Modular graph functions and asymptotic expansions of Poincar\'e series

Understand

In this note we study $SL(2,\mathbb{Z})$-invariant functions such as modular graph functions or coefficient functions of higher derivative corrections in type IIB string theory.

• The functions solve inhomogeneous Laplace equations and we choose to represent them as Poincar\'e series.
• In this way we can combine different methods for asymptotic expansions and obtain the perturbative and non-perturbative contributions to their zero Fourier modes.
• In the case of the higher derivative corrections, these terms have an interpretation in terms of perturbative string loop effects and pairs of instantons/anti-instantons.