Amplituhedron Positive Geometry Conjecture

Prove that for every integer triple (n, k, m) with k + m ≤ n and for generic choices of the n × (k + m) matrix Z whose maximal minors are positive, the amplituhedron A_{n,k,m}(Z)—defined as the image of the totally nonnegative Grassmannian Gr(k,n)_{≥0} under the rational map induced by Z—is a positive geometry in the sense of Arkani-Hamed, Bai, and Lam.

Background

The amplituhedron A_{n,k,m}(Z), introduced to compute scattering amplitudes in N=4 super Yang–Mills theory, is constructed as the image of the nonnegative Grassmannian Gr(k,n)_{≥0} under a linear map induced by a matrix Z with all positive maximal minors. Positive geometries, defined in Arkani-Hamed, Bai, and Lam (2017), provide a framework where canonical forms have poles and residues encoding physical properties. Establishing that amplituhedra are positive geometries would unify geometric and physical structures across all relevant parameters.

While a number of special cases are known (e.g., k=1 where the amplituhedron is a cyclic polytope, k+m=n where it is isomorphic to Gr_{≥0}(k,n), and k=m=2), the general statement for arbitrary n, k, m remains unresolved. The authors verify genus-zero compatibility with the Brown–Dupont Hodge-theoretic framework in the known cases, but also show that for k ≥ 3 and large n the associated pairs have positive genus, indicating that Brown–Dupont’s criteria may not cover all amplituhedra even if they are positive geometries in the original sense.