Generalised wavefunction coefficients and acyclonesto-cosmohedra

TL;DR

This paper extends the geometric framework of scattering amplitudes and wavefunction coefficients from associahedra and cosmohedra to a broader class called acyclonestohedra, revealing new factorization properties and generalizations.

Contribution

It introduces acyclonestohedra as a generalization of associahedra and cosmohedra, and explores their canonical forms related to scattering amplitudes and wavefunction coefficients.

Findings

Acyclonestohedra encode factorization properties similar to associahedra.

Constructs truncations called acyclonesto-cosmohedra with generalized wavefunction coefficients.

Provides evidence that acyclonesto-cosmohedra are sections of graph cosmohedra.

Abstract

Scattering amplitudes of $\operatorname{tr}(\phi^3)$ 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.