Generalised wavefunction coefficients and acyclonesto-cosmohedra

TL;DR

The paper broadens the positive geometry framework for physics by introducing acyclonestohedra, a wide class of polytopes that generalise the Stasheff associahedron and graph associahedra via oriented building sets and oriented matroids. It extends the cosmohedron construction to acyclonestohedra, defining acyclonesto-cosmohedra and amplitubes that encode amplitude- and wavefunction-like objects with locality and unitarity built in. The authors develop ABHY-like realizations, provide explicit examples (including Stasheff, simplex, permutohedron, and diamond/bowtie posets), and demonstrate that acyclonesto-cosmohedra can be obtained as sections of graph cosmohedra, pointing to a unifying geometric framework for cosmological correlators and scattering amplitudes in exotic kinematics. The work lays groundwork for applying these structures to cosmology and quantum field theory, and outlines future directions such as correlator polytopes and broader nested-set generalisations.

Abstract

Scattering amplitudes of $\operatorname{tr}(φ^3)$ theory can be encoded as the canonical form of the Stasheff associahedron. Similarly, the flat-space wavefunction coefficients of the same theory are captured by the recently proposed cosmohedron, a non-simple polytope associated to the Stasheff associahedron; unitarity and locality of the amplitudes and wavefunction coefficients are then encoded in the factorisation properties of faces of these polytopes. In this paper, we argue that these desirable properties of the Stasheff associahedron are shared by a wider class of polytopes called acyclonestohedra and generalise the cosmohedron construction to arbitrary acyclonestohedra. Acyclonestohedra are generalisations of Stasheff associahedra and graph associahedra defined on the data of a partially ordered set or, more generally, an acyclic realisable matroid on a building set. When the acyclonestohedron is associated to a partially ordered set, it may be interpreted as arising from Chan-Paton-like factors that are only (cyclically) partially ordered, rather than (cyclically) totally ordered as for the ordinary open string. In this paper, we argue that the canonical forms of acyclonestohedra encode scattering-amplitude-like objects that factorise onto themselves, thereby extending recent results for graph associahedra, and construct truncations of acyclonestohedra into acyclonesto-cosmohedra whose canonical forms may be interpreted as encoding a generalisation of the cosmological wavefunction coefficients. As a byproduct, we provide evidence that acyclonesto-cosmohedra can be obtained as sections of graph cosmohedra.

Figures (9)

• Figure 1: The acyclonestohedron (left) and associated acyclonesto-cosmohedron (right) for nestings of the poset $K_{2,3}$. The acyclonestohedron is shown as a realisation in sack, and both polytopes as realisations here in Figure \ref{['fig:RealK23']}. One vertex of each polytope is circled, with the corresponding maximal nesting $\tau$ of $K_{2,3}$ shown below and maximal nested nesting $(\tau, \mathcal{N})$ shown above.
• Figure 2: Venn diagram of different classes of polytopes whose faces factorise. Acyclonestohedra include all other classes shown. Operahedra, as defined in laplante-anfossi, are poset associahedra when the Hasse diagram of the poset is a tree. In that case tubings on the line graph are all compatible with the poset and they are thus included in the intersection of graph associahedra and poset associahedra.
• Figure 3: The maximal nested nestings here are on the building set from a path graph or poset as described in Example \ref{['ex:stasheff']}. They are paired with their corresponding pictures of subpolygon collections from Arkani-Hamed:2024jbp.
• Figure 4: Explicit realisations for the acyclonestohedron and its associated acyclonesto-cosmohedron for the $K_{2,3}$ poset discussed in Example \ref{['ex:K23']}.
• Figure 5: The poset associahedron (left) and poset cosmohedron (right) for the poset depicted in Figure $1$ of galashin. Both polytopes are obtained as sections of the graph associahedron/cosmohedron for the line graph of the Hasse diagram.

Theorems & Definitions (22)

• Definition 1: postnikovfkfs
• Example 1: cd
• Definition 2
• Example 2: Oriented matroid from matrix
• Example 3: Oriented matroid from hyperplane arrangement
• Example 4: Oriented matroid from directed graph
• Definition 3: mpp
• Example 5
• Example 6: galashin
• Example 7: Stasheff associahedron