Geometrical Construction of Supertwistor Theory Shikoku-Seminar Developments of Q.F.T. & String Theory Jul.28 - Aug.1 2008 Kazuki Hasebe Takuma National College of Technology arXiv:0805.2644 Introduction: Twistor Program Roger Penrose (1967) Space-time is taken to be a secondary construction from the more primitive twistor notions. From ``The Road to Reality’’ Space-Time Event Incidence Relation Twistor Space Incidence Relation 4D Minkowski-space Twistor-space Light (Null-line) Projective complex-line Non-local transformation Massless particle and Twistor Free particle Massless particle Helicity Pauli-Lubanski spin-vector Hopf Map: Template of Twistor Heinz Hopf (1931) Topological map from sphere to sphere in different dimensions. 1st Hopf map 2nd Hopf map 3rd Hopf map 1st Hopf Map 1st Hopf Map Hopf spinor Incidence Relation 2nd Hopf Map 2nd Hopf map 2nd Hopf spinor S.C. Zhang & J.P. Hu (2001) Direct Relation to Twistor Incidence Relation Constraint is Hermitian (space-time is real) Null Twistor Helicity zero Idea of Supertwistor A. Ferber (1978) Complexified space-time Fermion coordinates Incidence relation Non-Hermitian Super-twistor Complex space-time is postulated. Fermion number can be even or odd integer. The SUSY Hopf Map The SUSY Hopf map C. Bartocci, U. Bruzzo, G. Landi (1987) Supertwistor Variables Super Incidence Relation Not-complexified : Super-Hermitian Supertwistor variables Even number :null-supertwistor Super Incidence Relation Minkowski-superspace Supertwistor-space Non-local super-transformation Supertwistor action and Quantization Supertwistor action Helicity should be even integer. Twistor function wave-function for mass-less particle Relation to Lowest Landau Level Dirac monopole U(1) connection One-particle action LLL-limit Analogies between Twistor and LLL LLL Twistor Massless Condition Enhanced Symmetry More Fundamental Quantity than Space-Time Holomorphicity, Incidence Relations Complex conjugation = Derivative Noncommutative Geometry Conclusion Geometrical construction of the supertwistor based on the SUSY Hopf map. Properties of this construction 1. Space-time is not complexified. 2. Even number of fermionic components of twistor is automatically incorporated. Close Analogy between LLL physics and Twistor Does it suggest something deeper??
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