All order alpha'-expansion of superstring trees from the Drinfeld associator

TL;DR

The paper addresses computing the alpha'-corrections to open superstring tree amplitudes for arbitrary multiplicity N. It develops a recursion based on the Knizhnik-Zamolodchikov equation and the Drinfeld associator to relate N-point disk integrals to (N−1)-point data, recast as matrix operations using representations of e0 and e1. This approach yields an order-by-order alpha'-expansion in terms of multiple zeta values and is illustrated with explicit N=4 and N=5 examples, including higher-depth MZVs arising from the associator. The method is universal across spacetime dimensions and supersymmetries, offering a scalable path to higher multiplicities and potentially to closed strings and higher-genus amplitudes.

Abstract

We derive a recursive formula for the alpha'-expansion of superstring tree amplitudes involving any number N of massless open string states. String corrections to Yang-Mills field theory are shown to enter through the Drinfeld associator, a generating series for multiple zeta values. Our results apply for any number of spacetime dimensions or supersymmetries and chosen helicity configurations.

Figures (1)

• Figure 1: Worldsheet with an auxiliary position $z_0$.