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Faces of Cosmological Polytopes

Lukas Kühne, Leonid Monin

TL;DR

The paper develops a detailed combinatorial framework for the faces of cosmological polytopes $P_G$ associated to graphs, linking their structure to the physics of positive geometries and the canonical form. It proves a concrete vertex-set criterion for when a collection of cosmological-polytope vertices forms a face, and identifies two families of minimal non-simplex faces corresponding to vertices and cycles. It then establishes recursive tools for trees via pyramid/bipyramid constructions, deriving a $f$-polynomial recurrence for path graphs and a clean volume growth ${\rm Vol}(P_T)=4^{e}$ for trees. Finally, it provides explicit counts for low-dimensional and simplex faces, including cycle and banana graphs, highlighting the precise combinatorial interplay between graph structure and polytope geometry and informing the computation of canonical forms in the associated positive geometries.

Abstract

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e. an undirected graph. In this paper, we initiate a combinatorial study of these polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the $f$-vector of cosmological polytopes of trees.

Faces of Cosmological Polytopes

TL;DR

The paper develops a detailed combinatorial framework for the faces of cosmological polytopes associated to graphs, linking their structure to the physics of positive geometries and the canonical form. It proves a concrete vertex-set criterion for when a collection of cosmological-polytope vertices forms a face, and identifies two families of minimal non-simplex faces corresponding to vertices and cycles. It then establishes recursive tools for trees via pyramid/bipyramid constructions, deriving a -polynomial recurrence for path graphs and a clean volume growth for trees. Finally, it provides explicit counts for low-dimensional and simplex faces, including cycle and banana graphs, highlighting the precise combinatorial interplay between graph structure and polytope geometry and informing the computation of canonical forms in the associated positive geometries.

Abstract

A cosmological polytope is a lattice polytope introduced by Arkani-Hamed, Benincasa, and Postnikov in their study of the wavefunction of the universe in a class of cosmological models. More concretely, they construct a cosmological polytope for any Feynman diagram, i.e. an undirected graph. In this paper, we initiate a combinatorial study of these polytopes. We give a complete description of their faces, identify minimal faces that are not simplices and compute the number of faces in specific instances. In particular, we give a recursive description of the -vector of cosmological polytopes of trees.
Paper Structure (13 sections, 15 theorems, 20 equations, 1 figure)

This paper contains 13 sections, 15 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Physical perspective
  3. Combinatorial perspective
  4. Our contribution
  5. Acknowledgments
  6. Preliminaries
  7. Face structure of cosmological polytopes
  8. General faces
  9. Special faces
  10. Trees
  11. Counting faces
  12. Low-dimensional faces
  13. Simplex faces

Key Result

Theorem 2.1

[arkani2017cosmological] Facets of $P_G$ are in bijection with connected subgraphs $H=(V_H,E_H)$ of $G$. Under this bijection, a subgraph $H$ corresponds to the facet $F_H$ with all vertices of $P_G$ except So in particular, if an edge $e=\{v,w\}$ is not in $E_H$ but both $v$ and $w$ are in $V_H$ both vertices $p_{e,v}$ and $p_{e,w}$ are not part of the facet $F_H$. Moreover, the facet $F_H$ is t

Figures (1)

  • Figure 1: The graph on the left corresponds to the cosmological polytope on the right.

Theorems & Definitions (33)

  • Definition 1.1: arkani2017cosmological
  • Remark 1.2
  • Example 1.3
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • Remark 3.4
  • ...and 23 more