Amplitude recursions with an extra marked point

TL;DR

This paper extends recursive methods for calculating Selberg integrals and string amplitudes from genus zero to genus one using an extra marked point and a differential equation framework.

Contribution

It introduces a genus-one recursive formalism for Selberg integrals and string amplitudes, generalizing previous genus-zero techniques with an additional marked point.

Findings

Recursive calculation of genus-one Selberg integrals established

Genus-one amplitudes derived from higher-point genus-zero amplitudes

Framework aligns with recent intersection theory and string theory results

Abstract

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 genus one 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.