Open-string integrals with multiple unintegrated punctures at genus one

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 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 socalled universal case is written in terms of the genus-one Drinfeld-Kohno algebra t 1,N 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.