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

Authors: Oliver Schlotterer, Yoann Sohnle, Yi-Xiao Tao

Pre-print: UUITP–33/25

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 SL2(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 SL2(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.