Orbifolds, Higher Spin Curves, and Algebraic Structures

本提案的目标是研究受高自旋结构的曲线模空间和映射成轨道的曲线模空间之间的类比启发的几何结构。前者与高KdV可积层次有关，后者与轨道的量子上同调有关。我们将分析由可积系统提出的代数结构和这些空间的拓扑结构。KdV方程是一个非线性方程，它模拟了水波在直线通道中的行为，是解相互兼容的方程层次中的第一个。这种层次结构及其推广与具有内部对称性的附加场的曲面密切相关。我们将调查这种关系及其后果。

DMS-0204824Takashi KimuraThe goal of this proposal is to investigate geometric constructions inspired by the analogy between the moduli space of curves together with higher spin structures and the moduli space of curves together with maps into orbifolds.The former is related to higher KdV integrable hierarchies while the latter is related to the quantum cohomology of orbifolds. We will analyze the resulting algebraic structures suggested by integrable systems and the topology of these spaces.The KdV equation is a nonlinear equation modeling the behavior of water waves in a straight channel and is the first of a hierarchy of equations whose solutions are mutually compatible. Such hierarchies and their generalizations are intimately related to surfaces endowed with additional fields possessinginternal symmetries. We will investigate this relationship and its ramifications.