Combinatorics of the Cosmohedron

Abstract

The cosmohedron was recently proposed as a polytope underlying the cosmological wavefunction for $\text{Tr}(Φ^3)$ 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.

Figures (24)

• Figure 1: An octagonal Matryoshka $M$ and a maximal Matryoshka $M'$ containing it.
• Figure 2: The $3$-cosmohedron as a blowup of the $3$-associahedron, and the Matryoshkas labeling some of its faces.
• Figure 3: The contribution of a tree graph (or equivalently a triangulation) to the wavefunction is determined by the bracketings of the graph (or equivalently the Matryoshkas of the triangulation). Here $k_{i,j}$ denotes the length between vertices $i$ and $j$, which corresponds to the length of the momenta flowing through the edges of the graph, and $\mathcal{P}$ stands for the perimeter of the subpolygons entering the Matryoshka.
• Figure 4: Two maximal Matryoshkas $M_1$ and $M_2$ on a hexagon.
• Figure 5: The Matryoshka $M_1$ and its containment poset $\tau_1$.

Theorems & Definitions (71)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Example 2.2
• Definition 2.3
• Theorem 2.4
• Definition 2.5
• Theorem 2.6
• Definition 2.7