Modular Graph Functions

TL;DR

Modular graph functions provide a bridge between genus-one string amplitudes and the theory of single-valued elliptic polylogarithms. The authors show that these modular functions are special values of single-valued elliptic MPLs evaluated at the elliptic identity, and propose a general conjecture bounding the depth and weight of the associated polylogarithms, proven for star graphs. The cusp (constant Fourier mode) coefficients of these functions are rational combinations of single-valued zeta values, revealing a deep arithmetic structure intertwined with Eisenstein-series components. This framework suggests a unified, algebraic approach to interrelations among modular graph functions and their potential connections to tree-level single-valued zeta values and KLT-like relations between open and closed string amplitudes, with broader implications for the basis of modular graphs at higher weights and vertex counts.

Abstract

In earlier work we studied features of non-holomorphic modular functions associated with Feynman graphs for a conformal scalar field theory on a two-dimensional torus with zero external momenta at all vertices. Such functions, which we will refer to as modular graph functions, arise, for example, in the low energy expansion of genus-one Type II superstring amplitudes. We here introduce a class of single-valued elliptic multiple polylogarithms, which are defined as elliptic functions associated with Feynman graphs with vanishing external momenta at all but two vertices. These functions depend on a coordinate, $ζ$, on the elliptic curve and reduce to modular graph functions when $ζ$ is set equal to $1$. We demonstrate that these single-valued elliptic multiple polylogarithms are linear combinations of multiple polylogarithms, and that modular graph functions are sums of single-valued elliptic multiple polylogarithms evaluated at the identity of the elliptic curve, in both cases with rational coefficients. This insight suggests the many interrelations between modular graph functions (a few of which were established in earlier papers) may be obtained as a consequence of identities involving multiple polylogarithms, and explains an earlier observation that the coefficients of the Laurent polynomial at the cusp are given by rational numbers times single-valued multiple zeta values.