Two dialects for KZB equations: generating one-loop open-string integrals

Abstract

Two different constructions generating the low-energy expansion of genus-one configuration-space integrals appearing in one-loop open-string amplitudes have been put forward in \rcites{Mafra:2019xms, *Mafra:2019ddf, Broedel:2019gba}. We are going to show that both approaches can be traced back to an elliptic system of Knizhnik--Zamolodchikov--Bernard(KZB) type on the twice-punctured torus. We derive an explicit all-multiplicity representation of the elliptic KZB system for a vector of iterated integrals with an extra marked point and explore compatibility conditions for the two sets of algebra generators appearing in the two differential equations.