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

Abstract

Any totally positive $(k+m)\times n$ matrix induces a map ${\pi }_{+}$ from the positive Grassmannian ${\mathrm{Gr}}_{+}(k,n)$ to the Grassmannian $\mathrm{Gr}(k,k+m)$, whose image is the amplituhedron ${\mathcal{A}}_{n,k,m}$ and is endowed with a top-degree form called the canonical form $\mathbf{\Omega }({\mathcal{A}}_{n,k,m})$. This construction was introduced by Arkani-Hamed and Trnka, where they showed that $\mathbf{\Omega }({\mathcal{A}}_{n,k,4})$ encodes scattering amplitudes in $\mathcal{N}=4$ super Yang-Mills theory. Moreover, the computation of $\mathbf{\Omega }({\mathcal{A}}_{n,k,m})$ 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 $\pi :\mathrm{Gr}(k,n)⇢\mathrm{Gr}(k,k+m)$. We then use this to explicitly describe a rational top-degree form ${\omega }_{n,k,m}$ (with simple poles) on the fibers and compute $\mathbf{\Omega }({\mathcal{A}}_{n,k,m})$ 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 $\mathbf{\Omega }({\mathcal{A}}_{n,k,m})$ 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.

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Cited by in corpus (4)

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• Combinatorics of Essential Sets for Positroids