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We propose a geometric characterisation of the topological string partitionfunctions associated to the local Calabi-Yau (CY) manifolds used in thegeometric engineering of $d=4$, $\mathcal{N}=2$ supersymmetric field theoriesof class $\mathcal{S}$. A quantisation of these CY manifolds definesdifferential operators called quantum curves. The partition functions areextracted from the isomonodromic tau-functions associated to the quantum curvesby expansions of generalised theta series type. It turns out that the partitionfunctions are in one-to-one correspondence with preferred coordinates on themoduli spaces of quantum curves defined using the Exact WKB method. Thecoordinates defined in this way jump across certain loci in the moduli space.The changes of normalisation of the tau-functions associated to these jumpsdefine a natural line bundle playing a key role in the geometriccharacterisation of the topological string partition functions proposed here.

From quantum curves to topological string partition functions II