Geometry of in-in correlators

TL;DR

This paper introduces in-in zonotopes, a new class of polytopes that organize scalar correlator contributions in flat space, revealing their boundary and factorization structures, and linking geometric properties to correlator evaluations.

Contribution

The paper defines in-in zonotopes, describes their geometric structure, and connects their properties to the factorization and evaluation of scalar in-in correlators.

Findings

In-in zonotopes encode correlator contributions.

Boundary structures factorize into graphical zonotopes.

Volume calculations reproduce 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.