Elliptic symbol calculus: from elliptic polylogarithms to iterated integrals of Eisenstein series

TL;DR

This work extends the symbol calculus from ordinary multiple polylogarithms to elliptic counterparts by leveraging Brown's motivic coaction, thereby unifying elliptic polylogarithms and iterated integrals of modular forms. It defines a holomorphic, non-periodic variant of elliptic MPLs $ ilde{\boldsymbol{\textGamma}}$ with a total differential and introduces a corresponding elliptic symbol using letters $\boldsymbol{\boldomega}_{ij}^{(n)}$, then generalizes the coaction to unipotent periods including iterated integrals of modular forms. A key result is that eMPLs evaluated at rational points can be expressed as iterated integrals of Eisenstein series for a congruence subgroup $\Gamma(N)$, enabling compact representations of elliptic hypergeometric functions and the equal-mass sunrise integral in two dimensions as such iterated Eisenstein integrals. The framework bridges elliptic function theory and multi-loop Feynman integrals, offering a path toward canonical bases and a diagrammatic coaction for elliptic amplitudes in perturbation theory.

Abstract

We present a generalization of the symbol calculus from ordinary multiple polylogarithms to their elliptic counterparts. Our formalism is based on a special case of a coaction on large classes of periods that is applied in particular to elliptic polylogarithms and iterated integrals of modular forms. We illustrate how to use our formalism to derive relations among elliptic polylogarithms, in complete analogy with the non-elliptic case. We then analyze the symbol alphabet of elliptic polylogarithms evaluated at rational points, and we observe that it is given by Eisenstein series for a certain congruence subgroup. We apply our formalism to hypergeometric functions that can be expressed in terms of elliptic polylogarithms and show that they can equally be written in terms of iterated integrals of Eisenstein series. Finally, we present the symbol of the equal-mass sunrise integral in two space-time dimensions. The symbol alphabet involves Eisenstein series of level six and weight three, and we can easily integrate the symbol in terms of iterated integrals of Eisenstein series.

Figures (2)

• Figure 1: Choice of contours according to eq. \ref{['eq:paths']}.
• Figure 2: Images under Abel's map of the branch points and punctures in the fundamental domain of the torus. Note that every point on the elliptic curve has two images on the torus; we have moved both images of the real line into the fundamental domain using periodicity.