Tau Functions from Joyce Structures

We argued in [Proc. Sympos. Pure Math., Vol. 103, American Mathematical Society, Providence, RI, 2021, 1-66, arXiv:1912.06504] that, when a certain sub-exponential growth property holds, the Donaldson-Thomas invariants of a 3-Calabi-Yau triangulated category should give rise to a geometric structure on its space of stability conditions called a Joyce structure. In this paper, we show how to use a Joyce structure to define a generating function which we call the $\tau$-function. When applied to the derived category of the resolved conifold, this reproduces the non-perturbative topological string partition function of [J. Differential Geom. 115 (2020), 395-435, arXiv:1703.02776]. In the case of the derived category of the Ginzburg algebra of the A$_2$ quiver, we obtain the Painlev\'e I $\tau$-function.