Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models

TL;DR

The work investigates the nonperturbative content of topological strings and $c=1$ matrix models, focusing on instanton corrections and their relation to large-order behavior in the $1/N$ expansion. It develops and employs Borel analysis and Schwinger-type integral representations (via GV invariants) to obtain exact all-loop instanton corrections and to connect them with Stokes phenomena and hyperasymptotics. A semiclassical eigenvalue-tunneling picture is established through multi-sheeted holomorphic potentials, with instantons mapped to D-brane effects (ZZ/FZZT/Toric branes) in the dual spacetime, especially in the double-scaling limit to the $c=1$ string. The results are reinforced by trans-series/Toda approaches, which corroborate the instanton expansions and provide a pathway to smoothing nonperturbative ambiguities, offering a coherent nonperturbative completion of the dual theories.

Abstract

We address the nonperturbative structure of topological strings and c=1 matrix models, focusing on understanding the nature of instanton effects alongside with exploring their relation to the large-order behavior of the 1/N expansion. We consider the Gaussian, Penner and Chern-Simons matrix models, together with their holographic duals, the c=1 minimal string at self-dual radius and topological string theory on the resolved conifold. We employ Borel analysis to obtain the exact all-loop multi-instanton corrections to the free energies of the aforementioned models, and show that the leading poles in the Borel plane control the large-order behavior of perturbation theory. We understand the nonperturbative effects in terms of the Schwinger effect and provide a semiclassical picture in terms of eigenvalue tunneling between critical points of the multi-sheeted matrix model effective potentials. In particular, we relate instantons to Stokes phenomena via a hyperasymptotic analysis, providing a smoothing of the nonperturbative ambiguity. Our predictions for the multi-instanton expansions are confirmed within the trans-series set-up, which in the double-scaling limit describes nonperturbative corrections to the Toda equation. Finally, we provide a spacetime realization of our nonperturbative corrections in terms of toric D-brane instantons which, in the double-scaling limit, precisely match D-instanton contributions to c=1 minimal strings.