Correlator Polytopes

TL;DR

This paper introduces the 'Correlatron', a new polytope that geometrically encodes correlation functions in cosmology, extending the framework of cosmohedra and associahedra to better understand wavefunctions and scattering amplitudes.

Contribution

The paper defines the correlatron polytope and its embedding, connecting correlation functions with geometric structures, and introduces graph correlahedra for fixed graph contributions.

Findings

Correlatron geometry captures cosmological correlation functions.

Canonical forms of polytopes compute correlators directly.

Graph correlahedra represent fixed graph contributions to correlators.

Abstract

Recently, "cosmohedra" have been introduced as polytopes underlying the cosmological wavefunction for conformally coupled Tr($\Phi^3$) theory in FRW cosmologies, generalizing associahedra for flat space scattering amplitudes. In this letter we show that correlation functions are also directly captured by a new polytope - the "Correlatron". The combinatorics of correlation functions is an interesting blend of flat space scattering amplitudes and wavefunctions. This is reflected in the correlatron geometry, which is a one-higher dimensional polytope sandwiched between cosmohedron and associahedron facets. We provide an explicit embedding for the correlatron, which is a natural extension of the "shaving" picture for cosmohedra to one higher dimension. As a byproduct, we also define "graph correlahedra" as polytopes for the contribution to correlators from any fixed graph. We show how the canonical form of these polytopes directly computes the graph correlator, without the power of two weights seen in previous geometric formulations. Finally, we give a prescription for extracting the full correlator from the canonical form of the correlatron.