One-loop open-string integrals from differential equations: all-order alpha'-expansions at n points

Understand

We study generating functions of moduli-space integrals at genus one that are expected to form a basis for massless $n$-point one-loop amplitudes of open superstrings and open bosonic strings.

• These integrals are shown to satisfy the same type of linear and homogeneous first-order differential equation w.r.t.
• the modular parameter $\tau$ which is known from the A-elliptic Knizhnik--Zamolodchikov--Bernard associator.
• The expressions for their $\tau$-derivatives take a universal form for the integration cycles in planar and non-planar one-loop open-string amplitudes.