Amplitubes: Graph Cosmohedra

TL;DR

This work generalises the geometric encoding of scattering amplitudes from the associahedron to graph-based polytopes, introducing graph cosmohedra and their associated amplitubes. Amplitudes are defined as sums over vertices of graph associahedra via maximal tubings (amplitubes), while cosmological wavefunctions are encoded by regional tubings leading to cosmological amplitubes. The authors provide ABHY-like embeddings for graph associahedra and graph cosmohedra, establish factorisation relations at boundaries, and develop a canonical-form framework for cosmohedra, including explicit examples for path, cycle, and complete graphs. The construction connects to broader mathematical structures such as permutohedra and permutoassociahedra and opens avenues toward correlahedra and generalized permutohedra in cosmological and scattering contexts.

Abstract

The tree-level scattering amplitudes for $\text{tr}(φ^3)$ theory can be interpreted as a sum over the vertices of a polytope known as the associahedron. For each graph $G$, there exists a natural generalisation of the associahedron, which is constructed by considering tubes and tubings of the underling graph. This family of polytopes are called graph associahedra. The classical associahedra then arise as the graph associahedron for the path graphs. It is therefore natural to associate to each graph associahedron an amplitude-like object, we refer to as the amplitube, defined via a sum over its vertices. Recently, also in the context of trace $\text{tr}(φ^3)$ theory, progress has been made towards defining a new geometric object, coined the cosmohedron, which computes not the amplitude, but the cosmological wavefunction as a sum over its vertices. This polytope can be constructed by consistently blowing up all boundaries of the associahedron to co-dimension one. Building on these results, in the present paper, we generalise the notion of the wavefunction for arbitrary graphs. These new expressions, which we call cosmological amplitubes, are defined via a sum over the vertices of a corresponding polytope, the graph cosmohedron. The graph cosmohedra are constructed by considering regions and regional tubings of the underlying graph which we introduce. Like the cosmohedron, the graph cosmohedra can be obtained by consistently blowing up all boundaries of the corresponding graph associahedron to co-dimension one. This new family of polytopes constitutes a vast generalisation of the cosmohedron, and we provide explicit embeddings for them, which builds upon an ABHY-like embedding for the graph associahedra.

Figures (10)

• Figure 1: An illustration of the four possible configurations of tubings: intersecting, adjacent, nested and non-adjacent non-intersecting tubes. The last two pairs of tubes are compatible.
• Figure 2: Embeddings of three-dimensional graph associahedra for the path graph, cycle graph, and complete graph on four vertices. These polytopes are examples of the three-dimensional associahedron, cyclohedron and permutohedron, respectively.
• Figure 3: The fifteen regions for the graph $P_3$.
• Figure 4: An example of the sub-region relation $\prec_{\text{reg}}$ for a subset of the regions of $P_3$.
• Figure 5: The sub-region relation on the set of regions contained in the regional tubing $\rho_v$ defined in \ref{['eq:reg_tubing_ex']}. This corresponds to a non-simple vertex of the corresponding cosmohedron for the graph $P_4$.