Amplitude recursions with an extra marked point

Understand

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.