Combinatorics of the Cosmohedron

TL;DR

This paper proves the combinatorial structure of the cosmohedron, a polytope related to the cosmological wavefunction, and explores its generalizations and applications to quantum field theory divergences.

Contribution

It confirms the bijection between cosmohedron faces and Matryoshka subdivisions, and introduces a broader class of chiseled polytopes with potential physics applications.

Findings

Confirmed the face-Matryoshka correspondence in cosmohedra

Generalized cosmohedra to wider classes of polytopes

Suggested applications to ultraviolet divergence analysis in quantum field theory

Abstract

The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(\Phi^3)$ theory. Its faces were conjectured to be in bijection with Matryoshkas, which are obtained from a subdivision of a polygon by sequentially wrapping groups of polygons into larger polygons. In this paper we prove the correctness of this construction, and elucidate its combinatorial structure. Cosmohedra generalize to a wider class of $\mathcal{X}$ in $Y$ polytopes, where we chisel a polytope from the family $\mathcal{X}$ at each vertex of a polytope $Y$. We sketch a new application of these chiseled polytopes to the physics of ultraviolet divergences in loop-integrated Feynman amplitudes.