Open-string integrals with multiple unintegrated punctures at genus one

We study integrals appearing in intermediate steps of one-loop open-string amplitudes, with multiple unintegrated punctures on the $A$-cycle a torus. construct vector such which closes after taking total differential respect to $N$ and modular parameter $\tau$. These are found satisfy elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, can be written as power series $\alpha$' -- string length squared terms polylogarithms (eMPLs). In $N$-puncture case, KZB equation reveals representation $B_{1,N}$, braid group strands torus, acting its solutions. write simplest these elements braiding one puncture around another obtain generating functions analytic continuations eMPLs. The equations so-called universal case is genus-one Drinfeld-Kohno algebra $\mathfrak{t}_{1,N} \rtimes \mathfrak{d}$, graded algebra. Our construction determines matrix representations various dimensions for several generators this grading up commuting terms.