TGD Variant of Twistor Story (Appendix & References)

Abstract

Twistor Grassmannian formalism has made a breakthrough in N=4 supersymmetric gauge theories and the Yangian symmetry suggests that much more than mere technical breakthrough is in question. Twistors seem to be tailor made for TGD but it seems that the generalization of twistor structure to that for 8-D imbedding space H=M⁴ x CP₂ is necessary. M⁴ (and S⁴ as its Euclidian counterpart) and CP₂ are indeed unique in the sense that they are the only 4-D spaces allowing twistor space with Kähler structure. The Cartesian product of twistor spaces P₃=SU(2,2)/SU(2,1) x U(1) and F₃ defines twistor space for the imbedding space H and  one can ask whether  this generalized twistor structure could allow to understand both quantum TGD and classical TGD defined by the extremals of Kähler action. In the following I summarize the background and develop a proposal for how to construct extremals of Kähler action in terms of the generalized twistor structure.  One ends up with a scenario in which space-time surfaces are lifted to twistor spaces by adding CP₁ fiber so that the twistor spaces give an alternative representation for generalized Feynman diagrams.

There is also a very closely analogy with superstring models. Twistor spaces replace Calabi-Yau manifolds and the modification recipe for Calabi-Yau manifolds by removal of singularities can be applied to remove self-intersections of twistor spaces and mirror symmetry emerges naturally. The overall important implication is that the methods of algebraic geometry used in super-string theories should apply in TGD framework. The physical interpretation is totally different in TGD. The landscape is replaced with twistor spaces of space-time surfaces having interpretation as generalized Feynman diagrams and twistor spaces as sub-manifolds of P₃ x F₃ replace Witten's twistor strings. The classical view about twistorialization of TGD makes possible a more detailed formulation of the previous ideas about the relationship between TGD and Witten's theory and twistor Grassmann approach. Furthermore, one ends up to a formulation of  the scattering amplitudes in terms of Yangian of the super-symplectic algebra relying on the idea that scattering amplitudes are sequences consisting of algebraic operations (product and co-product) having interpretation as vertices in the Yangian extension of super-symplectic algebra. These sequences connect given initial and final states and having minimal length. One can say that Universe performs calculations.

Appendix: Some mathematical details about Grasmannian formalism

References