In this paper we construct a family of cohomology classes on the moduli space of stable curves generalizing Witten's r-spin classes. They are parameterized by a phase space which has one extra dimension and in genus 0 they correspond to the extended r-spin classes appearing in the computation of intersection numbers on the moduli space of open Riemann surfaces, while when restricted to the usual smaller phase space, they give in all genera the product of the top Hodge class by the r-spin class. They do not form a cohomological field theory, but a more general object which we call F-CohFT, since in genus 0 it corresponds to a flat F-manifold. For $r=2$ we prove that the partition function of such F-CohFT gives a solution of the discrete KdV hierarchy. Moreover the same integrable system also appears as its double ramification hierarchy.

在本文中，我们在推广Witten的r-自旋类的稳定曲线的模空间上构造了一组同调类。它们由具有一个额外维数的相空间参数化，并且在归类0中，它们对应于扩展R-自旋类，该类出现在开放Riemann曲面的模空间上相交数的计算中，而当限制在通常较小的相空间中时，它们在所有类别中都以r -spin类给出顶级Hodge类的乘积。它们不构成同调场论，而是一个更一般的对象，我们称为F-CohFT，因为在0属中，它对应于平坦的F流形。为了$\mathrm{[R}=\mathrm{2个}$我们证明了这种F-CohFT的分区函数给出了离散KdV层次结构的解决方案。而且，相同的可积系统也表现为其双重分支层次。