Non-Perturbative Corrections to 3d BPS Indices and Topological Strings

For a 3d gauged linear sigma model parametrized by a Kahler manifold X, the 3d BPS index defines a q-series that can be analytically continued in the Kahler modulus by standard methods. It is argued that an SL(2,Z)-transform of the Birkhoff connection matrix captures non-perturbative corrections to the 3d GLSM. As an application, a 3d lift of the standard 2d GLSM for the resolved conifold is shown to provide a world-volume dual for the non-perturbative topological string on the resolved conifold. The perturbative 3d BPS index computes the Gopakumar-Vafa partition function, while the analytic continuation matches existing proposals for a non-perturbative completion of the topological string.