Ellipsitomic Associators

TL;DR

This paper introduces ellipsitomic associators using operad theory, reviews their connection to Drinfeld associators, and links them to elliptic functions and Eisenstein series, expanding the understanding of elliptic associators.

Contribution

It develops the concept of ellipsitomic associators via operad theory and establishes their existence through monodromy of the universal ellipsitomic KZB connection.

Findings

Existence of ellipsitomic associators proved

Connection to Eisenstein series and elliptic multiple zeta values established

Operadic framework for elliptic associators developed

Abstract

We develop a notion of ellipsitomic associators by means of operad theory. We take this opportunity to review the operadic point-of-view on Drinfeld associators and to provide such an operadic approach for elliptic associators too. We then show that ellipsitomic associators do exist, using the monodromy of the universal ellipsitomic KZB connection, that we introduced in a previous work. We finally relate the KZB ellipsitomic associator to certain Eisenstein series associated with congruence subgroups of $SL_2(\mathbb{Z})$, and to twisted elliptic multiple zeta values.