Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-28T13:38:16.725Z Has data issue: false hasContentIssue false
  • English
  • Français

The generalized Penrose-Ward transform

Published online by Cambridge University Press:  24 October 2008

Michael G. Eastwood
Michael G. Eastwood
Affiliation:
Mathematical Institute, 24–29 St Giles', Oxford 0X1 3LB
Get access Rights & Permissions [Opens in a new window]

Extract

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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 1 , January 1985 , pp. 165 - 187
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Atiyah, M. F. and Ward, R. S.. Instantons and algebraic geometry. Commun. Math. Phys. 55 (1977), 111–124.CrossRefGoogle Scholar
[2]Atiyah, M. F., Hitchin, N. J. and Singer, I. M.. Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond. A 362 (1978), 425–461.Google Scholar
[3]Atiyah, M. F.. Geometry of Yang-Mills Fields (Lezioni Fermiane, Pisa: Scuola Normale Superiore, 1979).Google Scholar
[4]Bott, R.. Homogeneous vector bundles. Ann. Math. 66 (1957), 203–248.CrossRefGoogle Scholar
[5]Buchdahx, N. P.. On the relative deRham sequence. Proc. A.M.S. 87 (1983), 363–366.CrossRefGoogle Scholar
[6]Buchdahl, N. P.. Analysis on analytic spaces and non-self-dual Yang-Mills fields. Trans. A.M.S. (To appear.)Google Scholar
[7]Eastwood, M. G., Penrose, R. and Wells, R. O. Jr. Cohomology and massless fields. Commun. Math. Phys. 78 (1981), 305–351.Google Scholar
[8]Eastwood, M. G. and Ginsberg, M. L.. Duality in twistor theory. Duke Math. J. 48 (1981), 177–196.CrossRefGoogle Scholar
[9]Eastwood, M. G.. The generalized twistor transform and unitary representations of SU(p, q). (To appear.)Google Scholar
[10]Eastwood, M. G.. Antitwistors. (To appear.)Google Scholar
[11]Henkin, G. M. and Manin, Yu. I.. Twistor description of classical Yang-Mills-Dirac fields. Phys. Lett. 95 B (1980), 405–408.Google Scholar
[12]Hitchin, N. J.. Linear field equations on self-dual spaces. Proc. Boy. Soc. Lond. A 370 (1980), 173–191.Google Scholar
[13]Hitchin, N. J.. Complex manifolds and Einstein's equations. In Twistor Geometry and Non-linear Systems, 4th Bulg. Summer Sch., Primorsko/Bulg. 1980. Lecture Notes in Math. vol. 970 (Springer-Verlag, 1982).Google Scholar
[14]Hitchin, N. J.. Monopoles and geodesies. Commun. Math. Phys. 83 (1982), 579–602.CrossRefGoogle Scholar
[15]Isenberg, J., Yasskin, P. B. and Green, P. S.. Non-self-dual gauge fields. Phys. Lett. 78 B (1978), 462–464.CrossRefGoogle Scholar
[16]Jozsa, R. O.. Models in categories and twistor theory. D.Phil, thesis, Oxford 1981.Google Scholar
[17]LeBrun, C. R.. The first formal neighbourhood of ambitwistor space for curved spacetime. Lett. Math. Phys. 6 (1982), 345–354.Google Scholar
[18]Leitbrbb, J.. The Penrose transform for bundles non-trivial on the general line. (To appear.)Google Scholar
[19]Leitebbb, J.. Subsheaves in bundles on Pn and the Penrose transform. (To appear.)Google Scholar
[20]Littlewood, D. E.. The Theory of Group Characters and Matrix Representations of Groups, 2nd ed. (Clarendon Press 1950).Google Scholar
[21]Manin, Yu. I.. Gauge Fields and Holomorphic Geometry. [Russian.] Current Problems in Math. 17 (Akad. Nauk. S.S.S.R. 3–55, 1981).Google Scholar
[22]Penrose, R.. Twistor theory, its aims and achievements. In Quantum Gravity: an Oxford Symposium (Clarendon Press, 1975), pp. 268–407.Google Scholar
[23]Penrose, R.. Non-linear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 31–52.Google Scholar
[24]Penbose, R. and Ward, R. S.. Twistors for fiat and curved space-time. In General Relativity and Gravitation, vol. it, ed. Held, A. (Plenum Press, 1980).Google Scholar
[25]Pool, R.. Yang-Mills fields and extension theory. Memoirs A.M.S. (To appear.)Google Scholar
[26]Salamon, S. M.. Quaternionic manifolds. D.Phil, thesis, Oxford, 1980.Google Scholar
[27]Salamon, S. M.. Quaternionic manifolds. In Symposia Mathematica, no. XXVI (Istituto Nationale di Alta Matematica Francesco Severi, Dist. Academic Press, 1982).Google Scholar
[28]Ward, R. S.. On self-dual gauge fields. Phys. Lett. 61 A (1977), 81–82.CrossRefGoogle Scholar
[29]Ward, R. S.. A Yang-Mills-Higgs monopole of charge 2. Commun. Math. Phys. 79 (1981), 317–325.Google Scholar
[30]Wells, R. O. Jr. Complex Geometry in Mathematical Physics (Presses de l'universite de Montreal, 1982).Google Scholar
[31]Witten, E.. An interpretation of classical Yang-Mills theory. Phys. Lett. 77 B (1978), 394–398.CrossRefGoogle Scholar
[32]Woodhotjse, N. M. J.. On self-dual gauge fields arising from twistor theory. Phys. Lett. 94 A (1983), 269–270.Google Scholar
[33]Wybourne, B. G.. Symmetry Principles and Atomic Spectroscopy (Wiley-Interscience, 1970).CrossRefGoogle Scholar