Mathematical Structures of Non-perturbative Topological String Theory: From GW to DT Invariants

Abstract

We study the Borel summation of the Gromov–Witten potential for the resolved conifold. The Stokes phenomena associated to this Borel summation are shown to encode the Donaldson–Thomas (DT) invariants of the resolved conifold, having a direct relation to the Riemann–Hilbert problem formulated by Bridgeland (Invent Math 216(1), 69–124, 2019). There exist distinguished integration contours for which the Borel summation reproduces previous proposals for the non-perturbative topological string partition functions of the resolved conifold. These partition functions are shown to have another asymptotic expansion at strong topological string coupling. We demonstrate that the Stokes phenomena of the strong-coupling expansion encode the DT invariants of the resolved conifold in a second way. Mathematically, one finds a relation to Riemann–Hilbert problems associated to DT invariants which is different from the one found at weak coupling. The Stokes phenomena of the strong-coupling expansion turn out to be closely related to the wall-crossing phenomena in the spectrum of BPS states on the resolved conifold studied in the context of supergravity by Jafferis and Moore (Wall crossing in local Calabi Yau manifolds, arXiv:0810.4909, 2008).

Quantum Curves, Resurgence and Exact WKB

“…This new difference equation (the third equation in (4.4)) is invisible to the perturbative expansion of the refined free energy, which is periodic in the closed modulus. We show in (4.15) that this new difference equation encodes the Stokes jumps of the unrefined topological string, obtained previously in [9]. In Section 4.3, we write down the analogous statements in the Nekrasov-Shatashvili (NS) limit.…”

“…Similar to [9], it was noted that there exists a natural generalization W eff ϑ ′ of the effective twisted superpotential W eff . The superpotential W eff ϑ ′ (x, ϵ) is defined in the same way as in equations (1.2) and (1.3), but now in terms of quantum periods that are Borel summed in an arbitrary direction ϑ ′ .…”

“…Moreover, following the ideas of [26,27], it was found in [9] that the exponentials of the Borel summations exp F top ρ along the ray ρ form a collection of local sections of the conformal limit of a certain hyperholomorphic line bundle previously considered in [5,84], whose transition functions are the exponentials of the above Stokes factors, leading to a new geometrical, nonperturbative picture of the topological string.…”

“…Mathematically, when the Borel resummation of the asymptotic series is considered it turns out that other sets of invariants are encoded in the Stokes jumps of different Borel resummations, see for example [31,44,45,55], see also [47] for a study of resurgence for the quantum dilogarithm function, which is a building block of many interesting objects in mathematical physics and in particular in Chern-Simons theories. In [9], the techniques of [47] were used to study the Borel resummation of the Gromov-Witten potential F top (λ, t) for the resolved conifold. Earlier results on the Borel resummation for the resolved conifold with different techniques and scope were obtained in [59,91].…”

“…The Borel transform has infinitely many singularities organized along rays coinciding with the rays ±R <0 Z γ , where Z γ denotes the central charge of a BPS state of charge γ. Different Borel resummations were defined along rays which avoid the singularities, and it was found that they experience Stokes jumps across the rays ±R <0 Z γ , with the BPS charge γ ∈ Γ contributing to the jump by [9]:…”