Resumming the string perturbation series

TL;DR

This paper investigates the limitations of Borel resummation in capturing non-perturbative effects in string theory, highlighting the role of complex instantons and membrane instantons in obtaining accurate results.

Contribution

It demonstrates that Borel summability alone is insufficient for non-perturbative completion, emphasizing the necessity of including complex instantons and non-perturbative sectors in string theory resummation.

Findings

Borel resummation does not match exact results due to complex instantons.

Complex instantons affect the non-perturbative completion of string series.

Membrane instantons are essential for accurate non-perturbative results.

Abstract

We use the AdS/CFT correspondence to study the resummation of a perturbative genus expansion appearing in the type II superstring dual of ABJM theory. Although the series is Borel summable, its Borel resummation does not agree with the exact non-perturbative answer due to the presence of complex instantons. The same type of behavior appears in the WKB quantization of the quartic oscillator in Quantum Mechanics, which we analyze in detail as a toy model for the string perturbation series. We conclude that, in these examples, Borel summability is not enough for extracting non-perturbative information, due to non-perturbative effects associated to complex instantons. We also analyze the resummation of the genus expansion for topological string theory on local $\mathbb P^1 \times \mathbb P^1$, which is closely related to ABJM theory. In this case, the non-perturbative answer involves membrane instantons computed by the refined topological string, which are crucial to produce a well-defined result. We give evidence that the Borel resummation of the perturbative series requires such a non-perturbative sector.