The recursive calculation of Selberg integrals by Aomoto and Terasoma using the Knizhnik–Zamolodchikov equation and the Drinfeld associator makes use of an auxiliary point and facilitates the recursive evaluation of string amplitudes at genus zero: open-string $N$‑point amplitudes can be obtained from those at $N-1$ points. We establish a similar formalism at genus one, which allows the recursive calculation of genus-one Selberg integrals using an extra marked point in a differential equation of Knizhnik–Zamolodchikov–Bernard type. Hereby genus-one Selberg integrals are related to genus-zero Selberg integrals. Accordingly, $N$‑point open-string amplitudes at one loop can be obtained from $(N+2)$‑point open-string amplitudes at tree level. The construction is related to and in accordance with various recent results in intersection theory and string theory.

中文翻译：

---

带有额外标记点的幅度递归

Aomoto 和 Terasoma 使用 Knizhnik-Zamolodchikov 方程和 Drinfeld 关联器对 Selberg 积分的递归计算利用了一个辅助点，并有助于在零属点处对弦振幅进行递归评估：可以获得开弦 $N$ 点振幅从那些在 $N-1$ 点。我们在第一类建立了一个类似的形式，它允许使用 Knizhnik-Zamolodchikov-Bernard 类型的微分方程中的一个额外的标记点来递归计算类一塞尔伯格积分。因此，属一塞尔伯格积分与属零塞尔伯格积分有关。因此，可以从树级的$(N+2)$-点开弦幅度获得一个循环的$N$-点开弦幅度。