Integral of depth zero to three basis of Modular Graph Functions

Abstract: Modular Graph Functions (MGFs) are SL(2,$\mathbb{Z}$)-invariant functions that emerge in the study of the low-energy expansion of the one-loop closed string amplitude. To find the string scattering amplitude, we must integrate MGFs over the moduli space of the torus. In this paper, we use the iterated integral representation of MGFs to establish a depth-dependent basis for them, where "depth" refers to the number of iterations in the integral. This basis has a suitable Laplace equation. We integrate this basis from depth zero to depth three over the fundamental domain of SL(2,$\mathbb{Z}$) with a cut-off.