作者：Alexandr Buryak Danil Gubarevich
代数曲线的模空间的拓扑结构和可积系统理论之间的深层关系的许多表现之一是Arsie、Lorenzoni、Rossi和第一作者最近的构建，他们将进化偏微分方程的可积系统与F-上同调场论（F-CohFT）相关联，它是满足某些自然分裂财产的曲线模空间上的上同调类的集合。通常，这些偏微分方程在色散参数中具有无限扩展，这是因为它们涉及任意大亏格曲线的模量空间的贡献。在本文中，对于每个秩$N\ge2$，我们给出了一个没有单位的F-CohFT族，对于该族，相关可积系统的方程在色散参数中具有有限展开。对于$N=2$，我们显式地计算这个可积系统的主流。
One of many manifestations of a deep relation between the topology of the moduli spaces of algebraic curves and the theory of integrable systems is a recent construction of Arsie, Lorenzoni, Rossi, and the first author associating an integrable system of evolutionary PDEs to an F-cohomological field theory (F-CohFT), which is a collection of cohomology classes on the moduli spaces of curves satisfying certain natural splitting properties. Typically, these PDEs have an infinite expansion in the dispersive parameter, which happens because they involve contributions from the moduli spaces of curves of arbitrarily large genus. In this paper, for each rank $N\ge 2$, we present a family of F-CohFTs without unit, for which the equations of the associated integrable system have a finite expansion in the dispersive parameter. For $N=2$, we explicitly compute the primary flows of this integrable system.
论文链接：http://arxiv.org/pdf/2303.13356v1
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