Triangulations of cosmological polytopes

TL;DR

This paper explores the triangulations of cosmological polytopes associated with Feynman diagrams, providing algebraic formulas for their canonical forms and extending volume computations for specific graph classes.

Contribution

It introduces algebraic techniques to identify formulas for canonical forms and proves the existence of regular unimodular triangulations for cosmological polytopes.

Findings

Canonical forms are obtained via algebraic formulas.

Regular unimodular triangulations exist for all cosmological polytopes.

Explicit volume formulas are derived for paths and cycles.

Abstract

A cosmological polytope is defined for a given Feynman diagram, and its canonical form may be used to compute the contribution of the Feynman diagram to the wavefunction of certain cosmological models. Given a subdivision of a polytope, its canonical form is obtained as a sum of the canonical forms of the facets of the subdivision. In this paper, we identify such formulas for the canonical form via algebraic techniques. It is shown that the toric ideal of every cosmological polytope admits a Gr\"obner basis with a squarefree initial ideal, yielding a regular unimodular triangulation of the polytope. In specific instances, including trees and cycles, we recover graphical characterizations of the facets of such triangulations that may be used to compute the desired canonical form. For paths and cycles, these characterizations admit simple enumeration. Hence, we obtain formulas for the normalized volume of these polytopes, extending previous observations of K\"uhne and Monin.