Triangulations and Canonical Forms of Amplituhedra: a fiber-based approach beyond polytopes

Understand

Any totally positive $(k+m)\times n$ matrix induces a map $\pi_+$ from the positive Grassmannian ${\rm Gr}_+(k,n)$ to the Grassmannian ${\rm Gr}(k,k+m)$, whose image is the amplituhedron $\mathcal{A}_{n,k,m}$ and is endowed with a top-degree form called the canonical form ${\bf\Omega}(\mathcal{A}_{n,k,m})$.

• This construction was introduced by Arkani-Hamed and Trnka, where they showed that ${\bf\Omega}(\mathcal{A}_{n,k,4})$ encodes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory.
• Moreover, the computation of ${\bf\Omega}(\mathcal{A}_{n,k,m})$ is reduced to finding the triangulations of $\mathcal{A}_{n,k,m}$.
• However, while triangulations of polytopes are fully captured by their secondary polytopes, the study of triangulations of objects beyond polytopes is still underdeveloped.