Correlator Polytopes

Abstract

Recently, "cosmohedra" have been introduced as polytopes underlying the cosmological wavefunction for conformally coupled Tr($Φ^3$) 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.

Figures (4)

• Figure 1: (Left)$2$-dimensional projection of the $3$-dimensional fan of the $5$-point correlatron. On top of the associahedron rays, $(i,j)$, and the cosmohedron rays, $(i,j)_B$, we have the bottom and top rays, $B$ and $T$. (Right) Realization of the (truncated) $5$-point correlatron, with the appropriate facet labellings in terms of sub-polygon tilings or single chords. In red we highlight the facet for chord $(1,4)$, with facet inequality $X_{1,4} \geq 0$; and in blue the facet corresponding to the triangle-tiling given by triangulation $\{(1,3),(1,4)\}$, labelled by $(P_{123},P_{134},P_{145})$, and with facet inequality $X_{1,3} + X_{1,4} + X_B\geq \epsilon_{\{(1,3),(1,4)\}}$. We highlight three vertices ($1, 2$ and $3$) on the right, and provide the explicit computation of their contribution to the correlator in appendix \ref{['sec:AppExamples']}.
• Figure 2: (Left) Graph correlahedra at $6$-points for triangulations of the hexagon whose dual graph is a chain. (Right) Graph correlahedra at $6$-points for triangulation $\{(1,3),(3,5),(1,5)\}$ (and cyclic). In both cases, the facets are associated with internal edges (red) or tubings enclosing more than a single site (green/purple).
• Figure 3: $5$-point graph correlahedron for the graph containing propagators $\{(1,3),(1,4)\}$ with respective facet labels and the map to the usual $x$ and $y$ energy variables.
• Figure 4: Left$2$-point one-loop Correlatron, with the respective facet labellings. Right Example of loop graph correlahedron for the one-loop triangle diagram.