Non-perturbative effects and wall-crossing from topological strings

We argue that the Gopakumar-Vafa interpretation of the topological string partition function can be used to compute and resum certain non-perturbative brane instanton effects of type II CY compactifications. In particular the topological string A-model encodes the non-perturbative corrections to the hypermultiplet moduli space metric from general D1/D(-1)-brane instantons in 4d N=2 IIB models. By introducing fluxes and/or orientifolds and/or D-branes, we describe the reduction to 4d N=1 models, and describe the computation of non-perturbative superpotential contributions from resummed brane instantons. We argue that the connection between non-perturbative effects and the topological string underlies the continuity and holomorphy of non-perturbative effects across lines of BPS stability. The computation of non-perturbative effects from the topological string requires a 3d circle compactification and T-duality, relating effects from particles and instantons, suggesting a realization of the Kontsevich-Soibelmann wall-crossing formula.