Communications in Number Theory and Physics

Volume 16 (2022)

Number 1

Amplitude recursions with an extra marked point

Pages: 75 – 158



Johannes Broedel (Eidgenössische Technische Hochschule (ETH) Zürich, Switzerland)

Andre Kaderli (Humboldt-Universität, Berlin, Germany)


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.


Selberg integrals, string scattering, KZ equation, KZB equation

A.K. would like to thank the IMPRS for Mathematical and Physical Aspects of Gravitation, Cosmology and Quantum Field Theory, of which he was a member and which rendered his studies possible. Furthermore, A.K. is supported by the Swiss Studies Foundation, to which he would like to express his gratitude.

Received 15 January 2020

Accepted 26 October 2021

Published 1 February 2022