Which Functions Admit a Positive Geometry? From Branch Cuts to String Amplitudes

TL;DR

This paper extends positive geometry to include functions with branch cuts, classifies their canonical forms using pseudogenus, and relates these geometries to string amplitudes and the KLT double copy.

Contribution

It introduces the notion of pseudogenus to classify meromorphic functions and generalizes positive geometries to include branch cuts and string amplitude structures.

Findings

Canonical forms can be expressed as dlog of pseudogenus zero functions.

Positive geometries with branch cuts relate to string and Kaluza-Klein towers.

Open and closed string amplitudes admit positive geometry interpretations.

Abstract

Positive geometry provides a geometric framework where physical observables are encoded as canonical forms associated to regions of kinematic space. In this paper we consider a generalisation to an infinite union of line segments, which allows us to capture canonical forms beyond rational functions. In the continuum limit of positive geometries, we show that we can generalise even further and describe positive geometries whose canonical forms contain branch cuts. We will constrain which functions can be obtained as the canonical form of one-dimensional positive geometries. We introduce the notion of the pseudogenus to classify meromorphic functions, and show that canonical forms can be written as the $\mathrm d\log$ of a function with pseudogenus zero. Furthermore, we argue that the spectrum encoded by a union of line segments is consistent with the presence of a stringy tower of states or a Kaluza-Klein tower with three or more compact directions only if nearly all such states do not contribute to the scattering amplitude. In addition, we show how the d log of both open and closed string amplitudes admits a positive geometry. This allows us to give a fully geometric interpretation for the KLT double copy at four points.