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

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.

22 pages, LaTeX

References in corpus (2)

• SL(2,Z)-invariance and D-instanton contributions to the $D^6{R}^4$ interaction
• Integral of two-loop modular graph functions

Cited by in corpus (6)

• The SAGEX Review on Scattering Amplitudes, Chapter 10: Selected topics on modular covariance of type IIB string amplitudes and their $\mathcal{N}=4$ supersymmetric Yang-Mills duals
• Poincaré series for modular graph forms at depth two. II. Iterated integrals of cusp forms
• Poincaré series for modular graph forms at depth two. I. Seeds and Laplace systems
• Basis Decompositions and a Mathematica Package for Modular Graph Forms
• To the cusp and back: Resurgent analysis for modular graph functions
• Integrating three-loop modular graph functions and transcendentality of string amplitudes