Non-perturbative effects and the refined topological string

TL;DR

The paper addresses non-perturbative corrections in ABJM theory on $S^3$, demonstrating that membrane instantons are fully determined by the NS limit of the refined topological string on local ${\mathbb P}^1\times{\mathbb P}^1$ via quantum periods. It establishes a precise dictionary between worldsheet instantons (topological-string data) and membrane instantons/bound states (NS-refined data), and proposes a first-principles, non-perturbative free energy for local Calabi–Yau geometries by combining unrefined and NS sectors with effective Kähler parameters. A key technical achievement is the identification of membrane instanton generating functions with quantum A-/B-periods and the framing of the HMO cancellation mechanism in terms of refined BPS invariants, ensuring pole-free, finite results. The work offers a generalizable framework for non-perturbative refined topological strings across local CYs and connects matrix-model results to quantum-period calculations, with potential implications for ABJ and related theories.

Abstract

The partition function of ABJM theory on the three-sphere has non-perturbative corrections due to membrane instantons in the M-theory dual. We show that the full series of membrane instanton corrections is completely determined by the refined topological string on the Calabi-Yau manifold known as local P1xP1, in the Nekrasov-Shatashvili limit. Our result can be interpreted as a first-principles derivation of the full series of non-perturbative effects for the closed topological string on this Calabi-Yau background. Based on this, we make a proposal for the non-perturbative free energy of topological strings on general, local Calabi-Yau manifolds.