Twistors allow an impressive organization of ordinary Feynman diagrams of gauge theories. Instead of calculating an immense number of individual diagrams you get their sum as single twistor diagram. The minimal function for twistor diagrams would be this kind of organization.
Twistor diagrams inspire also more ambitious ideas. The notion of plane wave is usually taken as given but twistors suggest as basic objects the analogs of light-rays which are waves completely localized in directions transverse to momentum direction. These are perfectly ok quantum objects since de-localization still takes place in the direction of momentum. Parton picture in QCD strongly suggest them physically. Also quantum classical correspondence becomes especially clear for them: quantum states in particle experiment would really look what they do look in laboratory. There are excellent reasons to expect that IR divergences of gauge theories are eliminated by transverse localization.
The condition that twistor structure exists in space-time is also quite a constraint and suggests strongly that higher dimensional theories should use M4× S type space so that the higher-dimensional space would not be dynamical. M4 of course has also other marvelous properties: light-cone boundary in M4 is metrically 2-D and allows generalized conformal invariance (I wonder how many times I have said this without absolutely any effect on colleagues: they simply cannot take me seriously for the fraction of minute needed to realize "Hey, this guy is right!").
In spirit of twistorialization program of Penrose I proposed some time ago how space-time surfaces representing preferred extremals of Kähler action in M4× CP2 and coding locally basic data for light rays (local momentum direction and polarization essential for twistor concept) could be lifted to holomorphic surfaces in 12-D T× CP2 or 10-D PT× CP2.
The surprise was that for surfaces which are not representable as graphs of a map M4→CP2 ("non-pertubative phase" for which QFT in M4 description does not make sense) the surfaces would have dimension higher than 4: D=6,8,10. Maybe there is a connection with branes of M-theory and TGD.
Twistors are also highly powerful idea generators. Twistor concept led through a rather funny interlude to the realization that QFT limit of TGD must be based on Dirac action coupled to gauge bosons without any YM action. The counterpart of YM action is generated radiatively so that all gauge couplings are predicted provided the loop integration can be carried out so that divergences disappear. Gauge boson propagator would have standard form apart for normalization factor which represents square of gauge coupling.
The basic problem is definition of the cutoff of momentum integration and zero energy ontology and p-adic length scale hypothesis force this cutoff physically and allow a geometric interpretation for it in terms of fractal hierarchy of causal diamonds within causal diamonds. Theory produces realistically the basic aspects of coupling constant evolution for standard model gauge couplings apart from gauge boson loops. The values of fine structure constant at electron and intermediate boson length scale fix the two parameters - call them a and b, characterizing the cutoff in hyperbolic angle to two very natural values. b is exponent and exactly equal to b=1/3 by argument based on analyticity (no fractional powers of logarithms). Second one is coefficient equal to a=0.22050469512552 if fine structure constant is required exactly in electron length scale (this means of course over accuracy). Taking analyticity argument seriously, one can say that fine structure constant is predicted in intermediate gauge boson length scale.
It turned out that massivation of gauge bosons occurs unless the hyperbolic cutoffs for time-like and space-like momenta are related in a unique manner. The hyperbolic cutoff is the ad hoc element of the model, and the next project is to find whether the proposed model in which quantum criticality would fix the UV cutoff in hyperbolic angle really does it and whether it leads to the hyperbolic cutoff forced by the values of fine structure constant at electron and intermediate gauge boson length scale.
This involves rather heavy numerical calculations using rather primitive tools [just MATLAB (afforded by a friend since Helsinki University long time ago found it impossible to help by providing program packages like Mathematica), no symbol manipulation packages, no young left-brainy students] and represents quite a challenge for my 58 year old badly right-halved brain.
I have organized the work on twistors and emergence of gauge boson propagators to two new chapters: Twistors, N=4 Super-Conformal Symmetry, and Quantum TGD and Quantum Field Theory Limit of TGD from Bosonic Emergence of "Towards M-matrix".