Overview on Elliptic Multiple Zeta Values

Abstract

We give an overview of some work on elliptic multiple zeta values. First defined by Enriquez as the coefficients of the elliptic KZB associator, elliptic multiple zeta values are also special values of multiple elliptic polylogarithms in the sense of Brown and Levin. Common to both approaches to elliptic multiple zeta values is their representation as iterated integrals on a once-punctured elliptic curve. Having compared the two approaches, we survey various recent results about the algebraic structure of elliptic multiple zeta values, as well as indicating their relation to iterated integrals of Eisenstein series, and to a special algebra of derivations.

Elliptic Feynman integrals and pure functions

“…Here we follow a slightly different path, and inspired by refs. [138,156] we define eMPLs by the iterated integral…”

Heterotic-string amplitudes at one loop: modular graph forms and relations to open strings

“…The dependence of the eMZVs on τ will usually be suppressed for ease of notation. See [105] for a comprehensive reference on eMZVs.…”

Elliptic polylogarithms and Feynman parameter integrals

“…(2.18) eMPLs were originally defined in refs. [25,68,69] as iterated integrals on a complex torus. This description is related to the one presented here through the relation between elliptic curves defined by the equation y 2 = P 4 (x) and a torus defined as the complex plane quotiented by a two-dimmensional lattice Λ = Z ω 1 + Z ω 2 .…”

“…Since elliptic curves are isomorphic to complex tori, eMPLs can be described as iterated integrals over functions related to the torus, and were originally defined as such in refs. [25,68,69]. In this context, eMPLs are defined as iterated integrals given by…”