On the geometry of the orthogonal momentum amplituhedron

TL;DR

This paper explores the geometric structure of the orthogonal momentum amplituhedron, revealing its boundary structure, Euler characteristic, and suggesting it is topologically a ball, with computational tools provided.

Contribution

It provides the full boundary stratification of the orthogonal momentum amplituhedron and links boundaries to orthogonal Grassmannian forests, advancing understanding of its geometry.

Findings

Boundaries labeled by orthogonal Grassmannian forests

Euler characteristic equals one, indicating a ball topology

Generated boundary enumeration function

Abstract

In this paper we study the orthogonal momentum amplituhedron $\mathcal{O}_k$, a recently introduced positive geometry that encodes the tree-level scattering amplitudes in ABJM theory. We generate the full boundary stratification of $\mathcal{O}_k$ and show that its boundaries can be labelled by so-called orthogonal Grassmannian forests (OG forests). We also determine the generating function for enumerating boundaries according to their dimension and show that the Euler characteristic of $\mathcal{O}_k$ equals one. This provides a strong indication that the orthogonal momentum amplituhedron is homeomorphic to a ball. This paper is supplemented with the Mathematica package "orthitroids" which contains useful functions for studying the positive orthogonal Grassmannian and the orthogonal momentum amplituhedron.