Joyce structures and poles of Painlev\'e equations

Joyce structures are a class of geometric structures that first arose in relation to Donaldson-Thomas theory. There is a special class of examples, called class $S[A_1]$, whose underlying manifold parameterises Riemann surfaces of some fixed genus equipped with a meromorphic quadratic differential with poles of fixed orders. We study two Joyce structures of this type using the isomonodromic systems associated to the Painlev\'e II and III$_3$ equations. We give explicit formulae for the Pleba\'nski functions of these Joyce structures, and compute several associated objects, including their tau functions, which we explicitly relate to the corresponding Painlev\'e tau functions. We show that the behaviour of the Joyce structure near the zero-section can be studied analytically through poles of Painlev\'e equations. The systematic treatment gives a blueprint for the study of more general Joyce structures associated to meromorphic quadratic differentials on the Riemann sphere.