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

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. We initiate the geometric study of subdivisions of $\mathcal{A}_{n,k,m}$ and provide a concrete birational parametrization of fibers of $\pi: {\rm Gr}(k,n)\dashrightarrow {\rm Gr}(k,k+m)$. We then use this to explicitly describe a rational top-degree form $\omega_{n,k,m}$ (with simple poles) on the fibers and compute ${\bf\Omega}(\mathcal{A}_{n,k,m})$ as a summation of certain residues of $\omega_{n,k,m}$. As main application of our approach, we develop a well-structured notion of secondary amplituhedra for conjugate to polytopes, i.e. when $n-k-1=m$ (even). We show that, in this case, each fiber of $\pi$ is parametrized by a projective space and its volume form $\omega_{n,k,m}$ has only poles on a hyperplane arrangement. Using such linear structures, for amplituhedra which are cyclic polytopes or conjugate to polytopes, we show that the Jeffrey-Kirwan residue computes ${\bf\Omega}(\mathcal{A}_{n,k,m})$ from $\omega_{n,k,m}$. Finally, we propose a more general framework of fiber positive geometries and analyze new families of examples such as fiber polytopes and Grassmann polytopes.