Cluster algebras and tilings for the m=4 amplituhedron

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $G{r}_{k,n}^{\ge 0}$ under the map $Z:G{r}_{k,n}^{\ge 0}\to G{r}_{k,k+m}$ induced by a positive linear map $Z:{\mathbb{R}}^n\to {\mathbb{R}}^{k+m}$. Motivated by a question of Hodges, Arkani-Hamed and Trnka introduced the amplituhedron as a geometric object whose tilings conjecturally encode the BCFW recursion for computing scattering amplitudes. More specifically, the expectation was that one can compute scattering amplitudes in $N=4$ SYM by tiling the $m=4$ amplituhedron $A_{n,k,4}(Z)$ - that is, decomposing the amplituhedron into tiles' (closures of images of $4k$-dimensional cells of $Gr_{k,n}^{\geq 0}$ on which ${Z}$ is injective) - and summing the volumes’ of the tiles. In this article we prove two major conjectures about the $m=4$ amplituhedron: $i)$ the BCFW tiling conjecture, which says that any way of iterating the BCFW recurrence gives rise to a tiling of the amplituhedron $A_{n,k,4}(Z)$; $ii)$ the cluster adjacency conjecture for BCFW tiles, which says that facets of tiles are cut out by collections of compatible cluster variables for $G{r}_{4,n}$. Moreover, we show that each BCFW tile is the subset of $G{r}_{k,k+4}$ where certain cluster variables have particular signs. Along the way, we construct many explicit seeds for $G{r}_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras.