Communications in Number Theory and Physics

Volume 13 (2019)

Number 3

Modular graph functions and asymptotic expansions of Poincaré series

Pages: 569 – 617



Daniele Dorigoni (Centre for Particle Theory and Department of Mathematical Sciences, Durham University, Durham, United Kingdom)

Axel Kleinschmidt (Max-Planck-Institut für Gravitationsphysik (Albert-Einstein-Institut), Potsdam, Germany; and International Solvay Institutes, Brussels, Belgium)


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 Poincaré 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.

We would like to thank Jens Funke, Herbert Gangl, Jan Gerken and Oliver Schlotterer for useful discussions. D.D. would like to thank the Albert Einstein Institute and in particular Hermann Nicolai for the hospitality and support during the various stages of this project. A.K. gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University at which part of the research for this paper was performed.

Received 30 March 2019

Accepted 11 June 2019

Published 8 August 2022