Partition functions from quantum curves

In joint work with I. Coman and E. Pomoni we had recently proposed a definition of the topological stringpartition functions for the local Calabi-Yau manifolds used inthe geometric engineering of N=2, d=4 class S field theories.The main goal of my talk will be to review this proposal, andto explain how the exact WKB method provides a crucialingredient. We thereby arrive at a fully non-perturbativeconstruction of the partition functions from the classicalcurves describing the relevant B-model backgrounds