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Good Triangulations of Cosmological Polytopes

Aenne Benjes, Kamillo Ferry, Benjamin Schröter

TL;DR

The paper addresses how cosmological polytopes $\, ext{C}_G$ associated to graphs encode wavefunction data via their lattice-point structure. It develops a geometric-combinatorial framework of good unimodular triangulations, together with a half-open decomposition that directly relates the Ehrhart $h^ extast$-polynomial to graph-theoretic invariants. A key result expresses $h^ extast(\, ext{C}_G; z)$ as a Tutte polynomial specialization, yielding a deletion-contraction recurrence, ultra log-concavity of coefficients, and a compact volume formula $ ext{vol}( ext{C}_G)=2^m T_G(2,1)$. The approach unifies and extends prior results for multitrees and multicycles, provides exact enumerations of maximal simplices via decorated graphs, and connects the cosmological polytopes' Ehrhart theory to classical graph polynomials, with substantial implications for understanding the combinatorial structure of cosmological wavefunctions.

Abstract

Cosmological polytopes of graphs are a geometric tool in physics to study wavefunctions for cosmological models whose Feynman diagram is given by the graph. After their recent introduction by Arkani-Hamed, Benincasa and Postnikov the focus of interest shifted towards their mathematical properties, e.g., their face structure and triangulations. Juhnke, Solus and Venturello used toric geometry to show that these polytopes have a so-called good triangulation that is unimodular. Based on these results Bruckamp et al. studied the Ehrhart theory of those polytopes and in particular the h*-polynomials of cosmological polytopes of multitrees and multicycles. In this article we complete this part of the story. We enumerate all maximal simplices in good triangulations of any cosmological polytope. Furthermore, we provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph. This settles several open questions and conjectures of Juhnke, Solus and Venturello as well as Bruckamp et al.

Good Triangulations of Cosmological Polytopes

TL;DR

The paper addresses how cosmological polytopes associated to graphs encode wavefunction data via their lattice-point structure. It develops a geometric-combinatorial framework of good unimodular triangulations, together with a half-open decomposition that directly relates the Ehrhart -polynomial to graph-theoretic invariants. A key result expresses as a Tutte polynomial specialization, yielding a deletion-contraction recurrence, ultra log-concavity of coefficients, and a compact volume formula . The approach unifies and extends prior results for multitrees and multicycles, provides exact enumerations of maximal simplices via decorated graphs, and connects the cosmological polytopes' Ehrhart theory to classical graph polynomials, with substantial implications for understanding the combinatorial structure of cosmological wavefunctions.

Abstract

Cosmological polytopes of graphs are a geometric tool in physics to study wavefunctions for cosmological models whose Feynman diagram is given by the graph. After their recent introduction by Arkani-Hamed, Benincasa and Postnikov the focus of interest shifted towards their mathematical properties, e.g., their face structure and triangulations. Juhnke, Solus and Venturello used toric geometry to show that these polytopes have a so-called good triangulation that is unimodular. Based on these results Bruckamp et al. studied the Ehrhart theory of those polytopes and in particular the h*-polynomials of cosmological polytopes of multitrees and multicycles. In this article we complete this part of the story. We enumerate all maximal simplices in good triangulations of any cosmological polytope. Furthermore, we provide a method to turn such a triangulation into a half-open decomposition from which we deduce that the h*-polynomial of a cosmological polytope is a specialization of the Tutte polynomial of the defining graph. This settles several open questions and conjectures of Juhnke, Solus and Venturello as well as Bruckamp et al.

Paper Structure

This paper contains 4 sections, 18 theorems, 33 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Good triangulations
  3. A decomposition into half-open simplices
  4. $h^\ast$-polynomials of cosmological polytopes

Key Result

Lemma 2.4

Let $G$ be the subgraph of the graph $\hat{G}$ on the same set of nodes where the edge $f$ with nodes $u$, $v$ is removed. Furthermore let $\mathcal{T}_{\hat{G}}$ be a good triangulation of the polytope $\mathcal{C}_{\hat{G}}$ which is induced by the height function $\psi$, and $\mathcal{T}_G$ be th

Figures (3)

  • Figure 1: A good triangulation $\mathcal{T}$ of the cosmological polytope of Example \ref{['ex:ex1']}.
  • Figure 2: The good triangulation $\mathcal{T}$ and the dual graph of Example \ref{['ex:ex2']}.
  • Figure 3: The poset of unions of cycles of the graph $\Theta_{a,b,c}$ with $a=1$, $b=2$ and $c=3$ and its five elements $\emptyset$, $C_{ab}$, $C_{ac}$, $C_{bc}$ and $E=C_{ab}\cup C_{ac}\cup C_{bc}$.

Theorems & Definitions (49)

  • Definition 2.1
  • Remark 2.2
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • Remark 2.7
  • ...and 39 more