To the cusp and back: resurgent analysis for modular graph functions

Modular graph functions arise in the calculation of low-energy expansion closed-string scattering amplitudes. For toroidal world-sheets, they are ${\rm SL}(2,\mathbb{Z})$-invariant torus complex structure that have to be integrated over moduli space inequivalent tori. We use methods from resurgent analysis construct non-perturbative corrections arising when argument modular function approaches cusp on this space. SL}(2,\mathbb{Z})$-invariance will turn strongly constrain behaviour sector expanded at origin