圆盘模空间的相交理论、开KdV与Virasoro代数
Intersection theory on moduli of disks, open KdV and Virasoro
AI总结：
该研究定义了稳定带点圆盘模空间的下降积分理论，证明其与开KdV层级τ函数、半Virasoro代数的关系，通过几何递推计算积分，得到适用于所有亏格的Witten猜想开版本。
摘要：
我们定义了稳定带点圆盘模空间上的下降积分理论。证明了下降积分是开KdV层级τ函数的系数，还证明了这些积分与半Virasoro代数表示之间的关系。构建该理论需要深入研究多值边界条件的同伦类，我们利用基于边界条件与模空间组合结构的几何递推来计算积分，同时还对定向性进行了详细分析。我们的开KdV约束与Virasoro约束唯一确定了高亏格开下降积分理论。由此，我们得到了关于稳定曲线Deligne-Mumford空间上下降积分的Witten猜想的开版本（适用于所有亏格）。
英文摘要：
We define a theory of descendent integration on the moduli spaces of stable pointed disks. The descendent integrals are proved to be coefficients of the $τ$-function of an open KdV heirarchy. A relation between the integrals and a representation of half the Virasoro algebra is also proved. The construction of the theory requires an in depth study of homotopy classes of multivalued boundary conditions. Geometric recursions based on the combined structure of the boundary conditions and the moduli space are used to compute the integrals. We also provide a detailed analysis of orientations. Our open KdV and Virasoro constraints uniquely specify a theory of higher genus open descendent integrals. As a result, we obtain an open analog (governing all genera) of Witten's conjectures concerning descendent integrals on the Deligne-Mumford space of stable curves.
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