Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Positive Genus Pairs from Amplituhedra

Joris Koefler, Dmitrii Pavlov, Rainer Sinn

TL;DR

The work probes the compatibility between Brown–Dupont’s mixed Hodge framework for positive geometries and amplituhedra, focusing on the genus of the pair $(\mathrm{Gr}(k,k+m), \partial_a \mathcal{A}_{n,k,m})$. It shows that all known positive-geometric amplituhedra yield genus-zero pairs in the BD sense for certain parameter choices ($k=1$, $k+m=n$, or $k=m=2$), while for $m=2$, $k\ge 3$ and sufficiently large $n$ the genus is strictly positive, indicating limitations of BD's genus-zero criterion. The authors also construct a genus-one positive geometry (an explicit semi-algebraic set in $\mathbb{P}^3$) to illustrate that positivity in pos_geom_2017 can coexist with nonzero genus in the BD framework, motivating potential adaptations. Overall, the results motivate refining or extending the BD framework to better align with pos_geom_2017 and to understand when genus-zero is essential for positivity in amplitudes.

Abstract

A main conjecture in the field of Positive Geometry states that amplituhedra are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology. In this paper, we show that this framework is consistent with the known results for amplituhedra but does not apply beyond those families. We provide an explicit example showing that the central assumption of the Brown-Dupont framework (namely to have a pair of genus zero) is not a necessary condition to be a positive geometry in the original sense of Arkani-Hamed, Bai, and Lam. This underscores the fact that our results do not immediately disqualify the amplituhedron from being a positive geometry.

Positive Genus Pairs from Amplituhedra

TL;DR

The work probes the compatibility between Brown–Dupont’s mixed Hodge framework for positive geometries and amplituhedra, focusing on the genus of the pair . It shows that all known positive-geometric amplituhedra yield genus-zero pairs in the BD sense for certain parameter choices (, , or ), while for , and sufficiently large the genus is strictly positive, indicating limitations of BD's genus-zero criterion. The authors also construct a genus-one positive geometry (an explicit semi-algebraic set in ) to illustrate that positivity in pos_geom_2017 can coexist with nonzero genus in the BD framework, motivating potential adaptations. Overall, the results motivate refining or extending the BD framework to better align with pos_geom_2017 and to understand when genus-zero is essential for positivity in amplitudes.

Abstract

A main conjecture in the field of Positive Geometry states that amplituhedra are positive geometries. It is motivated by examples showing that the canonical forms of certain amplituhedra compute scattering amplitudes in particle physics. In recent work, Brown and Dupont introduced a new framework, based on mixed Hodge theory, connecting canonical forms and de Rham cohomology. In this paper, we show that this framework is consistent with the known results for amplituhedra but does not apply beyond those families. We provide an explicit example showing that the central assumption of the Brown-Dupont framework (namely to have a pair of genus zero) is not a necessary condition to be a positive geometry in the original sense of Arkani-Hamed, Bai, and Lam. This underscores the fact that our results do not immediately disqualify the amplituhedron from being a positive geometry.
Paper Structure (8 sections, 18 theorems, 54 equations, 1 figure)

This paper contains 8 sections, 18 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A pinch of Positive Geometry
  4. A hint of Hodge Theory
  5. Amplituhedra that make genus zero pairs
  6. The $m=2$ amplituhedra $\mathcal{A}_{n,k,2}$ make positive genus pairs
  7. The $m=4$ amplituhedra make positive genus pairs
  8. A genus one positive geometry

Key Result

Proposition 2.1

For $m=2$ the algebraic boundary $\partial_a \mathcal{A}_{n,k,2}$ of $\mathcal{A}_{n,k,2}$, with $n \geq k+2$, is the union of the following hyperplane sections of $\mathop{\mathrm{Gr}}\nolimits(k,k+2)$: where $Y\in \mathop{\mathrm{Gr}}\nolimits(k,k+2)$ and $i\in [n]$, with indices read cyclically.

Figures (1)

  • Figure 1: Example of a positive geometry with genus one.

Theorems & Definitions (43)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Remark 2.7
  • Definition 2.8
  • Remark 2.9
  • Remark 2.10
  • ...and 33 more