The understanding of twistor structure of imbedding space and its relationship to the consctruction of extremals of Kähler action is certainly greatest breakthroughs in the mathematical understanding of TGD for years. One can say that the good physics provided by TGD can be now combined with the marvelous mathematics produced by string theorists by replacing Calabi-Yau manifolds with twistor spaces assignable to space-time surfaces and representable as sub-manifolds of the twistor space CP3× F3 of imbedding space M4× CP2 (strictly speaking M4 twistor space is the non-compact space SU(2,2)/SU(2,1) ×U(1)). What it so wonderful is that the enormous collective knowhow involved with algebraic geometry becomes avaiblable in TGD and now this mathematics makes sense physically. My sincere hope is that also colleagues would finally realize that TGD is the only way out from the recent dead alley in fundamental physics.
Below is the introduction of the article article Classical part of twistor story. I will later add some key pieces of the article.
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× 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 CP3 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 having as lines space-time surfaces with Euclidian signature of induced metric and having wormhole contacts as basic building bricks.
There is also a very close 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 basic problem of TGD has been indeed the lack of existing mathematical methods to realize quantitatively the view about space-time as 4-surface.
The physical interpretation is totally different in TGD. Twistor space has space-time as base-space rather than forming with it Cartesian factors of a 10-D space-time. The Calabi-Yau landscape is replaced with the space of twistor spaces of space-time surfaces having interpretation as generalized Feynman diagrams and twistor spaces as sub-manifolds of CP3× F3 replace Witten's twistor strings. The space of twistor spaces is the lift of the "world of classical worlds" (WCW) by adding the CP1 fiber to the space-time surfaces so that the analog of landscape has beautiful geometrization.