Zero mode of the Fourier series of some modular graphs from Poincare series

TL;DR

This paper analyzes specific modular graph functions on a torus, expressing their zero modes via Poincare series to simplify the calculation of asymptotic expansions, revealing their eigenvalue properties with source terms.

Contribution

It introduces a method to express certain modular graph functions as Poincare series, facilitating the analysis of their zero modes and asymptotic behavior.

Findings

Eigenvalue equations with source terms involving Eisenstein series.

Explicit Poincare series representations for modular graph functions.

Simplified calculation of asymptotic expansion terms.

Abstract

We consider specific linear combinations of two loop modular graph functions on the toroidal worldsheet with $2s$ links for $s=2, 3$ and $4$. In each case, it satisfies an eigenvalue equation with source terms involving $E_{2s}$ and $E_s^2$ only. On removing certain combinations of $E_{2s}$ and $E_s^2$ from it, we express the resulting expression as an absolutely convergent Poincare series. This is used to calculate the power behaved terms in the asymptotic expansion of the zero mode of the Fourier expansion of these graphs in a simple manner.