Joyce structures from quadratic differentials on the sphere

Abstract

Motivated by known examples of Joyce structures on spaces of meromorphic quadratic differentials, we consider the isomonodromic deformations of particular second-order linear ODEs with rational potential. We show the infinitesimal isomonodromic deformations are the kernel of a closed $2$-form arising from the intersection pairing of an algebraic curve defined by the potential. This observation enables us to construct Joyce structures on a class of moduli spaces of meromorphic quadratic differentials on the Riemann sphere, and provides a new, geometric description of the hyper-K\"ahler structures of previously computed examples. We focus on the case of moduli of quadratic differentials with poles of odd orders, where we obtain a complex hyper-K\"ahler metric with homothetic symmetry. We also include an example corresponding to the moduli space of quadratic differentials with four simple poles, which is a version of the classical isomonodromy problem that leads to the Painlev\'{e} VI equation.