A note on the Drinfeld associator for genus-zero superstring amplitudes in twisted de Rham theory

TL;DR

This paper connects string amplitude corrections to twisted de Rham theory, providing a combinatorial framework and algorithms for recursive calculations of genus-zero superstring amplitudes using the Drinfeld associator.

Contribution

It offers a new mathematical description of string amplitude recursions via twisted de Rham theory and intersection numbers, enabling efficient computational algorithms.

Findings

Derived combinatorial expressions for Lie algebra generators

Developed algorithms using directed graph adjacency matrices

Facilitated recursive calculations of string amplitudes

Abstract

The string corrections of tree-level open-string amplitudes can be described by Selberg integrals satisfying a Knizhnik-Zamolodchikov (KZ) equation. This allows for a recursion of the $\alpha'$-expansion of tree-level string corrections in the number of external states using the Drinfeld associator. While the feasibility of this recursion is well-known, we provide a mathematical description in terms of twisted de Rham theory and intersection numbers of twisted forms. In particular, this leads to purely combinatorial expressions for the matrix representation of the Lie algebra generators appearing in the KZ equation in terms of directed graphs. This, in turn, admits efficient algorithms for symbolic and numerical computations using adjacency matrices of directed graphs and is a crucial step towards analogous recursions and algorithms at higher genera.