Elliptic modular graph forms, equivariant iterated integrals and single-valued elliptic polylogarithms

TL;DR

The work develops a unifying generating-series framework to study genus-one elliptic modular graph forms (eMGFs) by solving their z- and $\tau$-dependent differential equations through equivariant iterated integrals. It then constructs single-valued elliptic multiple polylogarithms (eMPLs) in one variable by a gauge transformation linked to zeta generators and Tsunogai derivations, and embeds these into non-holomorphic modular forms via modular-frame conjugations. A central result is the explicit equivalence between z-dependent equivariant series and $\tau$-dependent eMPLs, yielding single-valued eMPLs as finite linear combinations of meromorphic eMPLs, their complex conjugates, svMZVs, and equivariant Eisenstein integrals, all organized by a rich Lie-algebraic structure. The framework also clarifies the cusp-expansion behavior and provides a path to count indecomposable eMGFs by analyzing shuffle-independent single-valued eMPLs, connecting deep number-theoretic objects (svMZVs, zeta generators) with the modular geometry of genus-one amplitudes. Overall, the paper furnishes a comprehensive algebraic and analytic toolkit to relate modular and elliptic polylogarithmic objects to genus-one string amplitudes, with implications for both mathematical structures and string-theory computations.

Abstract

The low-energy expansion of genus-one string amplitudes produces infinite families of non-holomorphic modular forms after each step of integrating over a point on the torus worldsheet which are known as elliptic modular graph forms (eMGFs). We solve the differential equations of eMGFs depending on a single point $z$ and the modular parameter $τ$ via iterated integrals over holomorphic modular forms which individually transform inhomogeneously under ${\rm SL}_2(\mathbb Z)$. Suitable generating series of these iterated integrals over $τ$, their complex conjugates and single-valued multiple zeta values (svMZVs) are combined to attain equivariant transformations under ${\rm SL}_2(\mathbb Z)$ such that their components are modular forms. Our generating series of equivariant iterated integrals for eMGFs is related to elliptic multiple polylogarithms (eMPLs) through a gauge transform of the flat Calaque-Enriquez-Etingof connection. By converting iterated $τ$-integrals to iterated integrals over points on a torus, we arrive at an explicit construction of single-valued eMPLs where all the monodromies in the points cancel. Each single-valued eMPL depending on a single point $z$ is found to be a finite combination of meromorphic eMPLs, their complex conjugates, svMZVs and equivariant iterated Eisenstein integrals. Our generating series determines the latter two admixtures via so-called zeta generators and Tsunogai derivations which act on the two generators $x$, $y$ of a free Lie algebra and where the coefficients of words in $x,y$ define the single-valued eMPLs.

Theorems & Definitions (42)

• Lemma 3.1
• Proposition 3.2
• proof
• Theorem 3.3
• proof
• Lemma 3.4
• proof
• Proposition 3.5
• proof
• Definition 3.6