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 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.