-self-dual Einstein manifolds with non-zero scalar curvature. We prove that if a sequence of manifolds satisfying the curvature inequality converges to a hyperk\"ahler limit X (in the $C^2$ pointed topology) then X cannot contain a holomorphic 2-sphere (for any of its hyperk\"ahler complex structures). In particular, this rules out the formation of bubbles modelled on ALE gravitational instantons in such families of metrics.
- Published
- 2017
25. Twistor spaces of hyperka¨hler manifolds with <f>S1</f>-actions
- Author
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Feix, Birte
- Subjects
- *
TWISTOR theory , *HOLOMORPHIC functions - Abstract
We shall describe the twistor space of a hyperka¨hler
4n -manifold with an isometricS1 -action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimensionn .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 theS1 -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]- Published
- 2003
- Full Text
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26. SASAKIAN TWISTOR SPINORS AND THE FIRST DIRAC EIGENVALUE
- Author
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Eui Chul Kim
- Subjects
Spinor ,General Mathematics ,010102 general mathematics ,Dirac (software) ,Dirac algebra ,Dirac operator ,01 natural sciences ,Twistor theory ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Eigenvalues and eigenvectors ,Mathematical physics ,Mathematics - Published
- 2016
27. Finite geometric toy model of spacetime as an error correcting code
- Author
-
Péter Lévay, Frédéric Holweck, Laboratoire Interdisciplinaire Carnot de Bourgogne [Dijon] (LICB), Université de Technologie de Belfort-Montbeliard (UTBM)-Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS), and Université de Bourgogne (UB)-Université de Technologie de Belfort-Montbeliard (UTBM)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
High Energy Physics - Theory ,Pure mathematics ,Twistors ,compactification ,gauge/gravity duality ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,toy model ,Boundary (topology) ,01 natural sciences ,model: geometrical ,Twistor theory ,twistor ,0103 physical sciences ,Minkowski space ,phase space: discrete ,String theory ,Boundary value problem ,010306 general physics ,qubit ,Mathematical Physics ,Physics ,Quantum geometry ,Quantum Physics ,010308 nuclear & particles physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,Linear subspace ,boundary condition ,[PHYS.PHYS.PHYS-GEN-PH]Physics [physics]/Physics [physics]/General Physics [physics.gen-ph] ,quantum gravity ,Quantum Geometry ,network ,Twistor correspondence ,Twistor space ,light cone ,Quantum Entanglement ,Stabilizer Codes - Abstract
A finite geometric model of space-time (which we call the bulk) is shown to emerge as a set of error correcting codes. The bulk is encoding a set of messages located in a blow up of the Gibbons-Hoffman-Wootters (GHW) discrete phase space for $n$-qubits (which we call the boundary). Our error correcting code is a geometric subspace code known from network coding, and the correspondence map is the finite geometric analogue of the Pl\"ucker map well-known form twistor theory. The $n=2$ case of the bulk-boundary correspondence is precisely the twistor correspondence where the boundary is playing the role of the twistor space and the bulk is a finite geometric version of compactified Minkowski space-time. For $n\geq 3$ the bulk is identified with the finite geometric version of the Brody-Hughston quantum space-time. For special regions on both sides of the correspondence we associate certain collections of qubit observables. On the boundary side this association gives rise to the well-known GHW quantum net structure. In this picture the messages are complete sets of commuting observables associated to Lagrangian subspaces giving a partition of the boundary. Incomplete subsets of observables corresponding to subspaces of the Lagrangian ones are regarded as corrupted messages. Such a partition of the boundary is represented on the bulk side as a special collection of space-time points. For a particular message residing in the boundary, the set of possible errors is described by the fine details of the light-cone structure of its representative space-time point in the bulk. The geometric arrangement of representative space-time points, playing the role of the variety of codewords, encapsulates an algebraic algorithm for recovery from errors on the boundary side., Comment: 64 pages, 9 figures, 4 tables, LaTex
- Published
- 2019
28. Octahedron of complex null rays and conformal symmetry breaking
- Author
-
Simone Speziale, Maciej Dunajski, Miklos Långvik, Centre de Physique Théorique - UMR 7332 (CPT), Aix Marseille Université (AMU)-Université de Toulon (UTLN)-Centre National de la Recherche Scientifique (CNRS), Apollo - University of Cambridge Repository, and Department of Physics
- Subjects
High Energy Physics - Theory ,space: Minkowski ,gr-qc ,FOS: Physical sciences ,alternative theories of gravity ,General Relativity and Quantum Cosmology (gr-qc) ,Loop quantum gravity ,01 natural sciences ,General Relativity and Quantum Cosmology ,Twistor theory ,twistor ,Conformal symmetry ,0103 physical sciences ,Minkowski space ,010306 general physics ,Mathematical physics ,Physics ,symplectic ,010308 nuclear & particles physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,hep-th ,Null (mathematics) ,parametrization ,115 Astronomy, Space science ,Manifold ,symmetry breaking: conformal ,High Energy Physics - Theory (hep-th) ,General relativity ,symmetry: conformal ,Twistor space ,Mathematics::Differential Geometry ,quantum gravity: loop space ,Symplectic geometry - Abstract
We show how the manifold $T^*SU(2, 2)$ arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of twelve complex light rays in the Minkowski space. We discuss a mechanism to break the conformal symmetry down to the twistorial parametrisation of $T^*SL(2, C)$ used in loop quantum gravity., The work of MD has been partially supported by STFC consolidated grant no. ST/P000681/1. ML acknowledges grant no. 266101 by the Academy of Finland.
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- 2019
29. General Schlesinger Systems and Their Symmetry from the View Point of Twistor theory
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Damiran Tseveenamijil and Hironobu Kimura
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Weyl group ,Pure mathematics ,Group (mathematics) ,010102 general mathematics ,Statistical and Nonlinear Physics ,01 natural sciences ,Twistor theory ,Algebra ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,Isomonodromic deformation ,0101 mathematics ,Symmetry (geometry) ,Hypergeometric function ,Abelian group ,Mathematical Physics ,Mathematics - Abstract
Isomonodromic deformation of linear differential equations on ℙ1 with regular and irregular singular points is considered from the view point of twistor theory. We give explicit form of isomonodromic deformation using the maximal abelian subgroup H of G = GLN+1(ℂ) which appeared in the theory of general hypergeometric functions on a Grassmannian manifold. This formulation enables us to obtain a group of symmetry for the nonlinear system which is an Weyl group analogue NG (H)/H.
- Published
- 2021
30. Totally complex submanifolds of a complex Grassmann manifold of 2-planes
- Author
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Kazumi Tsukada
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Pure mathematics ,Mathematics::Complex Variables ,Complex projective space ,010102 general mathematics ,Mathematical analysis ,Structure (category theory) ,Kähler manifold ,01 natural sciences ,Linear subspace ,Twistor theory ,Computational Theory and Mathematics ,Grassmannian ,0103 physical sciences ,Cotangent bundle ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Geometry and Topology ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
A complex Grassmann manifold G 2 ( C m + 2 ) of all 2-dimensional complex subspaces in C m + 2 has two nice geometric structures – the Kahler structure and the quaternionic Kahler structure. We study totally complex submanifolds of G 2 ( C m + 2 ) with respect to the quaternionic Kahler structure. We show that the projective cotangent bundle P ( T ⁎ C P m + 1 ) of a complex projective space C P m + 1 is a twistor space of the quaternionic Kahler manifold G 2 ( C m + 2 ) . Applying the twistor theory, we construct maximal totally complex submanifolds of G 2 ( C m + 2 ) from complex submanifolds of C P m + 1 . Then we obtain many interesting examples. In particular we classify maximal homogeneous totally complex submanifolds. We show the relationship between the geometry of complex submanifolds of C P m + 1 and that of totally complex submanifolds of G 2 ( C m + 2 ) .
- Published
- 2016
31. Twistor fishnets
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Tim Adamo and Sumer Jaitly
- Subjects
High Energy Physics - Theory ,Statistics and Probability ,Physics ,Scalar (mathematics) ,FOS: Physical sciences ,General Physics and Astronomy ,Statistical and Nonlinear Physics ,Conformal map ,Scattering amplitude ,Twistor theory ,High Energy Physics::Theory ,Scaling limit ,High Energy Physics - Theory (hep-th) ,Conformal symmetry ,Modeling and Simulation ,Twistor space ,Mathematical Physics ,Gauge symmetry ,Mathematical physics - Abstract
Four-dimensional conformal fishnet theory is an integrable scalar theory which arises as a double scaling limit of $\gamma$-deformed maximally supersymmetric Yang-Mills. We give a perturbative reformulation of $\gamma$-deformed super-Yang-Mills theory in twistor space, and implement the double scaling limit to obtain a twistor description of conformal fishnet theory. The conformal fishnet theory retains an abelian gauge symmetry on twistor space which is absent in space-time, allowing us to obtain cohomological formulae for scattering amplitudes that manifest conformal invariance. We study various classes of scattering amplitudes in twistor space with this formalism., Comment: 39 pages, 11 figures. v2: references added; v3: published version
- Published
- 2020
32. The all-loop conjecture for integrands of reggeon amplitudes in $$ \mathcal{N}=4 $$ SYM
- Author
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A. E. Bolshov, A.I. Onishchenko, and L. V. Bork
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Physics ,Nuclear and High Energy Physics ,Wilson loop ,Conjecture ,010308 nuclear & particles physics ,Perturbative QCD ,Superspace ,01 natural sciences ,Loop (topology) ,Twistor theory ,Scattering amplitude ,0103 physical sciences ,Twistor space ,010306 general physics ,Mathematical physics - Abstract
In this paper we present the all-loop conjecture for integrands of Wilson line form factors, also known as reggeon amplitudes, in$$ \mathcal{N}=4 $$N=4SYM. In particular we present a new gluing operation in momentum twistor space used to obtain reggeon tree-level amplitudes and loop integrands starting from corresponding expressions for on-shell amplitudes. The introduced gluing procedure is used to derive the BCFW recursions both for tree-level reggeon amplitudes and their loop integrands. In addition we provide predictions for the reggeon loop integrands written in the basis of local integrals. As a check of the correctness of the gluing operation at loop level we derive the expression for LO BFKL kernel in$$ \mathcal{N}=4 $$N=4SYM.
- Published
- 2018
33. Notes on Scattering Amplitudes as Differential Forms
- Author
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Chi Zhang and Song He
- Subjects
High Energy Physics - Theory ,Physics ,Nuclear and High Energy Physics ,Spinor ,010308 nuclear & particles physics ,Pushforward (homology) ,FOS: Physical sciences ,Superspace ,01 natural sciences ,Amplituhedron ,Supersymmetric Gauge Theory ,Moduli space ,Twistor theory ,High Energy Physics - Theory (hep-th) ,Grassmannian ,0103 physical sciences ,lcsh:QC770-798 ,Differential and Algebraic Geometry ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,Twistor space ,Scattering Amplitudes ,010306 general physics ,Mathematical physics - Abstract
Inspired by the idea of viewing amplitudes in ${\cal N}=4$ SYM as differential forms on momentum twistor space, we introduce differential forms on the space of spinor variables, which combine helicity amplitudes in any four-dimensional gauge theory as a single object. In this note we focus on such differential forms in ${\cal N}=4$ SYM, which can also be thought of as "bosonizing" superamplitudes in non-chiral superspace. Remarkably all tree-level amplitudes in ${\cal N}=4$ SYM combine to a $d\log$ form in spinor variables, which is given by pushforward of canonical forms of Grassmannian cells, the tree forms can also be obtained using BCFW or inverse-soft construction, and we present all-multiplicity expression for MHV and NMHV forms to illustrate their simplicity. Similarly all-loop planar integrands can be naturally written as $d\log$ forms in the Grassmannian/on-shell-diagram picture, and we expect the same to hold beyond the planar limit. Just as the form in momentum twistor space reveals underlying positive geometry of the amplituhedron, the form in terms of spinor variables strongly suggests an "amplituhedron in momentum space". We initiate the study of its geometry by connecting it to the moduli space of Witten's twistor-string theory, which provides a pushforward formula for tree forms in ${\cal N}=4$ SYM., Comment: 24 pages, a dozen figures, references added
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- 2018
- Full Text
- View/download PDF
34. The holographic dual of the Penrose transform
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Yasha Neiman
- Subjects
High Energy Physics - Theory ,Nuclear and High Energy Physics ,Spacetime symmetries ,FOS: Physical sciences ,Duality (optimization) ,General Relativity and Quantum Cosmology (gr-qc) ,AdS-CFT Correspondence ,01 natural sciences ,General Relativity and Quantum Cosmology ,Twistor theory ,High Energy Physics::Theory ,0103 physical sciences ,Penrose transform ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,010306 general physics ,Mathematical physics ,Higher Spin Symmetry ,Physics ,Conformal Field Theory ,Spacetime ,010308 nuclear & particles physics ,Partition function (mathematics) ,Higher Spin Gravity ,AdS/CFT correspondence ,High Energy Physics - Theory (hep-th) ,lcsh:QC770-798 ,Twistor space - Abstract
We consider the holographic duality between type-A higher-spin gravity in AdS_4 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., 82 pages, 1 figure; v2: JHEP version - expanded exposition and references, improved the discussion of contact effects; v3: corrected signs, in light of later work on boundary-local/twistor dictionary
- Published
- 2018
35. Twistor spaces of hypercomplex manifolds are balanced
- Author
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Artour Tomberg
- Subjects
Twistor theory ,Pure mathematics ,Hypercomplex number ,General Mathematics ,Metric (mathematics) ,Structure (category theory) ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Hermitian matrix ,Hyperkähler manifold ,Manifold ,Mathematics - Abstract
A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkahler. In the paper “Non-Hermitian Yang–Mills connections” [13] , Kaledin and Verbitsky proved that the twistor space of a hyperkahler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.
- Published
- 2015
36. Conformal higher-spin symmetries in twistor string theory
- Author
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D.V. Uvarov
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Conformal anomaly ,FOS: Physical sciences ,String field theory ,Global symmetry ,Superalgebra ,Twistor theory ,Non-critical string theory ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,Twistor string theory ,Quantum electrodynamics ,Mathematics::Quantum Algebra ,lcsh:QC770-798 ,lcsh:Nuclear and particle physics. Atomic energy. Radioactivity ,Twistor space ,Mathematics::Representation Theory ,Mathematical physics - Abstract
It is shown that similarly to massless superparticle, classical global symmetry of the Berkovits twistor string action is infinite-dimensional. We identify its superalgebra, whose finite-dimensional subalgebra contains $psl(4|4,\mathbb R)$ superalgebra. In quantum theory this infinite-dimensional symmetry breaks down to $SL(4|4,\mathbb R)$ one., 20 pages, LaTeX. v2: section 3 undergone major revision, others - minor improvements including correction of typos. Version accepted to Nuclear Physics B
- Published
- 2014
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- View/download PDF
37. On deformations of the dispersionless Hirota equation
- Author
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Wojciech Kryński
- Subjects
Mathematics - Differential Geometry ,Class (set theory) ,Integrable system ,010308 nuclear & particles physics ,010102 general mathematics ,General Physics and Astronomy ,01 natural sciences ,Dispersionless equation ,Twistor theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,Geometry and Topology ,0101 mathematics ,Mathematical Physics ,Mathematical physics ,Mathematics - Abstract
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.
- Published
- 2017
38. On form factors and correlation functions in twistor space
- Author
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Matthias Wilhelm, Laura Koster, Vladimir Mitev, and Matthias Staudacher
- Subjects
Physics ,High Energy Physics - Theory ,Nuclear and High Energy Physics ,Pure mathematics ,Wilson loop ,010308 nuclear & particles physics ,Wilson ,Form factor (quantum field theory) ,FOS: Physical sciences ,Inverse ,AdS-CFT Correspondence ,'t Hooft and Polyakov loops ,01 natural sciences ,Supersymmetric gauge theory ,Twistor theory ,Scattering amplitude ,AdS/CFT correspondence ,High Energy Physics::Theory ,High Energy Physics - Theory (hep-th) ,Correlation function ,0103 physical sciences ,Twistor space ,Scattering Amplitudes ,010306 general physics - Abstract
In this paper, we continue our study of form factors and correlation functions of gauge-invariant local composite operators in the twistor-space formulation of N=4 super Yang-Mills theory. Using the vertices for these operators obtained in our recent papers arXiv:1603.04471 and arXiv:1604.00012, we show how to calculate the twistor-space diagrams for general N^kMHV form factors via the inverse soft limit, in analogy to the amplitude case. For general operators without $\dot\alpha$ indices, we then reexpress the NMHV form factors from the position-twistor calculation in terms of momentum twistors, deriving and expanding on a relation between the two twistor formalisms previously observed in the case of amplitudes. Furthermore, we discuss the calculation of generalized form factors and correlation functions as well as the extension to loop level, in particular providing an argument promised in arXiv:1410.6310., Comment: 24+17 pages, 18 figures; v2: typos corrected, matches published version
- Published
- 2017
39. Twistor theory at fifty: from contour integrals to twistor strings
- Author
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Michael Atiyah, Maciej Dunajski, Lionel Mason, Dunajski, Maciej [0000-0002-6477-8319], and Apollo - University of Cambridge Repository
- Subjects
High Energy Physics - Theory ,Mathematics - Differential Geometry ,Instanton ,General Mathematics ,Holomorphic function ,General Physics and Astronomy ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Space (mathematics) ,01 natural sciences ,String (physics) ,General Relativity and Quantum Cosmology ,Twistor theory ,Theoretical physics ,High Energy Physics::Theory ,integrable systems ,0103 physical sciences ,FOS: Mathematics ,010306 general physics ,Review Articles ,Physics ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,010308 nuclear & particles physics ,General Engineering ,twistor theory ,Newtonian limit ,twistor strings ,Cohomology ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,High Energy Physics - Theory (hep-th) ,Differential Geometry (math.DG) ,self-duality ,Twistor space ,Mathematics::Differential Geometry ,instantons ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
We review aspects of twistor theory, its aims and achievements spanning thelast five decades. In the twistor approach, space--time is secondary with events being derived objects that correspond to compact holomorphic curves in a complex three--fold -- 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 (ASD) 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 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., Comment: Minor corrections, and additional references. Final version, to appear in the Proceedings of the Royal Society A. 49 pages, 6 Figures. Dedicated to Roger Penrose and Nick Woodhouse at 85 and 67 years
- Published
- 2017
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40. Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity
- Author
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Yannick Herfray, Laboratoire de Physique de l'ENS Lyon ( Phys-ENS ), École normale supérieure - Lyon ( ENS Lyon ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)
- Subjects
High Energy Physics - Theory ,effective Lagrangian: chiral ,General relativity ,FOS: Physical sciences ,integrability ,01 natural sciences ,[ PHYS.HTHE ] Physics [physics]/High Energy Physics - Theory [hep-th] ,Twistor theory ,High Energy Physics::Theory ,twistor ,SU(2) theory ,0103 physical sciences ,Euclidean geometry ,general relativity ,010306 general physics ,Mathematical Physics ,Mathematics ,Mathematical physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,010308 nuclear & particles physics ,Complex line ,Graviton ,Statistical and Nonlinear Physics ,Hermitian matrix ,Connection (mathematics) ,High Energy Physics - Theory (hep-th) ,SU(2) ,gravitation ,Twistor space ,Mathematics::Differential Geometry - Abstract
This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-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 we also review all known chiral Lagrangians for GR., This is the version published in J.Math.Phys. As compare to the previous version, some paragraph were rewritten to make the text easier to read and some typos corrected
- Published
- 2017
41. On Exact Solutions and Perturbative Schemes in Higher Spin Theory
- Author
-
Carlo Iazeolla, Per Sundell, and Ergin Sezgin
- Subjects
High Energy Physics - Theory ,Physics ,Spacetime ,010308 nuclear & particles physics ,Space time ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,Twistor theory ,High Energy Physics - Theory (hep-th) ,0103 physical sciences ,Homogeneous space ,Twistor space ,Anti-de Sitter space ,Invariant (mathematics) ,010306 general physics ,Spin-½ ,Mathematical physics - Abstract
We review various methods for finding exact solutions of higher spin theory in four dimensions, and survey the known exact solutions of (non)minimal Vasiliev's equations. These include instanton-like and black hole-like solutions in (A)dS and Kleinian spacetimes. A perturbative construction of solutions with the symmetries of a domain wall is described as well. Furthermore, we review two proposed perturbative schemes: one based on perturbative treatment of the twistor space field equations followed by inverting Fronsdal kinetic terms using standard Green's functions; and an alternative scheme based on solving the twistor space field equations exactly followed by introducing the spacetime dependence using perturbatively defined gauge functions. Motivated by the need to provide a higher spin invariant characterization of the exact solutions, aspects of a proposal for a geometric description of Vasiliev's equation involving an infinite dimensional generalization of anti de Sitter space is revisited and improved., Comment: 45 pages. Clarifying remarks and references are added. Version published in Universe, Special Issue on Higher Spin Gauge Theories
- Published
- 2017
- Full Text
- View/download PDF
42. Twistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structures
- Author
-
Ljudmila Kamenova
- Subjects
Twistor theory ,Pure mathematics ,Simply connected space ,Differentiable manifold ,Twistor space ,Geometry and Topology ,Quaternion ,Surface (topology) ,Mathematical Physics ,Hyperkähler manifold ,Manifold ,Mathematics - Abstract
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kahler and non-Kahler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kahler and non-Kahler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkahler manifold has the same property.
- Published
- 2017
43. Hessian of the natural Hermitian form on twistor spaces
- Author
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Noël Le Du, Christophe Mourougane, Guillaume Deschamps, Laboratoire de mathématiques de Brest ( LM ), Université de Brest ( UBO ) -Institut Brestois du Numérique et des Mathématiques ( IBNM ), Université de Brest ( UBO ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,[ MATH.MATH-CV ] Mathematics [math]/Complex Variables [math.CV] ,Closed manifold ,53C28 ,53C26 ,32Q45 ,General Mathematics ,Invariant manifold ,twistor space ,01 natural sciences ,Pseudo-Riemannian manifold ,Twistor theory ,symbols.namesake ,Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,FOS: Mathematics ,Hermitian manifold ,0101 mathematics ,Complex Variables (math.CV) ,hyperkähler manifold ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,4-dimensional Riemannian manifold ,Ricci curvature ,strong KT manifolds ,Mathematics ,quaternionic Kähler manifold ,Mathematics - Complex Variables ,010102 general mathematics ,Mathematical analysis ,Holonomy ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Mathematics::Geometric Topology ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,symbols ,Twistor space ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics::Differential Geometry - Abstract
International audience; We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic Kähler manifold of non-positive scalar curvature. We check that no non-Kähler strong KT manifold occurs as such a twistor space.
- Published
- 2017
44. Harmonic Almost Hermitian Structures
- Author
-
Johann Davidov
- Subjects
Harmonic coordinates ,Pure mathematics ,010308 nuclear & particles physics ,010102 general mathematics ,Mathematical analysis ,Harmonic map ,Riemannian manifold ,01 natural sciences ,Manifold ,Twistor theory ,0103 physical sciences ,Hermitian manifold ,Twistor space ,Mathematics::Differential Geometry ,0101 mathematics ,Complex manifold ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
This is a survey of old and new results on the problem when a compatible almost complex structure on a Riemannian manifold is a harmonic section or a harmonic map from the manifold into its twistor space. In this context, special attention is paid to the Atiyah-Hitchin-Singer and Eells-Salamon almost complex structures on the twistor space of an oriented Riemannian four-manifold.
- Published
- 2017
45. Twistor construction of asymptotically hyperbolic Einstein–Weyl spaces
- Author
-
Aleksandra Borówka
- Subjects
Mathematics - Differential Geometry ,conformal Cartan connection ,Pure mathematics ,Holomorphic function ,Conformal map ,Space (mathematics) ,Surface (topology) ,53C28, 32L25, 53A30, 53C25 ,Twistor theory ,minitwistor space ,Differential Geometry (math.DG) ,Computational Theory and Mathematics ,Jones–Tod correspondence ,Cartan connection ,FOS: Mathematics ,asymptotically hyperbolic Einstein–Weyl manifold ,Twistor space ,Geometry and Topology ,Analysis ,Quotient ,Mathematics - Abstract
Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary.
- Published
- 2014
46. Holonomies of gauge fields in twistor space 6: Incorporation of massive fermions
- Author
-
Yasuhiro Abe
- Subjects
High Energy Physics - Theory ,Physics ,Particle physics ,Nuclear and High Energy Physics ,High Energy Physics::Lattice ,Scalar (mathematics) ,Holonomy ,FOS: Physical sciences ,Fermion ,Gluon ,Scattering amplitude ,Twistor theory ,High Energy Physics - Phenomenology ,Theoretical physics ,High Energy Physics - Phenomenology (hep-ph) ,High Energy Physics - Theory (hep-th) ,Twistor space ,S-matrix - Abstract
Following the previous paper arXiv:1205.4827, we formulate an S-matrix functional for massive fermion ultra-helicity-violating (UHV) amplitudes, i.e., scattering amplitudes of positive-helicity gluons and a pair of massive fermions. The S-matrix functional realizes a massive extension of the Cachazo-Svrcek-Witten (CSW) rules in a functional language. Mass-dimension analysis implies that interactions among gluons and massive fermions should be decomposed into three-point massive fermion subamplitudes. Namely, such interactions are represented by combinations of three-point UHV and next-to-UHV (NUHV) vertices. This feature is qualitatively different from the massive scalar amplitudes where the number of involving gluons can be arbitrary., Comment: 24 pages; v2. references added; v3. supplemental paragraph inserted below (3.42), typos corrected, published version
- Published
- 2014
- Full Text
- View/download PDF
47. Twistor transforms of quaternionic functions and orthogonal complex structures
- Author
-
Simon Salamon, Caterina Stoppato, and Graziano Gentili
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics - Complex Variables ,Applied Mathematics ,General Mathematics ,Parabola ,53C28, 30G35, 53C55, 14J26 ,Function (mathematics) ,Domain (mathematical analysis) ,Twistor theory ,Mathematics - Algebraic Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Klein quadric ,Complex Variables (math.CV) ,Quartic surface ,Quaternion ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
The theory of slice regular functions of a quaternion variable is applied to the study of orthogonal complex structures on domains \Omega\ of R^4. When \Omega\ is a symmetric slice domain, the twistor transform of such a function is a holomorphic curve in the Klein quadric. The case in which \Omega\ is the complement of a parabola is studied in detail and described by a rational quartic surface in the twistor space CP^3., Comment: Some explanation added in section 1, other minor amendments and reformatting; to appear in JEMS
- Published
- 2014
48. The first Chern class and conformal area for a twistor holomorphic immersion
- Author
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Kazuyuki Hasegawa
- Subjects
Unit sphere ,Pure mathematics ,Twistor holomorphic surface ,Chern class ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Conformal area ,Twistor lift ,Physics::History of Physics ,Twistor theory ,Normal bundle ,Immersion (mathematics) ,Twistor space ,Mathematics::Symplectic Geometry ,Euler class ,First Chern class ,Mathematics - Abstract
We obtain an inequality involving the first Chern class of the normal bundle and the conformal area for a twistor holomorphic surface. Using this inequality, we can improve an inequality obtained by T. Friedrich for the Euler class of the normal bundle of a twistor holomorphic surface in the four-dimensional space form. Moreover, as a corollary, we see that the area of a superminimal surface in the unit sphere is an integer multiple of {Mathematical expression}, which is essentially proved by E. Calabi. © 2014 Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg., in Press
- Published
- 2014
49. ‐Algebras, the BV Formalism, and Classical Fields
- Author
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Tommaso Macrelli, Martin Wolf, Christian Sämann, Branislav Jurčo, and Lorenzo Raspollini
- Subjects
0303 health sciences ,Batalin–Vilkovisky formalism ,030302 biochemistry & molecular biology ,General Physics and Astronomy ,Classical field theory ,Twistor theory ,High Energy Physics::Theory ,03 medical and health sciences ,Formalism (philosophy of mathematics) ,Theoretical physics ,Differential geometry ,Fiber bundle ,Twistor space ,Gauge theory ,030304 developmental biology ,Mathematics - Abstract
We summarise some of our recent works on L∞‐algebras and quasi‐groups with regard to higher principal bundles and their applications in twistor theory and gauge theory. In particular, after a lightning review of L∞‐algebras, we discuss their Maurer–Cartan theory and explain that any classical field theory admitting an action can be reformulated in this context with the help of the Batalin–Vilkovisky formalism. As examples, we explore higher Chern–Simons theory and Yang–Mills theory. We also explain how these ideas can be combined with those of twistor theory to formulate maximally superconformal gauge theories in four and six dimensions by means of L∞‐quasi‐isomorphisms, and we propose a twistor space action.
- Published
- 2019
50. Twistor Space for Rolling Bodies
- Author
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Pawel Nurowski and Daniel An
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Circle bundle ,Statistical and Nonlinear Physics ,Symmetry group ,Twistor theory ,Differential Geometry (math.DG) ,Homogeneous space ,FOS: Mathematics ,Twistor space ,Configuration space ,Symmetry (geometry) ,Mathematical Physics ,Distribution (differential geometry) ,Mathematics - Abstract
On a natural circle bundle T(M) over a 4-dimensional manifold M equipped with a split signature metric g, whose fibers are real totally null selfdual 2-planes, we consider a tautological rank 2 distribution D obtained by lifting each totally null plane horizontally to its point in the fiber. Over the open set where g is not antiselfdual, the distribution D is (2,3,5) in T(M). We show that if M is a Cartesian product of two Riemann surfaces (S1,g1) and (S2,g2), and if g=g1--g2, then the circle bundle T(S1 x S2) is just the configuration space for the physical system of two solid bodies B1 and B2, bounded by the surfaces S1 and S2 and rolling on each other. The condition for the two bodies to roll on each other `without slipping or twisting' identifies the restricted velocity space for such a system with the tautological distribution D on T(S1 x S2). We call T(S1 x S2) the twistor space, and D the twistor distribution for the rolling bodies. Among others we address the following question: "For which pairs of bodies does the restricted velocity distribution (which we identify with the twistor distribution D) have the simple Lie group G2 as its group of symmetries?" Apart from the well known situation when the boundaries S1 and S2 of the two bodies have constant curvatures whose ratio is 1:9, we unexpectedly find three different types of surfaces, which when bounding a body that rolls `without slipping or twisting' on a plane, have D with the symmetry group G2. Although we have found the differential equations for the curvatures of S1 and S2 that gives D with G2 symmetry, we are unable to solve them in full generality so far., An extended version of a talk given on 5th of September 2012 by one of the authors at the conference `The interaction of geometry and representation theory. Exploring new frontiers". The conference was organized by A. Cap, A. L. Carey, A. R. Gover, C. R. Graham and J. Slovak and took place at the Erwin Schrodinger Institute in Vienna, Austria; http://www.mat.univie.ac.at/~cap/esiprog/Nurowski.pdf
- Published
- 2013
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