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 α ′-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.

中文翻译：

---

关于扭曲De Rham理论中属零超弦幅度的Drinfeld关联的注释

可以通过满足Knizhnik–Zamolodchikov（KZ）方程的Selberg积分来描述树级开放字符串幅度的字符串校正。这允许使用Drinfeld关联器递归在外部状态数中的树级字符串校正的α'扩展。尽管这种递归的可行性众所周知，但我们根据扭曲的德·拉姆理论和扭曲形式的交点数提供了数学描述。特别是，这导致出现在有向图上的KZ方程中出现的Lie代数生成器的矩阵表示的纯粹组合表达式。反过来，这也允许使用有向图的邻接矩阵进行符号和数值计算的高效算法，并且是迈向更高类的类似递归和算法的关键一步。