Positive Genus Pairs from Amplituhedra

TL;DR

This paper investigates the geometric properties of amplituhedra, showing they generally form positive genus pairs, and explores the implications for their role as positive geometries in scattering amplitudes.

Contribution

It proves that amplituhedra typically form positive genus pairs and provides explicit examples, expanding understanding of their geometric structure beyond genus zero cases.

Findings

Amplituhedra form positive genus pairs in general.

Explicit genus one example from positive geometry in projective space.

Positive geometry can have higher genus pairs but still relate to genus zero pairs in other varieties.

Abstract

A main conjecture in the field of Positive Geometry states that amplituhedra, which are certain semi-algebraic sets in the Grassmannian, are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. Beyond a small number of special cases, this conjecture is still open. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology via genus zero pairs. We give short proofs that the amplituhedron gives rise to a genus zero pair in the cases when it is known to be a positive geometry. However, in the general case we show that amplituhedra inside the Grassmannian give rise to pairs of strictly positive genus. We provide an explicit example of a genus one pair arising from a positive geometry in projective space, showing that having genus zero is not a necessary condition to be a positive geometry. Finally, we show that this positive geometry still gives rise to a genus zero pair in a different ambient variety.