Developments of Q.F.T. & String Theory

Geometrical Construction of Supertwistor Theory

Jul.28 - Aug.1 2008

Kazuki Hasebe

Takuma National College of Technology

Shikoku-Seminar

arXiv:0805.2644

Introduction: Twistor ProgramRoger Penrose (1967)

From ``The Road to Reality’’

Space-time is taken to be a secondary construction from the more primitive twistor notions.

Space-Time Event Twistor Space

Incidence Relation

Incidence Relation

Non-local transformation

Light

Projective complex-line

(Null-line)

4D Minkowski-space Twistor-space

Massless particle and Twistor

Massless particle

Free particle

Pauli-Lubanski spin-vector

Helicity

Hopf Map: Template of Twistor

Topological map from sphere to sphere in different dimensions.

Heinz Hopf (1931)

1st Hopf map

2nd Hopf map

3rd Hopf map

1st Hopf Map

Incidence Relation

Hopf spinor

1st Hopf Map

2nd Hopf Map

2nd Hopf map

S.C. Zhang & J.P. Hu (2001)

2nd Hopf spinor

Direct Relation to Twistor

Incidence Relation

Constraint

Null Twistor Helicity zero

is Hermitian (space-time is real)

Idea of Supertwistor

Complex space-time is postulated.

Fermion coordinates

Complexified space-time

Super-twistor

Incidence relation

A. Ferber (1978)

Non-Hermitian

Fermion number can be even or odd integer.

The SUSY Hopf MapC. Bartocci, U. Bruzzo, G. Landi (1987)

The SUSY Hopf map

Supertwistor VariablesSuper Incidence Relation

Supertwistor variables

Not-complexified

Even number

: Super-Hermitian

:null-supertwistor

Super Incidence Relation

Minkowski-superspace Supertwistor-space

Non-local super-transformation

Supertwistor action and Quantization

wave-function for mass-less particle

should be even integer.

Supertwistor action

Twistor function

Helicity

Relation to Lowest Landau Level

U(1) connection

LLL-limit

One-particle action

Dirac monopole

Analogies between Twistor and LLL

Complex conjugation = Derivative

Twistor LLL

More Fundamental Quantity than Space-Time

Massless Condition

Noncommutative Geometry

Holomorphicity, Incidence Relations

Enhanced Symmetry

Conclusion Geometrical construction of the supertwistor based on the SUSY Hopf map.

Properties of this construction

Close Analogy between LLL physics and Twistor

1. Space-time is not complexified.

2. Even number of fermionic components of twistor is automatically incorporated.

Does it suggest something deeper??