Geometrical Construction of Supertwistor Theory ShikokuSeminar Developments of

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 ar. Xiv: 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 4 D 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. 1 st Hopf map 2 nd Hopf map 3 rd Hopf map

1 st Hopf Map Hopf spinor Incidence Relation

2 nd Hopf Map 2 nd Hopf map 2 nd 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 Complexified space-time A. Ferber (1978) Fermion coordinates Incidence relation Non-Hermitian Super-twistor l Complex space-time is postulated. l 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 Twistor l Massless Condition Enhanced Symmetry l More Fundamental Quantity than Space-Time Holomorphicity, Incidence Relations l Complex conjugation = Derivative Noncommutative Geometry

Conclusion l Geometrical construction of the supertwistor based on the SUSY Hopf map. l Properties of this construction 1. Space-time is not complexified. 2. Even number of fermionic components of twistor is automatically incorporated. l Close Analogy between LLL physics and Twistor Does it suggest something deeper? ?