The Penrose transform. Its interaction with representation theory.

*(English)*Zbl 0726.58004
Oxford Mathematical Monographs. Oxford etc.: Clarendon Press. xv, 213 p. £25.00 (1989).

As one of the most impressive successes of his twistor theory, Roger Penrose showed how solutions of conformally invariant zero rest mass field equations with arbitrary spin on a four dimensional space-time can be expressed as contour integrals of free holomorphic functions over lines of a three-dimensional complex projective space. He subsequently realized that the freedom available in the function for a fixed solution was precisely that of Cech representative of a sheaf cohomology class. The resulting isomorphism between a sheaf cohomology group on a region of projective space and solutions of a zero mass field equation on a region of the space-time has become known as the Penrose transform. In recent years this notion has been extensively investigated and has important consequences for the generalization of twistors to higher dimensions and their extension to curved spaces.

In the present monograph the authors study the analogue of the Penrose transform when the conformal group is replaced by an arbitrary complex semisimple Lie group G, and the resulting relation to the representation theory of reductive Lie groups and algebras. In this context the space- time and projective space are replaced by suitable open subsets X and Z of compact complex homogeneous spaces and flag varieties of G, and the transform relates the cohomology of a homogeneous vector bundle on Z in terms of kernels and cokernels of invariant differential operators on sections of homogeneous vector bundles on X. This is an exciting prospect: physically it gives a systematic exposition of twistor theory in dimensions greater than four, which opens the possibility of applications in Kaluza-Klein type theories, supergravity, and string theories; mathematically it provides representation theory with a new construction, i.e. a geometric globalization of Zuckerman’s derived functor, the construction of unitary representations on cohomology groups (without the complications encountered in \(L^ 2\)-cohomology), and the study of homomorphisms between Verma modules.

The authors consider both the physical and mathematical aspects of the Penrose transform. They explicitly presuppose no prior acquaintance with twistor theory and only a minimum of representation theory, however - as the reader may readily expect - the prerequisites for their discussion are non-trivial. Their exposition is masterfully presented in a lively and lucid manner, and the result is a tour de force of mathematical thinking which should be welcomed by mathematical physicists and representation theorists. Contents include twelve chapters: Introduction, Lie algebras and flag manifolds, Homogeneous vector bundles on G/P, The Weyl group, its actions and Hasse diagrams, The Bott-Borel-Weil theorem, Realizations of G/P, The Penrose transform in principle, The Bernstein- Gelfand-Gelfand resolution, The Penrose transform in practice, Constructing unitary representations, Module structures on cohomology, and Conclusions and outlook. The monograph concludes with an exhaustive ten page bibliography.

In the present monograph the authors study the analogue of the Penrose transform when the conformal group is replaced by an arbitrary complex semisimple Lie group G, and the resulting relation to the representation theory of reductive Lie groups and algebras. In this context the space- time and projective space are replaced by suitable open subsets X and Z of compact complex homogeneous spaces and flag varieties of G, and the transform relates the cohomology of a homogeneous vector bundle on Z in terms of kernels and cokernels of invariant differential operators on sections of homogeneous vector bundles on X. This is an exciting prospect: physically it gives a systematic exposition of twistor theory in dimensions greater than four, which opens the possibility of applications in Kaluza-Klein type theories, supergravity, and string theories; mathematically it provides representation theory with a new construction, i.e. a geometric globalization of Zuckerman’s derived functor, the construction of unitary representations on cohomology groups (without the complications encountered in \(L^ 2\)-cohomology), and the study of homomorphisms between Verma modules.

The authors consider both the physical and mathematical aspects of the Penrose transform. They explicitly presuppose no prior acquaintance with twistor theory and only a minimum of representation theory, however - as the reader may readily expect - the prerequisites for their discussion are non-trivial. Their exposition is masterfully presented in a lively and lucid manner, and the result is a tour de force of mathematical thinking which should be welcomed by mathematical physicists and representation theorists. Contents include twelve chapters: Introduction, Lie algebras and flag manifolds, Homogeneous vector bundles on G/P, The Weyl group, its actions and Hasse diagrams, The Bott-Borel-Weil theorem, Realizations of G/P, The Penrose transform in principle, The Bernstein- Gelfand-Gelfand resolution, The Penrose transform in practice, Constructing unitary representations, Module structures on cohomology, and Conclusions and outlook. The monograph concludes with an exhaustive ten page bibliography.

Reviewer: J.D.Zund (Las Cruces)

##### MSC:

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |

58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |

22E55 | Representations of Lie and linear algebraic groups over global fields and adèle rings |

22E46 | Semisimple Lie groups and their representations |

58Z05 | Applications of global analysis to the sciences |

83E15 | Kaluza-Klein and other higher-dimensional theories |

83E30 | String and superstring theories in gravitational theory |

83E50 | Supergravity |