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Ehrhart theory of cosmological polytopes

Justus Bruckamp, Lina Goltermann, Martina Juhnke, Erik Landin, Liam Solus

TL;DR

The paper develops the Ehrhart-theoretic study of cosmological polytopes $\mathcal{C}_G$ associated to graphs, tying canonical forms for cosmological wavefunctions to combinatorial geometry. It combines toric-geometry with unimodular triangulations arising from square-free Gröbner bases of the toric ideal $I_{\mathcal{C}_G}$, and uses half-open decompositions to extract the $h^*$-polynomial. Key contributions include multiplicativity of $h^*$ under disjoint unions and $1$-sums, a tight lower bound on $h^*$ coefficients, and exact $h^*$-polynomials for multitrees and multicycles; a degree and a Gorenstein (palindromic) criterion are also established, with a full characterization of when $\mathcal{C}_G$ is Gorenstein. The paper also proposes a universal combinatorial formula for $h^*(\mathcal{C}_G; z)$ in terms of triangulation data and conjectures an upper bound $h^*(\mathcal{C}_G; z) \preccurlyeq (1+3z)^{|E|}$, pointing to broad future applications and generalizations to other graph families. Overall, these results generalize known normalized-volume formulas and provide a framework for estimating computation complexity in wavefunction calculations via geometric invariants.

Abstract

The cosmological polytope of a graph $G$ was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram $G$. Basic results in the theory of positive geometries dictate that this wavefunction may be computed as a sum of rational functions associated to the facets in a triangulation of the cosmological polytope. The normalized volume of the polytope then provides a complexity estimate for these computations. In this paper, we examine the (Ehrhart) $h^\ast$-polynomial of cosmological polytopes. We derive recursive formulas for computing the $h^\ast$-polynomial of disjoint unions and $1$-sums of graphs. The degree of the $h^\ast$-polynomial for any $G$ is computed and a characterization of palindromicity is given. Using these observations, a tight lower bound on the $h^\ast$-polynomial for any $G$ is identified and explicit formulas for the $h^\ast$-polynomials of multitrees and multicycles are derived. The results generalize the existing results on normalized volumes of cosmological polytopes. A tight upper bound and a combinatorial formula for the $h^\ast$-polynomial of any cosmological polytope are conjectured.

Ehrhart theory of cosmological polytopes

TL;DR

The paper develops the Ehrhart-theoretic study of cosmological polytopes associated to graphs, tying canonical forms for cosmological wavefunctions to combinatorial geometry. It combines toric-geometry with unimodular triangulations arising from square-free Gröbner bases of the toric ideal , and uses half-open decompositions to extract the -polynomial. Key contributions include multiplicativity of under disjoint unions and -sums, a tight lower bound on coefficients, and exact -polynomials for multitrees and multicycles; a degree and a Gorenstein (palindromic) criterion are also established, with a full characterization of when is Gorenstein. The paper also proposes a universal combinatorial formula for in terms of triangulation data and conjectures an upper bound , pointing to broad future applications and generalizations to other graph families. Overall, these results generalize known normalized-volume formulas and provide a framework for estimating computation complexity in wavefunction calculations via geometric invariants.

Abstract

The cosmological polytope of a graph was recently introduced to give a geometric approach to the computation of wavefunctions for cosmological models with associated Feynman diagram . Basic results in the theory of positive geometries dictate that this wavefunction may be computed as a sum of rational functions associated to the facets in a triangulation of the cosmological polytope. The normalized volume of the polytope then provides a complexity estimate for these computations. In this paper, we examine the (Ehrhart) -polynomial of cosmological polytopes. We derive recursive formulas for computing the -polynomial of disjoint unions and -sums of graphs. The degree of the -polynomial for any is computed and a characterization of palindromicity is given. Using these observations, a tight lower bound on the -polynomial for any is identified and explicit formulas for the -polynomials of multitrees and multicycles are derived. The results generalize the existing results on normalized volumes of cosmological polytopes. A tight upper bound and a combinatorial formula for the -polynomial of any cosmological polytope are conjectured.

Paper Structure

This paper contains 13 sections, 23 theorems, 66 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The toric ideal of a cosmological polytope and triangulations
  3. Ehrhart theory and half-open decompositions
  4. Toric ideals and Gröbner bases
  5. Unimodular triangulations of cosmological polytopes
  6. General results on h*-polynomials of cosmological polytopes
  7. Graph operations for computing h*-polynomials
  8. Bounds for the h*-polynomial
  9. Degree of the h*-polynomial
  10. h*-polynomials of multicycles and multitrees
  11. Multicycles
  12. Multitrees
  13. Other graphs

Key Result

Theorem 2.1

Let $P \subseteq \mathbb{R}^n$ be a lattice polytope that admits a unimodular triangulation $\mathcal{T}$. Let $\Delta_1,\ldots,\Delta_m$ denote the maximal simplices of $\mathcal{T}$, and let $q \in P$ be in general position relative to $P$, and to all $\Delta_i$, $i \in [m]$. Then

Figures (4)

  • Figure 1: A multicycle
  • Figure 2: Fundamental obstructions
  • Figure 3: Cyclic obstructions
  • Figure 4: Example for a multitree of type $(1,2,3,3,3,5)$.

Theorems & Definitions (46)

  • Theorem 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • ...and 36 more