Faces of Cosmological Polytopes

TL;DR

This paper provides a combinatorial analysis of cosmological polytopes, describing their faces, minimal non-simplex faces, and face counts, especially for tree-structured polytopes, advancing understanding of their geometric properties.

Contribution

It offers a complete face description, identifies minimal non-simplex faces, and derives recursive formulas for face counts of cosmological polytopes, particularly for trees.

Findings

Complete face descriptions of cosmological polytopes.

Identification of minimal faces that are not simplices.

Recursive formulas for face counts of tree-structured polytopes.

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.