Tau-functions, topological partition functions and cluster varieties

The goal of my second talk in this workshop will be to introduce the geometric characterisation of some topological string partition functions proposed in recent work of I. Coman, P. Longhi and myself. It combines three main ingredients: (i) Exact WKB, (ii) cluster Poisson varieties, (iii) tau-functions. The tau-functions are interpreted as holomorphic sections of a natural (hyper-)holomorphic line-bundle. The case of the resolved conifold will be discussed as an example of a local CY manifold in which this proposal is realised