The KITP program The Harmony of Scattering Amplitudes is still underway and there have been many wonderful talks on the geometry of scattering amplitudes in twistor space. The twistor approach allows one to look at scattering processes in a more algebraic geometrical fashion where as Ed Witten noted (arXiv:hep-th/0312171), one should focus on holomorphic curves.
At the most basic level, one is interested in degree one genus zero curves. In the complex case, such curves are copies of CP^1, 2-spheres. Witten argued that n-particle MHV amplitudes with two particles of negative helicity and n-2 with positive helicity localize on such degree one genus zero curves. This is the special case of Witten's more general conjecture that the twistor version of the n particle scattering amplitude is nonzero only if the points are supported on an algebraic curve in twistor space of degree d=q-1+l (where q is number of negative helicity particles and l is the number of loops). So for example, the tree level ++--- amplitude is nonzero on a curve of degree d=2-1+0=1, a degree one genus zero curve, a 2-sphere, as expected (see diagram above).
More recently, there is a more combinatorial way to view the MHV (and N^kMHV) amplitudes. This approach allows one to use associahedra, bubble diagrams and chorded polygons, for example. Below is a chorded polygon for the ++--- amplitude, and up to rotation and CPT transformation, is the only one contributing to the amplitude. For the --+++ amplitude there is another such polygon, so for the n=5 MHV amplitudes only (2(n-3))!/(n-3)!(n-2)!=2 total chorded polygons contribute.
In such chorded polygons the chords physically correspond to twistor fields exchanged between degree one genus zero instantons. So given an n-point N^kMHV amplitude, one can draw many different chorded polygons (given by Catalan number C_{n-2}), but those which contribute are those that have no internal twistor field triangles, and describe configurations where each genus zero curve in the process has at least two points with different helicities. Below is a diagram for a non-contributing chorded polygon for the n=8 NNMHV amplitude (note the "illegal" internal twistor field triangle).
As Nima Arkani-Hamed has noted, the twistor approach to scattering amplitudes is revealing a deeper mathematical unity that Feynman diagrams obscure. The mathematics so far involves the Riemann moduli space of surfaces of genus g with n marked points, Gromov-Witten invariants, symplectic geometry, quantum cohomology and motives.
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