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.
Highlights
Still show up at the LHC via small deviations in key cross sections, distributions and decay rates
In this paper we have presented for the first time an explicit construction of the coaction and the symbol map that are applicable to elliptic multiple polylogarithms
Our construction for the coaction is general enough to be applied well to this class of functions. It has provided us with a very efficient computational tool to transform linear combinations of eMPLs evaluated at rational points into linear combinations of iterated integrals of Eisenstein series
Summary
Before we discuss how to extend (some of) the algebraic properties of polylogarithms beyond genus zero, we present a concise review of ordinary multiple polylogarithms, as well as their symbols and coaction and how to use them to work out relations among MPLs. The material is well known, see, e.g., ref. [76] for a pedagogical review
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
