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

Authors: Johannes Broedel, André Kaderli, Oliver Schlotterer Preprint number: UUITP-22/20 Abstract: Two different constructions generating the low-energy expansion of genus-one moduli-space integrals appearing in one-loop open-string amplitudes have been put forward in refs. [1,2]. We are going to show that both approaches can be traced back to an elliptic Knizhinik--Zamolodchikov--Bernard (KZB) system 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.