高阶休眠算符的算术提升与二维TQFT
Arithmetic liftings and 2d TQFT for dormant opers of higher level
AI总结：
本文研究正特征下休眠算符的模空间，构造了紧化模空间，证明了n=2时的平展性并导出典范对角提升，还发现次数函数构成2d TQFT，提供了计数这类算符和二阶微分方程的有效方法。
摘要：
本文是算符计数几何领域的一项进展，将该主题的研究推进到我们此前工作的基础之上。受正特征下线性微分方程计数问题的启发，我们从算术和组合的角度研究了算符的模空间。我们构造了一个紧化模空间，用于分类特征p>0的带点稳定曲线上的休眠$\text{PGL}_n^{(N)}$-算符（即N阶休眠$\text{PGL}_n$-算符）。核心结果之一是当n=2时该空间的平展性，这一结论通过对相关形变空间的细致分析得以证明。这一性质使得一般曲线上的每个休眠$\text{PGL}_2^{(N)}$-算符都存在一种到特征$p^N$的算术提升，我们称之为典范对角提升。另一方面，这种平展性还意味着秩2情形下模空间的次数函数满足由底曲线的各类粘合态射所决定的因子分解性质。也就是说，该次数函数构成了一个2d TQFT（即二维拓扑量子场论）；这使得我们可以用三价图上的边编号以及广义有理多面体内部的格点来描述休眠$\text{PGL}_2^{(N)}$-算符。这些结果为计算这类对象以及特征$p^N$下具有完备解系的二阶微分方程的数量提供了一种有效方法。
英文摘要：
This manuscript represents an advance in the enumerative geometry of opers that takes the subject beyond our previous work. Motivated by a counting problem of linear differential equations in positive characteristic, we investigate the moduli space of opers from arithmetic and combinatorial points of view. We construct and investigate the moduli space classifying dormant $\mathrm{PGL}_n^{(N)}$-opers (i.e., dormant $\mathrm{PGL}_n$-opers of level $N$) on pointed stable curves in characteristic $p>0$. One of the key results is the generic étaleness of that space for $n=2$, which is proved by obtaining a detailed understanding of relevant deformation spaces. This fact induces a certain arithmetic lifting of each dormant $\mathrm{PGL}_2^{(N)}$-oper on a general curve to characteristic $p^N$; this lifting is called the canonical diagonal lifting. On the other hand, the generic étaleness also implies that the degree function for the moduli spaces in the rank $2$ case satisfies factorization properties determined by various gluing morphisms of the underlying curves. That is to say, the degree function forms a $2$d TQFT (= a $2$-dimensional topological quantum field theory); it leads us to describe dormant $\mathrm{PGL}_2^{(N)}$-opers in terms of edge numberings on trivalent graphs, as well as lattice points inside generalized rational polytopes. These results yield an effective way of computing the numbers of such objects and $2$nd order differential equations in characteristic $p^N$ with a full set of solutions.
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