Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On cosmological polytopes, their canonical forms and their duals

Anna Birkemeyer, Torben Donzelmann, Mieke Fink, Martina Juhnke

Abstract

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.

On cosmological polytopes, their canonical forms and their duals

Abstract

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.
Paper Structure (8 sections, 15 theorems, 49 equations, 2 figures)

This paper contains 8 sections, 15 theorems, 49 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tubings
  4. Cosmological polytopes
  5. Canonical forms and volumes of dual polytopes
  6. Construction of the dual cosmological polytope
  7. Computing the canonical form
  8. Computing the canonical form via the boundary complex

Key Result

Theorem 1.1

gao2024 Let $P$ be a $d$-dimensional polytope in $\mathbb{R}^d \subseteq \mathbb{P}^d$. Then the canonical form of $P$ is given by where $(P-x)^\circ$ denotes the dual of the shifted polytope $P-x$.

Figures (2)

  • Figure 1: The star graph $K_{1,3}$
  • Figure :

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Remark 2.2: Different tubing definitions
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5: Cosmological polytope arkani17
  • Theorem 2.6
  • ...and 25 more