Abstract
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.
Highlights
As we will show in this paper, we find a simple integral representation of (1.1) by applying the generalized Borel resummation to the genus expansion
We show that the resummed free energy contains the Gopakumar-Vafa formula and the non-perturbative membrane instanton correction, proposed in [6]
An advantage of the generalized Borel resummation is that one can write down exact integral representations
Summary
One useful way to make sense of such divergent series is to introduce the Borel sum. We can consider a generalization of the Borel sum, known as the moment method, by using some moment μn of the weight function μ(ζ), dζ μ(ζ)ζn = μn,. From this sum Bμ[f ](ζ), we can define the resummation associated with the moment μn dζ μ(ζ)Bμ[f ](gζ). The usual Borel sum is a special case of the moment method, corresponding to the choice μ(ζ) = e−ζ and μn = n!. The genus-g contribution usually grows as (2g)!, and it leads to a divergent series. It is useful to consider a moment method associated with the integral representation of B2g.
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