A single theta-lift construction generates both two-loop modular graph functions and S-dual modular functions as rational combinations of generalised Eisenstein series with all cusp-form L-values cancelling.
To the cusp and back: Resurgent analysis for modular graph functions
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abstract
Modular graph functions arise in the calculation of the low-energy expansion of closed-string scattering amplitudes. For toroidal world-sheets, they are ${\rm SL}(2,\mathbb{Z})$-invariant functions of the torus complex structure that have to be integrated over the moduli space of inequivalent tori. We use methods from resurgent analysis to construct the non-perturbative corrections arising when the argument of the modular graph function approaches the cusp on this moduli space. ${\rm SL}(2,\mathbb{Z})$-invariance will in turn strongly constrain the behaviour of the non-perturbative sector when expanded at the origin of the moduli space.
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Modular Features of Superstring Scattering Amplitudes: Generalised Eisenstein Series and Theta Lifts
A single theta-lift construction generates both two-loop modular graph functions and S-dual modular functions as rational combinations of generalised Eisenstein series with all cusp-form L-values cancelling.