Applications of complex manifold techniques to elementary particle physics

Part of the Lecture Notes in Physics book series (LNP, volume 97)


Holomorphic Function Minkowski Space Twistor Space Generalize Twistor Twistor Cohomology 
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  1. 1.
    The Kerr theorem is discussed in Penrose (1967), to which the reader is referred for further details.Google Scholar
  2. 2.
    For an account of the 27 lines on the cubic in P3 see, for example, Mumford (1976). In the case of twister theory the quadric surface in P3 is also of special interest. Associated with such a surface is a solution of the twister equation of valence two: \(\nabla ^{A(A'} \xi ^{B'C')} = 0\). This differential equation is rather curious inasmuch as it admits non-trivial solutions in curved spacetime. In particular, the Kerr solution of Einstein's equations possesses a solution of the valence two twister equation. For discussion related to this matter see Carter (1968), Walker and Penrose (1970), Hughston, Penrose, Sommers, and Walker (1972), Hughston and Sommers (1973a and 1973b), and Sommers (1973).Google Scholar
  3. 3.
    These results, which were first described in Twister Newsletter in 1977, appear in Penrose (1979).Google Scholar
  4. 4.
    For more extensive accounts of the material described in this section see, for example, iozsa (1976), Pratt (1977), Moore (1978), Burnett-Stuart (1978), Weber (1978), Ward (1979), and Wells (1979). There are also to be found numerous Twister Newsletter articles on the subject.Google Scholar
  5. 5.
    It is also interesting to note that the geodesic shearfree condition generalizes in an interesting way to the cases for which m is greater than one. One obtains the following formula: \((\xi ^{[a'} \nabla ^{b']b} \xi ^{[c'} )\xi ^{d']} = 0\) As will be described elsewhere, there exists an analogue of the Kerr theorem which allows one to solve this equation using complex analytic methods.Google Scholar
  6. 6.
    I am indebted to M. Eastwood for a number of illuminating discussions in connection with the material of Section 10.6.Google Scholar

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© Springer-Verlag 1979

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