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Abstract

We study integrals appearing in intermediate steps of one-loop open-string






amplitudes, with multiple unintegrated punctures on the $A$-cycle of a torus.






We construct a vector of such integrals which closes after taking a total






differential with respect to the $N$ unintegrated punctures and the modular






parameter $\tau$. These integrals are found to satisfy the elliptic






Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power






series in $\alpha$' -- the string length squared -- in terms of elliptic






multiple polylogarithms (eMPLs). In the $N$-puncture case, the KZB equation






reveals a representation of $B_{1,N}$, the braid group of $N$ strands on a






torus, acting on its solutions. We write the simplest of these braid group






elements -- the braiding one puncture around another -- and obtain generating






functions of analytic continuations of eMPLs. The KZB equations in the






so-called universal case is written in terms of the genus-one Drinfeld-Kohno






algebra $\mathfrak{t}_{1,N} \rtimes \mathfrak{d}$, a graded algebra. Our






construction determines matrix representations of various dimensions for






several generators of this algebra which respect its grading up to commuting






terms.