Motivic amplitudes and cluster coordinates

In this paper we study motivic amplitudes — objects which contain all of the essential mathematical content of scattering amplitudes in planar SYM theory in a completely canonical way, free from the ambiguities inherent in any attempt to choose particular functional representatives. We find that the cluster structure on the kinematic configuration space Conf n (ℙ 3 ) underlies the structure of motivic amplitudes. Specifically, we compute explicitly the coproduct of the two-loop seven-particle MHV motivic amplitude $\mathcal{A}_{7,2}^{\mathcal{M}}$ and find that like the previously known six-particle amplitude, it depends only on certain preferred coordinates known in the mathematics literature as cluster $\mathcal{X}$ - coordinates on Conf n (ℙ 3 ). We also find intriguing relations between motivic amplitudes and the geometry of generalized associahedrons, to which cluster coordinates have a natural combinatoric connection. For example, the obstruction to $\mathcal{A}_{7,2}^{\mathcal{M}}$ being expressible in terms of classica