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

TL;DR

This paper demonstrates that two different approaches to generating low-energy expansions of genus-one integrals in one-loop open-string amplitudes are fundamentally connected through an elliptic KZB system on the twice-punctured torus.

Contribution

It reveals the underlying elliptic KZB structure linking two previously separate constructions for open-string integrals.

Findings

Both approaches are shown to originate from an elliptic KZB system.

An explicit all-multiplicity representation of the elliptic KZB system is derived.

Compatibility conditions between the algebra generators of the two approaches are explored.

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.