Although twistor theory originated as an approach to the unification of quantum theory and general relativity, twistor correspondences and their generalizations have provided powerful mathematical tools for studying problems in differential geometry, nonlinear equations, and representation theory. At the same time, the theory continues to offer promising new insights into the nature of quantum theory and gravitation.
Further Advances in Twistor Theory, Volume III: Curved Twistor Spaces is actually the fourth in a series of books compiling articles from Twistor Newsletter-a somewhat informal journal published periodically by the Oxford research group of Roger Penrose. Motivated both by questions in differential geometry and by the quest to find a twistor correspondence for general Ricci-flat space times, this volume explores deformed twistor spaces and their applications.
Articles from the world's leading researchers in this field-including Roger Penrose-have been written in an informal, easy-to-read style and arranged in four chapters, each supplemented by a detailed introduction. Collectively, they trace the development of the twistor programme over the last 20 years and provide an overview of its recent advances and current status.
Arvustused
" In summary, these articles contain many interesting facts and provocative ideas that do not otherwise appear in the published literature." -Mathematical Reviews
Chapter 1: The nonlinear graviton and related constructions, III.1.1 The
Nonlinear Graviton and Related Constructions, III.1.2 The Good Cut Equation
Revisited, III.1.3 Sparling-Tod Metric = Eguchi-Hanson, III.1.4 The Wave
Equation Transfigured, III.1.5 Conformal Killing Vectors and Reduced Twistor
Spaces, III.1.6 An Alternative Interpretation of Some Nonlinear Gravitons,
III.1.7 -Space from a Different Direction, III.1.8 Complex Quaternionic
Kähler Manifolds, III.1.9 A.L.E. Gravitational Instantons and the
Icosahedron, III.1.10 The Einstein Bundle of a Nonlinear Graviton, III.1.11
Examples of Anti-Self-Dual Metrics, III.1.12 Some Quaternionically Equivalent
Einstein Metrics, III.1.13 On the Topology of Quaternionic Manifolds,
III.1.14 Homogeneity of Twistor Spaces, III.1.15 The Topology of
Anti-Self-Dual 4-Manifolds, III.1.16 Metrics with S.D. Weyl Tensor from
Painlevé-VI, III.1.17 Indefinite Conformally-A.S.D. Metrics on S2 × S2,
III.1.18 Cohomology of a Quaternionic Complex, III.1.19 Conformally Invariant
Differential Operators on Spin Bundles, III.1.20 A Twistorial Construction of
(1, 1)-Geodesic Maps, III.1.21 Exceptional Hyper-Kähler Reductions, III.1.22
A Nonlinear Graviton from the Sine-Gordon Equation, III.1.23 A Recursion
Operator for A.S.D. Vacuums and ZRM Fields on A.S.D Backgrounds,
Chapter 2:
Spaces of complex null geodesies, III.2.1 Introduction to Spaces of Complex
Null Geodesies, III.2.2 Null Geodesics and Conformai Structures, III.2.3
Complex Null Geodesics in Dimension Three, III.2.4 Null Geodesics and Contact
Structures, III.2.5 Heaven with a Cosmological Constant, III.2.6 Some Remarks
on Non-Abelian Sheaf Cohomology, III.2.7 Superstructure versus Formal
Neighbourhoods, III.2.8 Formal Thickenings of Ambitwistors for Curved
Space-Time, III.2.9 Deformations of Ambitwistor Space, III.2.10 Ambitwistors
and Yang-Mills Fields in Self-Dual Space-Times, III.2.11 Superambitwistors,
III.2.12 Formal Neighbourhoods, Supermanifolds and Relativised Algebras,
III.2.13 Quaternionic Geometry and the Future Tube, III.2.14 Deformation of
Ambitwistor Space and Vanishing Bach Tensors, III.2.15 Formal Neighbourhoods
for Curved Ambitwistors, III.2.16 Towards an Ambitwistor Description of
Gravity,
Chapter 3: Hypersurface twistors and Cauchy-Riemann manifolds,
III.3.1 Introduction to Hypersurface Twistors and Cauchy-Riemann Structures,
III.3.2 A Review of Hypersurface Twistors, III.3.3 Twistor CR Manifolds,
III.3.4 Twistor CR Structures and Initial Data, III.3.5 Visualizing Twistor
CR Structures, III.3.6 The Twistor Theory of Hypersurfaces in Space-Time,
III.3.7 Twistors, Spinors and the Einstein Vacuum Equations, III.3.8 Einstein
Vacuum Equations, III.3.9 On Bryant's Condition for Holomorphic Curves in
CR-Spaces, III.3.10 The Hill-Penrose-Sparling C.R.-Folds, III.3.11 The
Structure and Evolution of Hypersurface Twistor Spaces, III.3.12 The
Chern-Moser Connection for Hypersurface Twistor CR Manifolds, III.3.13 The
Constraint and Evolution Equations for Hypersurface CR Manifolds, III.3.14 A
Characterization of Twistor CR Manifolds, III.3.15 The Kähler Structure on
Asymptotic Twistor Space, III.3.16 Twistor CR manifolds for Algebraically
Special Space-Times, III.3.17 Causal Relations and Linking in Twistor Space,
III.3.18 Hypersurface Twistors, III.3.19 A Twistorial Approach to the Full
Vacuum Equations, III.3.20 A Note on Causal Relations and Twistor Space,
Chapter 4: Towards a twistor description of general space-times, III.4.1
Towards a Twistor Description of General Space-Times; Introductory Comments,
III.4.2 Remarks on the Sparling and Eguchi-Hanson (Googly?) Gravitons,
III.4.3 A New Angle on the Googly Graviton, III.4.4 Concerning a Fourier
Contour Integral, III.4.5 The Googly Maps for the Eguchi-Hanson/Sparling-Tod
Graviton, III.4.6 Physical Left-Right Symmetry and Googlies, III.4.7 On the
Geometry of Googly Maps, III.4.8 A Prosaic Approach to Googlies, III.4.9 More
on Googlies, III.4.10 A Note on Sparling's 3-Form, III.4.11 Remarks on
Curved-Space Twistor Theory and Googlies, III.4.12 Relative Cohomology,
Googlies and Deformations of I, III.4.13 Is the Plebanski Viewpoint Relevant
to the Googly Problem?, III.4.14 Note on the Geometry of the Googly Mappings,
III.4.15 Exponentiating a Relative H2, III.4.16 The Complex Structure of
Deformed Twistor Space, III.4.17 Local Twistor Transport at J+ : An Approach
to the Googly, III.4.18 An Approach to a Coordinate Free Calculus at J,
III.4.19 Twistor Theory for Vacuum Space-Time: A New Approach, III.4.20
Twistors as Charges for Spin 3/2 in Vacuum, III.4.21 Light Cone Cuts and
Yang-Mills Holonomies: a New Approach, III.4.22 Twistor as Spin 3/2 Charges
Continued: SL(3, ) Bundles, III.4.23 The Most General (2,2) Self-Dual
Vacuum: A Googly Approach, III.4.24 A Comment on the Preceding Article,
III.4.25 Spin 3/2 Fields and Local Twistors, III.4.26 Another View of the
Spin 3/2 Equation, III.4.27 The Bach Equations as an Exact Set of Spinor
Fields, III.4.28 A Novel Approach to Quantum Gravity, III.4.29 Twistors and
the Time-Irreversibility of State-Vector Reduction, III.4.30 Twistors and
State-Vector Reduction, Bibliography, Index
St Peters College and the Mathematical Institute, Oxford, Kings College London, Instytut Matematyki, Uniwersytet Jagielloski Kraków, Center for Mathematical Sciences, Munich University of Technology, Munich
