Geometry of in-in correlators

TL;DR

The paper defines in-in zonotopes, a family of polytopes whose facets organize in-in diagram contributions for flat-space scalar correlators. By expressing $\mathcal{I}(G)$ as a Minkowski sum $ \mathcal{I}(G) = \mathcal{H}(G)\oplus\mathcal{Z}(G)$ and exploiting facet factorization, the authors show that the canonical form evaluated at the origin reproduces the correlator, equivalently via the volume of the dual polytope $\mathcal{I}^\circ(G)$. Through explicit examples (e.g. $K_1$, $P_2$, and the complete graph) they demonstrate the geometric encoding of wavefunction decompositions as subdivisions of dual polytopes. The results connect diagrammatic in-in combinatorics to polytope geometry and suggest links to cosmohedra, Schwinger parametrizations, and kinematic-flow structures in cosmological correlators.

Abstract

We introduce a family of polytopes -- in-in zonotopes -- whose boundary structure organizes the contributions to scalar equal-time correlators in flat space computed via the in-in formalism. We provide explicit Minkowski sum and facet descriptions of these polytopes, and show that their boundaries factorize into products of graphical zonotopes and lower-dimensional in-in zonotopes, thereby mimicking the factorization structure of the correlators themselves. Evaluating their canonical forms at the origin -- equivalently, calculating the volume of the dual polytope -- reproduces the correlator. Finally, in a simple example, we show that the wavefunction decomposition of the correlator corresponds to a subdivision of the dual polytope.

Figures (4)

• Figure 1: The in-in zonotope $\mathcal{I}(P_2)$ with boundaries labelled by in-in diagrams.
• Figure 2: The in-in zonotope $\mathcal{I}(P_2)$ with vertices labelled by in-in diagrams.
• Figure 3: The in-in zonotope $\mathcal{I}(K_3)$ with vertices labelled by in-in diagrams.
• Figure 4: The dual in-in zonotope $\mathcal{I}^\circ(P_2)$ subdivided according to the wavefunction decomposition.