Nonperturbative completion of the string genus expansion
In plain words
String theory computes processes as a sum over surfaces with more and more holes, and the sum diverges like factorials. Which nonperturbative effects complete it, and whether the result is unique, is known only in toy models and special settings.
Precise statement
String perturbation theory in $g_{s}$ grows like $(2g)!$ at genus $g$ (Shenker 1990), which points to effects of order $\exp(-1/g_{s})$ from D-branes. For the type II superstring in $10\mathrm{D}$ flat space, determine whether the genus expansion of the four-graviton amplitude at fixed kinematics is resurgent, and whether D-instanton sectors give a unique median-resummed result.
What would settle it
An exact transseries for the flat-space four-graviton amplitude, with D-instanton coefficients fixed and the resummed result matched to an independent nonperturbative computation.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Resurgent transseries are known for minimal strings, JT-gravity matrix models and $N=4$ super-Yang-Mills integrated correlators (2021-2024); the flat-space critical superstring amplitude remains open.
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