Resumming the string perturbation series

  • Alba Grassi
  • Marcos Mariño
  • Szabolcs ZakanyEmail author
Open Access
Regular Article - Theoretical Physics


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 \( {\mathrm{\mathbb{P}}}^1\times {\mathrm{\mathbb{P}}}^1 \), 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.


Nonperturbative Effects 1/N Expansion Topological Strings Matrix Models 


Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.


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© The Author(s) 2015

Authors and Affiliations

  1. 1.Département de Physique Théorique et section de MathématiquesUniversité de GenèveGenèveSwitzerland

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