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 τ. These integrals are found to satisfy the elliptic Knizhnik-Zamolodchikov-Bernard (KZB) equations, and can be written as a power series in α′ — the string length squared- in terms of elliptic multiple polylogarithms (eMPLs). In the N-puncture case, the KZB equation reveals a representation of B1,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 ⋊ \( \mathfrak{d} \), 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.
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Kaderli, A., Rodriguez, C. Open-string integrals with multiple unintegrated punctures at genus one. J. High Energ. Phys. 2022, 159 (2022). https://doi.org/10.1007/JHEP10(2022)159
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DOI: https://doi.org/10.1007/JHEP10(2022)159