Ambitwistors and Conformal Gravity

  • Claude LeBrun
Part of the NATO ASI Series book series (NSSB, volume 245)


In a landmark paper [13] of the 1970’s, Roger Penrose described a remarkable and unexpected connection between Einstein’s equations and the theory of complex manifolds. The twistor correspondence which he detailed there gives a way of producing all self-dual complex solutions of these equations in terms of the global structure of an associated complex manifold called the twistor space. Unfortunately, the most interesting solutions from the view-point of physics are not the self-dual ones, but rather those of Lorentz signature. If this had been the end of the story, one might have therefore been tempted to conclude that this correspondence was merely a mathematical curiosity with no bearing on physics. Fortunately, however, the Penrose twistor correspondence is but one aspect of a rather more complicated story, which I will endeavor to recount here. Indeed, it turns out that the general complex solution of the 4-dimensional Einstein equations can also be described in terms of complex deformation theory, albeit in terms of non-reduced complex spaces rather than complex manifolds.


Line Bundle Complex Manifold Twistor Space Null Geodesic Conformal Class 
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  1. [1]
    V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978MATHGoogle Scholar
  2. [2]
    R.J. Baston and L.J. Maison, Conformai Gravity, the Einstein Equations and Spaces of Complex Null Geodesics, Class. Quantum Grav.4 (1987) 815–826MathSciNetADSMATHCrossRefGoogle Scholar
  3. [3]
    L.L. Chau and C.S. Lim, Geometrical Constraints and Equations of Motion in Extended Supergravity, Phys. Rev. Lett. 56 (1986) 294–297MathSciNetADSCrossRefGoogle Scholar
  4. [4]
    M.G. Eastwood, The Penrose Transform for Curved Ambitwistor Space, Qu. J. Math. Oxford (2) 39 (1988) 427–441MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    M.G. Eastwood and C.R. LeBrun, Thickenings and Supersymmetric Extensions of Complex Manifolds, Am. J. Math.108 (1986) 1177–1192MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    C.R. Graham, Formal Conformai Normal Form, to appear.Google Scholar
  7. [7]
    J. Isenberg, P. Yasskin, and P. Green, Non-Self-Dual Gauge Fields, Phys. Lett. 77B (1978) 462–464ADSGoogle Scholar
  8. [8]
    K. Kodaira, On Complete Systems of Compact Complex Submanifolds of a Complex Manifold, Am. J. Math. 85 (1962) 79–94MathSciNetCrossRefGoogle Scholar
  9. [9]
    C.R. LeBrun, The First Formal Neighbourhood of Ambitwistor Space for. Curved Space-time, Lett. Math. Phys. 6 (1982) 345–354MathSciNetADSMATHCrossRefGoogle Scholar
  10. [10]
    C.R. LeBrun, Spaces of Complex Null Geodesics in Complex-Riemannian Geometry, Trans. Am. Math. Soc.. 278 (1983) 209–231MathSciNetMATHCrossRefGoogle Scholar
  11. [11]
    C.R. LeBrun, Thickenings and Gauge Fields, Class. Quantum Gravity3, (1986) 1039–1059MathSciNetMATHCrossRefGoogle Scholar
  12. [12]
    C.R. LeBrun, Thickenings and Conformai Gravity, preprint, 1989.Google Scholar
  13. [13]
    R. Penrose, Non-linear Gravitons and Curved Twistor Theory, Gen.Rel. Grav. 7 (1976) 31–52MathSciNetADSMATHCrossRefGoogle Scholar
  14. [14]
    R. Penrose and W. Rindler, Spinors and Space-time, vol. 2, Cambridge U.P. 1986CrossRefGoogle Scholar
  15. [15]
    E. Witten, An Interpretation of Classical Yang-Mills Theory, Phys. Lett. 77B (1978) 394–398ADSGoogle Scholar
  16. [16]
    E. Witten, Twistor-like transform in ten dimensions, Nuc. Phys. B266 (1986) 245–264MathSciNetADSMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Claude LeBrun
    • 1
    • 2
  1. 1.Mathematics DepartmentSUNY Stony BrookUSA
  2. 2.School of MathematicsInstitute for Advanced StudyUSA

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