矩阵积分中关联函数的代数结构研究

In the formal matrix integral, the topological expansions of correlation functions are generating functions of some surface invariants, such as enumerating discrete surfaces. They have rich structures. The study on algebras implicit in the structure is one of intersting problems for a long time. The program will explore the algebras on the following two sides. First, applying the procedure in studying topological recursion for simple ramification, we try to give an Hopf algebra in generalized topological recursion for arbitrary ramification. Second, we aim to present the correspondence between cluster algebras and topological expansions by the aid of geometric explainations for terms in topological expansions. The Hopf algebra in generalized topological recursion allows to explore the quantum origin of toplogical recursion. It can also improve our comprehensions to topological recursion. On the other side, the correspondence between cluter algebras and topological expansions, will give a maneuverable approach to compute the invariants in cluster algebras by topological recursion.

在形式矩阵积分中，关联函数的拓扑展开项，是计数离散曲面等曲面不变量的生成函数。它们有着丰富的结构。这些结构所蕴含的代数，一直以来是大家关心的问题之一。本项目从两个角度，研究拓扑展开中的代数结构。第一，采用在单分歧点拓扑递归中定义Hopf代数结构的研究方法，给出任意分歧点代数曲线上拓扑递归的Hopf代数结构；第二，利用拓扑展开项对应离散曲面的几何解释，给出cluster代数与关联函数拓扑展开项之间的关系。在一般的拓扑递归中定义Hopf代数，可以进一步研究拓扑递归结构的量子背景，探索拓扑递归结构的来源；另一方面，将拓扑展开项与cluster代数对应起来，拓扑递归公式给出了一种计数cluster代数中一些离散变量的简便方法。

在形式矩阵积分中，关联函数的拓扑展开项是计数离散曲面等曲面不变量的生成函数，它们有着丰富的代数结构。不同亏格的关联函数之间满足圈方程。本项目主要研究了多圈方程背后的代数结构，它包含了Virasoro代数作为子代数。对推广的拓扑递归结构，通过定义多顶点图上的积与余积，证明了存在Hopf代数结构。这些结果丰富了关联函数背后的代数结构，有助于加深对它的认识。