In this paper, we show that it is always possible to deform a differential equation ∂xΨ(x) = L(x)Ψ(x) with L(x)∈sl2(C)(x) by introducing a small formal parameter ℏ in such a way that it satisfies the topological type properties of Bergere, Borot, and Eynard [Annales Henri Poincare 16(12), 2713–2782 (2015)]. This is obtained by including the former differential equation in an isomonodromic system and using some homogeneity conditions to introduce ℏ. The topological recursion is then proved to provide a formal series expansion of the corresponding tau-function whose coefficients can thus be expressed in terms of intersections of tautological classes in the Deligne–Mumford compactification of the moduli space of surfaces. We present a few examples including any Fuchsian system of sl2(C)(x) as well as some elements of Painleve hierarchies.

中文翻译：

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有理微分系统的等向变形和拓扑递归重构：sl2 情况

在本文中，我们表明总是可以通过引入一个小的形式参数ℏ来使微分方程 ∂xΨ(x) = L(x)Ψ(x) 变形为 L(x)∈sl2(C)(x)以这样的方式满足 Bergere、Borot 和 Eynard 的拓扑类型属性 [Annales Henri Poincare 16(12), 2713–2782 (2015)]。这是通过将前一个微分方程包含在等单系统中并使用一些齐次条件引入 ℏ 来获得的。然后证明拓扑递归提供了相应 tau 函数的形式级数展开，其系数因此可以根据表面模空间的 Deligne-Mumford 紧缩中的重言类的交集来表示。我们提供了一些示例，包括 sl2(C)(x) 的任何 Fuchsian 系统以及 Painleve 层次结构的一些元素。