Quantum Curves, Resurgence and Exact WKB

TL;DR

This paper investigates the non-perturbative quantum geometry of topological strings on the resolved conifold, employing resurgence and exact WKB methods to analyze difference equations, Borel transforms, and spectral networks, revealing deep links with Donaldson-Thomas invariants and quantum integrable systems.

Contribution

It introduces new finite difference equations for refined topological string partition functions and develops a resurgence framework connecting open and closed string sectors.

Findings

Derived new difference equations for refined partition functions.

Identified Borel plane singularities related to Donaldson-Thomas invariants.

Connected resurgence analysis with spectral networks and quantum integrable systems.

Abstract

We study the non-perturbative quantum geometry of the open and closed topological string on the resolved conifold and its mirror. Our tools are finite difference equations in the open and closed string moduli and the resurgence analysis of their formal power series solutions. In the closed setting, we derive new finite difference equations for the refined partition function as well as its Nekrasov-Shatashvili (NS) limit. We write down a distinguished analytic solution for the refined difference equation that reproduces the expected non-perturbative content of the refined topological string. We compare this solution to the Borel analysis of the free energy in the NS limit. We find that the singularities of the Borel transform lie on infinitely many rays in the Borel plane and that the Stokes jumps across these rays encode the associated Donaldson-Thomas invariants of the underlying Calabi-Yau geometry. In the open setting, the finite difference equation corresponds to a canonical quantization of the mirror curve. We analyze this difference equation using Borel analysis and exact WKB techniques and identify the 5d BPS states in the corresponding exponential spectral networks. We furthermore relate the resurgence analysis in the open and closed setting. This guides us to a five-dimensional extension of the Nekrasov-Rosly-Shatashvili proposal, in which the NS free energy is computed as a generating function of $q$-difference opers in terms of a special set of spectral coordinates. Finally, we examine two spectral problems describing the corresponding quantum integrable system.