Kinematic flow from the flow of cuts

TL;DR

This paper develops a geometric framework for understanding the kinematic flow in conformally coupled scalars within power-law FRW cosmologies, using combinatorial and graphical methods to derive differential equations without bulk physics.

Contribution

It introduces a geometric perspective on the kinematic flow, relating it to positive geometries, decorated graphs, and zonotopes, providing a new way to derive and compute the associated differential equations.

Findings

Derived the kinematic flow from hyperplane arrangements and decorated graphs.

Established a connection between residues of the FRW-form and graphical zonotopes.

Provided a closed-form formula for the differential equations governing the cut basis.

Abstract

The wavefunction coefficients of conformally coupled scalars in power-law FRW cosmologies satisfy differential equations governed by a set of simple combinatorial rules known as the kinematic flow. In this paper we derive the kinematic flow, expressed using a set of differential forms referred to as the cut basis, from a geometric perspective, relying solely on the cosmological hyperplane arrangement and without invoking bulk physics. Each element of the cut basis corresponds to the positive geometry associated to an independent cut of the physical FRW-form and can be labeled by decorating (minors of) the truncated Feynman graph with an acyclic orientation. We provide a straightforward prescription to associate a logarithmic differential form to each element of the cut basis by considering its corresponding decorated graph. Moreover, we show that the residues of the physical FRW-form are canonical forms of certain graphical zonotopes labeled by the same set of decorated graphs. These zonotopes control the cut combinatorics -- flow of cuts -- of the physical FRW-form and the cut basis (by construction). Using the theory of relative twisted cohomology and intersection theory, we derive a closed form formula for the differential equations of the cut basis. We also introduce combinatorial rules that compute the kinematic differential of any basis element without explicit calculation. The combinatorics of our differential equations is a natural consequence of the flow of cuts and is equivalent (up to rescaling) to the kinematic flow for the recently studied time integral basis. In particular, our differential equations decouple into exponentially many sectors, one for each way of cutting a subset of edges of the graph.