Integral of two-loop modular graph functions

TL;DR

This paper evaluates the integral of two-loop modular graph functions over the fundamental domain using recent Poincaré series results, advancing understanding of their mathematical properties.

Contribution

It provides a new method to compute integrals of two-loop modular graph functions leveraging recent developments in Poincaré series theory.

Findings

Explicit integral formulas derived for two-loop modular graph functions

Enhanced understanding of their modular properties and behavior

Potential applications in string theory and mathematical physics

Abstract

The integral of an arbitrary two-loop modular graph function over the fundamental domain for $SL(2,Z)$ in the upper half plane is evaluated using recent results on the Poincar\'e series for these functions.