Open-string integrals with multiple unintegrated punctures at genus one

TL;DR

This paper develops a mathematical framework for integrals in one-loop open-string amplitudes with multiple punctures on a torus, revealing their relation to elliptic polylogarithms and braid group representations.

Contribution

It constructs a vector of integrals satisfying elliptic KZB equations, expressed via elliptic multiple polylogarithms, and explores their connection to braid group actions and algebraic structures.

Findings

Integrals satisfy elliptic KZB equations.

Representation of braid group on solutions.

Expressed integrals as power series in string length squared.

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.