Elliptic hyperlogarithms

TL;DR

This paper introduces elliptic hyperlogarithms as a new basis for the algebra of holomorphic multivalued functions on elliptic curves minus a finite set, providing an alternative to existing elliptic hyperlogarithm bases.

Contribution

It establishes that functions from string theory, denoted , form a basis for the algebra of holomorphic multivalued functions on elliptic curves minus a finite set, stable under integration.

Findings

Functions  form a basis for the algebra.

The basis is alternative to previous elliptic hyperlogarithm bases.

The basis is stable under integration.

Abstract

Let $\mathcal E$ be a complex elliptic curve and $S$ be a non-empty finite subset of $\mathcal E$. We show that the functions $\tilde\Gamma$ introduced in arXiv:1712.07089 out of string theory motivations give rise to a basis of the minimal algebra $A_{\mathcal E\smallsetminus S}$ of holomorphic multivalued functions on $\mathcal E\smallsetminus S$ which is stable under integration, introduced in arXiv:2212.03119; this basis is alternative to the basis of $A_{\mathcal E\smallsetminus S}$ constructed in loc. cit. using elliptic analogues of the hyperlogarithm functions.