Resummations and non-perturbative corrections

We consider a generalization of the Borel resummation, which turns out to be equivalent to the standard Borel resummation. We apply it to the simplest large N duality between the pure Chern-Simons theory and the topological string on the resolved conifold, and obtain a simple integral formula for the free energy. Expanding this integral representation around the large radius point at finite string coupling g(s), we find that it includes not only the M-theoretic resummation a la Gopakumar and Vafa, but also a non-perturbative correction in g(s). Remarkably, the obtained non-perturbative correction is in perfect agreement with a proposal for membrane instanton corrections in arXiv: 1306.1734. Various other examples are also presented.