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

TL;DR

This paper develops a fiber-based geometric approach to study amplituhedra, extending triangulation concepts beyond polytopes, and provides explicit formulas for their canonical forms, with applications to certain classes like cyclic and conjugate polytopes.

Contribution

It introduces a birational parametrization of fibers of the amplituhedron map and defines a rational top-degree form, enabling new methods to compute canonical forms beyond traditional triangulations.

Findings

Explicit fiber parametrization for amplituhedra

Residue-based computation of canonical forms

Development of secondary amplituhedra for conjugate polytopes

Abstract

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.