The Magic Number Conjecture for the $m=2$ amplituhedron and Parke-Taylor identities

Abstract

The amplituhedron $A_{n,k,m}$ is a geometric object introduced in the context of scattering amplitudes in $N=4$ super Yang Mills. It generalizes the positive Grassmannian (when $n=k+m$), cyclic polytopes (when $k=1$), and the bounded complex of the cyclic hyperplane arrangement (when $m=1$). Of substantial interest are the tilings of the amplituhedron, which are analogous to triangulations of a polytope. Karp, Williams and Zhang (2020) observed that the known tilings of $A_{n,k,2}$ have cardinality $\left(\genfrac{}{}{0ex}{}{n-2}{k}\right)$ and the known tilings of $A_{n,k,4}$ have cardinality the Narayana number $\frac{1}{n-3}\left(\genfrac{}{}{0ex}{}{n-3}{k+1}\right)\left(\genfrac{}{}{0ex}{}{n-3}{k}\right)$; generalizing these observations, they conjectured that for even $m$ the tilings of $A_{n,k,m}$ have cardinality the MacMahon number, the number of plane partitions which fit inside a $k\times (n-k-m)\times \frac{m}{2}$ box. We refer to this prediction as the `Magic Number Conjecture'. In this paper we prove the Magic Number Conjecture for the $m=2$ amplituhedron: that is, we show that each tiling of $A_{n,k,2}$ has cardinality $\left(\genfrac{}{}{0ex}{}{n-2}{k}\right)$. We prove this by showing that all positroid tilings of the hypersimplex ${\Delta }_{k+1,n}$ have cardinality $\left(\genfrac{}{}{0ex}{}{n-2}{k}\right)$, then applying T-duality. In addition, we give combinatorial necessary conditions for tiles to form a tiling of $A_{n,k,2}$; we give volume formulas for Parke-Taylor polytopes and certain positroid polytopes in terms of circular extensions of cyclic partial orders; and we prove new variants of the classical Parke-Taylor identities.

44 pages, 14 figures. v2: minor changes