Lorentzian contours for tree-level string amplitudes

TL;DR

This paper introduces Lorentzian contours on moduli spaces for tree-level string amplitudes, enabling direct numerical computations in physical kinematics for various string interactions.

Contribution

It develops generalized Pochhammer contours based on associahedra, making the analytic structure of string amplitudes explicit and facilitating numerical calculations.

Findings

First numerical computations of open and closed string amplitudes in physical kinematics.

Contours work for any number and type of external strings.

Provides a code for practical computations.

Abstract

We engineer compact contours on the moduli spaces of genus-zero Riemann surfaces that achieve analytic continuation from Euclidean to Lorentzian worldsheets. These generalized Pochhammer contours are based on the combinatorics of associahedra and make the analytic properties of tree-level amplitudes entirely manifest for any number and type of external strings. We use them in practice to perform first numerical computations of open and closed string amplitudes directly in the physical kinematics for $n=4,5,6,7,8,9$. We provide a code that allows anyone to do such computations.