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  • Mathematical Proceedings of the Cambridge Philosophical Society, Volume 97, Issue 1
  • January 1985, pp. 165-187

The generalized Penrose-Ward transform

  • Michael G. Eastwood (a1)
  • DOI: http://dx.doi.org/10.1017/S030500410006271X
  • Published online: 24 October 2008
Abstract

The Penrose transform is an integral geometric method of interpreting elements of various analytic cohomology groups on open subsets of complex projective 3-space as solutions of linear differential equations on the Grassmannian of 2-planes in 4-space. The motivation for such a transform comes from interpreting this Grassmannian as the complexification of the conformal compactification of Minkowski space, the differential equations being the massless field equations of various helicities. This transform is a cornerstone of twistor theory [22, 24, 30], but the methods generalize considerably as will be explained in this article. Closely related is the Ward correspondence[28], a non-linear version of a special case of the Penrose transform. It also admits a rather more general treatment. The object of this article is to explain the general case and the natural connection between the Penrose and Ward approaches.

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[1]M. F. Atiyah and R. S. Ward . Instantons and algebraic geometry. Commun. Math. Phys. 55 (1977), 111124.

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[12]N. J. Hitchin . Linear field equations on self-dual spaces. Proc. Boy. Soc. Lond. A 370 (1980), 173191.

[14]N. J. Hitchin . Monopoles and geodesies. Commun. Math. Phys. 83 (1982), 579602.

[15]J. Isenberg , P. B. Yasskin and P. S. Green . Non-self-dual gauge fields. Phys. Lett. 78 B (1978), 462464.

[17]C. R. LeBrun . The first formal neighbourhood of ambitwistor space for curved spacetime. Lett. Math. Phys. 6 (1982), 345354.

[23]R. Penrose . Non-linear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 3152.

[28]R. S. Ward . On self-dual gauge fields. Phys. Lett. 61 A (1977), 8182.

[29]R. S. Ward . A Yang-Mills-Higgs monopole of charge 2. Commun. Math. Phys. 79 (1981), 317325.

[31]E. Witten . An interpretation of classical Yang-Mills theory. Phys. Lett. 77 B (1978), 394398.

[32]N. M. J. Woodhotjse . On self-dual gauge fields arising from twistor theory. Phys. Lett. 94 A (1983), 269270.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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