Joyce structures and poles of Painlevé equations
Tom Bridgeland, Fabrizio Del Monte
TL;DR
This work extends the theory of Joyce structures to meromorphic quadratic-differential settings, focusing on Painlevé III$_3$ and Painlevé II in class $S[A_1]$. It provides explicit Plebański functions, linear Joyce connections, and a coherent tau-function framework that recovers and normalizes the corresponding Painlevé tau functions, while revealing a deep link to spectral-curve data and topological string theory. The constructions give a systematic blueprint for building Joyce structures from pencils of projective structures and isomonodromic deformations, with concrete computations illustrating regularity on the zero section and the role of apparent singularities. The results bridge nonlinear hyperkähler geometry, isomonodromy, and DT/topological-string perspectives, offering a path to generalize these Joyce structures to broader meromorphic-differential data on the Riemann sphere.
Abstract
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 Painlevé II and III$_3$ equations. We give explicit formulae for the Plebański functions of these Joyce structures, and compute several associated objects, including their tau functions, which we explicitly relate to the corresponding Painlevé tau functions. We show that the behaviour of the Joyce structure near the zero-section can be studied analytically through poles of Painlevé equations. The systematic treatment gives a blueprint for the study of more general Joyce structures associated to meromorphic quadratic differentials on the Riemann sphere.