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

TL;DR

This paper extends the symbol calculus to elliptic polylogarithms and Eisenstein series, enabling new relations, analysis of their structure, and applications to hypergeometric functions and Feynman integrals.

Contribution

It introduces a formalism for elliptic symbol calculus based on a coaction on periods, generalizing the non-elliptic case, and applies it to various elliptic functions and integrals.

Findings

Elliptic symbol alphabet involves Eisenstein series for certain congruence subgroups.

Hypergeometric functions can be expressed as iterated integrals of Eisenstein series.

The symbol of the sunrise integral involves Eisenstein series of level six and weight three.

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.