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

TL;DR

This work studies non-perturbative corrections to 3d BPS indices in 3d ${\mathcal N}=2$ GLSMs with Higgs branches $X$ by analytic continuation of the governing $q$-difference system. It introduces the Birkhoff connection matrix $P$ and shows that an $SL(2,\mathbb{Z})$ duality transform of $P$ yields non-perturbative corrections; in the conifold case this reproduces a world-volume dual to the non-perturbative topological string. The perturbative 3d BPS index reproduces the Gopakumar--Vafa partition function ${\cal F}_{GV}$ and the analytic continuation delivers a non-perturbative completion consistent with proposals in the literature, linking quantum K-theory correlators to topological-string data. The results provide a concrete 3d/5d duality framework that connects 3d world-volume BPS counting, $q$-difference equations, and non-perturbative topological string physics, with explicit structures for the conifold geometry.

Abstract

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.