Representations of the Flat Space Wavefunction

TL;DR

This paper develops and proves three representations of the flat space wavefunction derived from a graph, crucial for cosmological correlator calculations, and confirms a conjecture on its partial fraction decomposition.

Contribution

The paper formulates three new representations of the flat space wavefunction and proves their correctness, linking it to the cosmological polytope and resolving a conjecture on partial fractions.

Findings

Wavefunction can be derived from the cosmological polytope's canonical form.

Confirmed a conjecture on partial fraction decomposition of the wavefunction.

Decomposition terms relate to connected subgraphs of the original graph.

Abstract

From any graph $G$ arises a flat space wavefunction, obtained by integrating a product of propagators associated to the vertices and edges of $G$. This function is a key ingredient in the computation of cosmological correlators, and several representations for it have been proposed. We formulate three such representations and prove their correctness. In particular, we show that the flat space wavefunction can be read off from the canonical form of the cosmological polytope, and we settle a conjecture of Fevola, Pimentel, Sattelberger, and Westerdijk regarding a partial fraction decomposition for the flat space wavefunction. The terms of the decomposition correspond to certain collections of connected subgraphs associated to $G$ and its spanning subgraphs, reflecting the fact that the flat space wavefunction contains information about how $G$ is connected.