M Theory Lesson 345

For some years now, kneemo and I have been thinking about the connection between the associahedra polytopes and $n$ particle scattering rules, such as BCFW recursion. Recall that the cells of an associahedron are labelled by rooted binary trees. The first four dimensions are (partly) given by the pictures: Now BCFW recursion (check out the new lectures at Perimeter) is applied to massless particle scattering where legs are labelled by momenta and helicity. We can turn a bunch of legs into a tree by creating a root from a choice of ordering on the legs. The question is, where do the signs go on the leaves of the associahedra labels? This is answered by considering that the terms in question are counted by Narayana numbers. For a given $n$, the sum of the Narayana numbers gives the total number of vertices on an associahedron. So we just pick the associahedron of the right dimension by adding a fixed set of signs to the sign sequences of our amplitudes.

For example, there are $6$ possible ways to write a length four sign sequence with two positive and two negative signs. These sequences are embedded in sequences of length $7$ in such a way that, after a root choice, they are mapped to the six trees of the $3d$ polytope that happen to have three left leaning leaves, as shown. In the rule of interest, this counts terms for $n = 7$. One choice in this case is to (i) append $--$ to the left of the length $4$ string and $+$ to the right, and (ii) delete the leftmost instance of $-+$, for the root. Moreover, since the case $n = 3$ would now correspond to the empty associahedron, there is a real sense in which geometry begins with four particles.