The twistor amplitudes relevant for N=4 SYM can be seen as functions on the 3(n-5) dimensional configuration space Conf_n(CP^3), the space of collections of n points in the projective 3-space CP^3, i.e., projective twistor space. Goncharov et al. assert that the cluster structure of the space Conf_n(CP^3) underlies the structure of amplitudes in N=4 SYM theory. It is pleasing to see polygon chordings, associahedra and Stasheff polytope mentioned in the paper. The chorded polygon approach to twistor string amplitudes was discussed on this blog back in 2011. It all makes better sense if one recalls Witten's original twistor string paper.

Another way to view it is with a chorded polygon, where the twistor field parts are represented as chords. As Goncharov et al. note, one can 'mutate' these polygons, and find equivalence classes of diagrams, count up the representative diagrams and it will tell you how many terms will show up in your amplitude. For higher polygons, where internal triangles appear, one can go back to the sphere picture and see why internal triangles violate Witten's rules for exchanged twistor fields, hence their diagrams don't contribute to the amplitudes.

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For the above diagram, imagine three spheres (degree one genus zero instantons) connected by three twistor field tubes that all meet together in a central location. How can one sensibly assign helicities to the ends of the twistor field tubes?

It would be neat to go beyond Conf_n(CP^3). To start, it is well known one can identify complex 4-space C^4 with quaternionic 2-space H^2, hence get a map from CP^3 to HP^1. Then we would be studying Conf_n(HP^1) and transforming lines with SL(2,H)~SO(5,1). Going a step higher and using octonionic 2-space O^2 gives a map to OP^1 and we can study Conf_n(OP^1) and mapping lines to lines with SL(2,O)~SO(9,1). We could also study quaternionic 3-space H^3 and map to HP^2 and study Conf_n(HP^2), mapping lines with SL(3,H)~SU*(6). For the octonions, O^3 gives OP^2 (the Cayley plane) and collineations are given by E6(-26) transformations, where collineations fixing a point in OP^2 are SO(9,1) transformations. The study of Conf_n(OP^2) would be tantalizing indeed and there are no higher projective spaces over the octonions beyond a plane. And what exactly would one be scattering in OP^2 anyway?