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 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 $Gr_{4,n}$. Moreover, we show that each BCFW tile is the subset of $Gr_{k, k+4}$ where certain cluster variables have particular signs. Along the way, we construct many explicit seeds for $Gr_{4,n}$ comprised of high-degree cluster variables, which may be of independent interest in the study of cluster algebras.

Figures (25)

• Figure 1: The chord diagram for a standard BCFW tile of $\mathcal{A}_{14,6,4}(Z)$.
• Figure 2: The seed associated to the standard BCFW tile in \ref{['fig:intr-chord']}. The mutable variables are circled; all other variables are frozen. The colors (red, green, blue) indicate the different cases of \ref{['def:seed']}.
• Figure 3: Left: the quiver $Q_{4,7}$ with vertices labeled by rectangles contained in a $4 \times 3$ rectangle. Middle: the rectangles seed $\Sigma_{4,7}$, where we identify $4$-element subsets of $[7]$ with Plücker coordinates. Frozen variables are boxed. Right: the cyclically shifted rectangles seed $\Sigma_{4,7}^1$. The following mutation sequence maps us from the seed in the middle, to the seed at the right: mutate at $2567, 3567, 2367, 3467, 2347, 3457$.
• Figure 4: The initial seed $\Sigma_0$ for $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}_{4,N_L}]\times \mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}_{4,N_R}]$, which is a disjoint union of a seed $\Sigma_0^{L}$ for $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}_{4,N_L}]$ (left) and a seed $\Sigma_0^R$ for $\mathbb{C}[\widehat{\mathop{\mathrm{Gr}}\nolimits}_{4,N_R}]$ (right). We use the notation $x':=x-1,x":=x-2$ and $x^{+i}:= x+i$. Mutable variables are in the shaded regions, the others are all frozen.
• Figure 5: The result $\Psi(\Sigma_0)$ of applying the product promotion to $\Sigma_0$. We use the notation $r_x:= \langle b \, c\, d\, n\rangle / \langle a \, b\, c\, x\rangle$ and $x':=x-1,x":=x-2,x^{+i}:= x+i$.

Theorems & Definitions (244)

• Theorem : BCFW tile theorem
• Theorem : Cluster adjacency for BCFW tiles
• Theorem : Sign description of BCFW tiles
• Theorem : BCFW tiling theorem
• Remark 2.1
• Definition 2.2: Positive Grassmannian
• Definition 2.3: Dihedral group on the Grassmannian
• Definition 2.4
• Definition 2.5: Chain polynomials
• Remark 2.6