标题： 点插入技术与开$r$-自旋理论 II: 零亏格中的交点理论 显示英文标题

标题： The point insertion technique and open $r$-spin theories II: intersection theories in genus-zero

摘要： 这些论文[5, 3, 6, 19, 20]开启了对开放$r$-自旋和开放FJRW交理论的研究，并将其与可积层次结构和镜像对称相关联。本文使用了一种新技术，即点插入技术，该技术在前一篇论文[36]中开发，用于定义新的开放r-自旋和开放FJRW交理论。这些新构造为之前猜想存在的理论提供了潜在的候选对象： $\bullet$ K. Hori [23] 预测了存在具有$\lfloor\frac{r}{2}\rfloor$种边界状态的开放$r$-自旋理论。 所构造的理论在[5, 3]中仅有一种类型的边界状态。 在本文中，我们描述了由$\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$标记的$\lfloor\frac{r}{2}\rfloor$个开放$r$自旋理论，其中第$\mathfrak{h}$个理论具有$\mathfrak{h}+1$个边界态。我们证明了$\mathfrak{h}=0$理论等价于 [5, 3] 构造，并计算了所有这些理论的所有交点数。$\bullet$在 [1] 中，K. Aleshkin 和 C.C.M. Liu 提出了五次费马 FJRW 理论存在的猜想。我们构建了这样的 FJRW 理论，并提供了证据表明这是所猜想的理论。我们还解释了如何使用点插入技术来构建其他满足相同普遍递归关系的开放枚举理论。 显示英文摘要

摘要： The papers [5, 3, 6, 19, 20] initiated the study of open $r$-spin and open FJRW intersection theories, and related them to integrable hierarchies and mirror symmetry. This paper uses a new technique, the point insertion technique, developed in the prequel [36], to define new open r-spin and open FJRW intersection theories. These new constructions provide potential candidates for theories whose existence was conjectured before: $\bullet$ K. Hori [23] predicted the existence of open $r$-spin theory with $\lfloor\frac{r}{2}\rfloor$ types of boundary states. The one constructed in [5, 3] has only one type of boundary state. In this work we describe $\lfloor\frac{r}{2}\rfloor$ open $r$-spin theories, labelled by $\mathfrak{h}\in\{0,\ldots,\lfloor\frac{r}{2}\rfloor-1\},$ where the $\mathfrak{h}$-th one has $\mathfrak{h}+1$ boundary states. We prove that the $\mathfrak{h}=0$ theory is equivalent to the [5, 3] construction, and calculate all intersection numbers for all these theories. $\bullet$ In [1] K. Aleshkin and C.C.M. Liu conjectured the existence of a quintic Fermat FJRW theory. We construct such an FJRW theory, and provide evidence that this is the conjectured theory. We also explain how the point insertion technique can be used for constructing other open enumerative theories, satisfying the same universal recursions.