标题： 乔伊斯结构与潘勒韦方程的奇点 显示英文标题

标题： Joyce structures and poles of Painlevé equations

摘要： Joyce结构是一类几何结构，它们最初与Donaldson-Thomas理论相关。有一类特殊的例子，称为类$S[A_1]$，其基础流形参数化了一些固定亏格的黎曼曲面，并配备了具有固定阶数极点的亚纯二次微分。我们研究了这类结构中的两种Joyce结构，这些结构与Painlevé II和III$_3$方程相关的等单调系统有关。我们给出了这些Joyce结构的Plebański函数的显式公式，并计算了几个相关对象，包括它们的tau函数，我们明确地将其与相应的Painlevé tau函数联系起来。我们展示了Joyce结构在零截面附近的性态可以通过Painlevé方程的极点通过解析方法来研究。这种系统的处理为研究更一般的Joyce结构提供了一个蓝图，这些结构与黎曼球面上的亚纯二次微分相关。 显示英文摘要

摘要： 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.