On cosmological polytopes, their canonical forms and their duals

TL;DR

This paper develops methods to compute the canonical form of cosmological polytopes for any graph, introduces explicit descriptions of their duals, and presents new triangulations, enhancing understanding of their geometric structure.

Contribution

It provides explicit coordinate descriptions of the dual cosmological polytope and introduces a novel triangulation, advancing the geometric analysis of cosmological polytopes.

Findings

Explicit coordinate description of the dual cosmological polytope

Two triangulations of the dual cosmological polytope

New expression for the canonical form of the cosmological polytope

Abstract

We compute the canonical form of the cosmological polytope for any graph in terms of the dual of the shifted cosmological polytope in two different ways. On the way, we provide an explicit coordinate description of the dual of the cosmological polytope. Moreover, we construct two triangulations of the dual cosmological polytope in terms of maximal and almost maximal tubings of the underlying graph. Though the existence of the first triangulation was already suggested by Arkani-Hamed, Benincasa and Postnikov, the second is completely new and, in particular, gives rise to a new expression of the canonical form of the cosmological polytope.