Representations of the Flat Space Wavefunction

TL;DR

The paper proves three representations of the flat space wavefunction $\psi_G$ for a graph $G$—bulk, boundary, and canonical form—bridging combinatorial tubings with geometric objects like the cosmological polytope. It confirms that the cosmological polytope canonical form aligns with the wavefunction via $\Omega_G = \psi_G dX dY$ and settles a conjecture about a partial fraction decomposition whose terms are indexed by connected subgraph collections. The results give explicit, interconnected formulas for $\psi_G$ in terms of spanning subgraphs, tubings, and polytope triangulations, enabling computation of cosmological correlators and revealing how $G$'s connectivity governs the wavefunction. The approach combines graph theory, polyhedral geometry, and algebraic tools (triangulations, adjoint polynomials) to unify multiple representations and provide new computational routes. The explicit $1$-loop bubble example illustrates the interplay between tubings, linear forms, and canonical data.

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.

Figures (5)

• Figure 1: Examples of a) a non-induced tube, b) nested tubes, c) overlapping tubes, d) incompatible (but not overlapping) tubes.
• Figure 2: Every complete tubing/admissible tubing of the "$1$-loop bubble", "$2$-site chain", and "$3$-site chain".
• Figure 3: Illustration of Lemma \ref{['lemma_order']} for the $3$-site graph, with vertices $\{1,2,3\}$ from left to right. Six total orders on the vertices induce five distinct admissible tubings, which in turn induce four distinct acyclic orientations.
• Figure 4: An example of a pair of overlapping tubes $\{S,T\}$ in the $3$-site chain giving rise to a relation among the linear forms.
• Figure 5: Illustration of the triangulation of the dual cosmological polytope for the $1$-loop bubble. Each vertex of the polytope corresponds to a tube, and each simplex in the triangulation corresponds to a complete tubing.

Theorems & Definitions (48)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Corollary 2.6
• Remark 2.7
• Lemma 2.8