The Penrose Transform: Its Interaction with Representation TheoryDuring the past two decades, Roger Penrose's twistor theory has been a continuing source of mathematical inspiration. It is an ambitious theory which aims to reformulate the foundations of physics using conformal and complex geometry. At the heart of twistor theory lies a geometrical transform now known as the Penrose transform. This book is an exposition of this transform in a fairly general setting. This setting is provided by complex homogeneous spaces and the mathematical input is taken from the representation theory of Lie groups. The book consists mainly of original research not published elsewhere and is intended for physicists and mathematicians with interests in geometry and symmetry. Whilst the material is presented in full generality, there are many examples throughout and special attention is directed towards the twistor theory of Minkowski space. |
Contents
Introduction | 1 |
Lie Algebras and Flag Manifolds | 10 |
Homogeneous Vector Bundles on GP | 21 |
Copyright | |
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action affine BGG resolution chapter cohomology group complex semisimple Lie compute conformal conjugate construction CS2n defined denote dimensions direct images discrete series double fibration dual Dynkin diagram element equations exact sequence example fibres finite dimensional flag manifolds flag varieties follows functor g-module geometric H¹(PT+ Hasse diagram helicity highest weight holomorphic homogeneous bundles homogeneous sheaves homogeneous vector bundles homomorphisms of Verma hypercohomology spectral sequence identified induced integral invariant differential operators irreducible representation isomorphism K-types lemma Lie algebra Lie group line bundle M. G. Eastwood massless fields metric Minkowski space nodes notation obtained orbit Penrose transform quadric Remark representation theory rest mass fields sheaf simple reflections spinors structure subgroup subvariety symplectic tangent bundle twistor space twistor theory twistor transform vanishes vector bundles vector fields Verma modules Ward correspondence Weyl group zero rest mass Ω¹ Ω²