Cluster algebras and tilings for the m=4 amplituhedron

Abstract

The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian $Gr_{k,n}^{\geq 0}$ under the map ${Z}: Gr_{k,n}^{\geq 0} \to Gr_{k,k+m}$ 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 in 2013 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. Also in 2013, Golden-Goncharov-Spradlin-Vergu-Volovich gave the first link between scattering amplitudes and cluster algebras, with Drummond-Foster-Gurdogan subsequently formulating the {cluster adjacency conjecture}. In this article we reveal and prove the deep mechanism behind `cluster phenomena' in tree-level scattering amplitudes. By connecting the BCFW recursion to a new cluster quasi-homomorphism on the Grassmannian $\Gr_{4,n}$, we prove the {cluster adjacency conjecture} for BCFW tiles, which says that each tile is a semialgebraic subset of the amplituhedron where a collection of compatible cluster variables take on definite signs. In particular, the facets of these tiles are cut out by compatible cluster variables. We also use the cluster description of BCFW tiles to prove the {BCFW tiling conjecture}, resolving the main original conjecture for the $m=4$ amplituhedron.