Amplituhedra and origami

Abstract

We establish a precise correspondence between points of momentum amplituhedra and origami crease patterns. As an application, we prove that the BCFW cells triangulate the momentum amplituhedron when all Mandelstam variables are nonnegative. As another application, we show that every weighted planar bipartite graph $Γ$ admits a t-embedding, i.e., an embedding of the planar dual of $Γ$ such that the sum of angles of white (equivalently, black) faces around each vertex is equal to $π$.

Figures (17)

• Figure 1: An origami crease pattern $\mathcal{T}$ (left) and its image under the origami map $\mathcal{O}$ (right). The angle condition is satisfied at each interior vertex of $\mathcal{T}$, which allows it to be folded consistently. Figure reproduced from Hull.
• Figure 2: A t-embedding (left) and a t-immersion (right).
• Figure 3: Applying the BCFW recursion (left) and recovering the point $\mathcal{T}(v^\ast_0)=\mathcal{O}(v^\ast_0)$ as the intersection of $\ell_{{j_0}}$ and $\rho$ (right); see \ref{['sec:intro:BCFW']}.
• Figure 4: The definition of the upstream wedge (left). The matching $\overleftarrow{M}(\hat{u}^\ast_n)$, the perfect orientation $\vec{\Gamma}_1$, and the standard Kasteleyn signs $\epsilon^{\operatorname{std}}$ (right).
• Figure 5: Examples of discrete holomorphic functions.

Theorems & Definitions (158)

• Theorem A
• Theorem B
• Definition 1.1
• Definition 1.2
• Definition 1.3: KLRRCLR1
• Theorem 1.4
• Corollary 1.5
• Definition 1.6: DFLP
• Remark 1.7
• Definition 1.8