A construction of single-valued elliptic polylogarithms

Abstract

We establish a general construction of single-valued elliptic polylogarithms as functions on the once-punctured elliptic curve. Our formalism is an extension of Brown's construction of genus-zero single-valued polylogarithms to the elliptic curve: the condition of trivial monodromy for solutions to the Knizhnik-Zamolodchikov-Bernard equation is expressed in terms of elliptic associators and involves two representations of a two-letter alphabet. Our elliptic single-valued condition reduces to Brown's genus-zero condition upon degeneration of the torus. We provide several examples for our construction, including the elliptic Bloch-Wigner dilogarithm.