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Taking the amplituhedron to the limit

Joris Koefler, Rainer Sinn

TL;DR

The paper defines the limit amplituhedron $\mathcal{A}^\infty_k$ by sending the particle number $n$ to infinity in the $m=2$ amplituhedron setup and proves it forms a positive geometry inside the Grassmannian $\mathrm{Gr}(k,k+2)$. It shows its algebraic boundary decomposes into two irreducible Chow hypersurfaces: the Chow of the rational normal curve $C_{k+1}$ and the Chow of the fixed secant line $\mathcal{S}_{01}$, with a detailed stratification of the iterated singular loci using incidence geometry and osculants. The residual arrangement is empty, established via density arguments showing $\mathcal{A}^\infty_k$ intersects each stratum densely; consequently, a canonical form exists (up to scaling) and is expressible through the Chow forms in the denominator, completing the positive geometry structure. Collectively, these results extend the pizza-slice intuition to general $k$ and provide a concrete geometric and algebro-geometric framework for the limit amplituhedron.

Abstract

The amplituhedron is a semialgebraic set given as the image of the non-negative Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters, namely the number of particles $n$, to infinity. We study this limit amplituhedron for $m = 2$ and any $k$, relating to the number of negative helcity particles. We determine its algebraic boundary in terms of Chow hypersurfaces. This hypersurface in the Grassmannian is stratified by singularities in terms of higher order secants of the rational normal curve. In conclusion, we show that the limit amplituhedron is a positive geometry with a residual arrangement that is empty.

Taking the amplituhedron to the limit

TL;DR

The paper defines the limit amplituhedron by sending the particle number to infinity in the amplituhedron setup and proves it forms a positive geometry inside the Grassmannian . It shows its algebraic boundary decomposes into two irreducible Chow hypersurfaces: the Chow of the rational normal curve and the Chow of the fixed secant line , with a detailed stratification of the iterated singular loci using incidence geometry and osculants. The residual arrangement is empty, established via density arguments showing intersects each stratum densely; consequently, a canonical form exists (up to scaling) and is expressible through the Chow forms in the denominator, completing the positive geometry structure. Collectively, these results extend the pizza-slice intuition to general and provide a concrete geometric and algebro-geometric framework for the limit amplituhedron.

Abstract

The amplituhedron is a semialgebraic set given as the image of the non-negative Grassmannian under a linear map subject to a choice of additional parameters. We define the limit amplituhedron as the limit of amplituhedra by sending one of the parameters, namely the number of particles , to infinity. We study this limit amplituhedron for and any , relating to the number of negative helcity particles. We determine its algebraic boundary in terms of Chow hypersurfaces. This hypersurface in the Grassmannian is stratified by singularities in terms of higher order secants of the rational normal curve. In conclusion, we show that the limit amplituhedron is a positive geometry with a residual arrangement that is empty.
Paper Structure (6 sections, 25 theorems, 58 equations, 1 figure, 2 tables)

This paper contains 6 sections, 25 theorems, 58 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Algebraic boundary
  3. Stratification
  4. Tangent spaces and incidences
  5. Strata
  6. Residual Arrangement

Key Result

Theorem 1.1

The limit amplituhedron $\mathcal{A}_k^\infty$ is a positive geometry $(\mathop{\mathrm{Gr}}\nolimits(k,k{+}2),\mathcal{A}^\infty_k)$ in the Grassmannian $\mathop{\mathrm{Gr}}\nolimits(k,k+2)$. Its algebraic boundary has two irreducible components, the Chow hypersurface of the rational normal curve

Figures (1)

  • Figure 1: The limit amplituhedron $\mathcal{A}^\infty_1$ with its algebraic boundary.

Theorems & Definitions (58)

  • Theorem 1.1
  • Example 1.2: Pizza slice
  • Remark 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • ...and 48 more