Motivic Amplitudes and Generalized Associahedra

With the ongoing amplituhedra hype, it's worthwhile to re-visit motivic amplitudes and associahedra once again.  Motivic amplitudes were created because of the plethora of functional identities amongst generalized polylogarithms that precludes the existence of any particular preferred or canonical functional representation (i.e. formula) for general multi-loop amplitudes.  Such amplitudes are elements of a Hopf algebra, acting as an algebra of functions on a motivic Galois group.  The group structure is contained in the coproduct of the Hopf algebra, enabling one to uncover the hidden Galois symmetries of the amplitudes.

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.

 Recall Witten's diagram for the n=5 MHV amplitude, shown above.  One can view it as spheres (degree one genus zero instantons) connected by a 'twistor field' tube by lines in CP^3 that exchange the twistor field, altogether giving an object topologically equivalent to a sphere. (In the twistor field tube, the helicities are reversed between the two ends because all fields attached to either instanton are considered to be outgoing and because crossing symmetry relates an incoming gluon of one helicity to an outgoing gluon of opposite helicity.  Also, each degree one instanton must have at least two opposite helicities attached to it.)  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 a 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).

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.

`
















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?