Operadic Structures in Topological Recursion

拓扑递归是几何学中的一个新发展，它将一个无限的量族（称为相关函数）分配给光谱曲线。虽然它在许多数学分支中扮演着重要的角色，如枚举几何，Gromov-Witten理论和随机矩阵理论，但它仍然是一个知之甚少的现象。该项目的目标是建立一个通用的数学框架下的拓扑递归。这种结构是通过新引入的拓扑递归（TR）运算符和TR合作运算符的概念来表达的，并且允许人们理解Eynard Orantin理论出现的枚举和几何问题的类型，计算这种问题的谱曲线的精确机制，以及相反地，与任意谱曲线相关联的具体几何或枚举问题。具体地说，Eynard-Orantin递归的组合结构很容易被看作是在TR操作上建模的，而在许多出现Eynard-Orantin递归的具体例子中，例如稳定曲线的模空间的相交数，Hurwitz理论，Kontsevich的矩阵积分等，有一个自然出现的TR合作。一般来说，TR操作数的拉普拉斯变换是用来确定谱曲线的。另一方面，费曼变换（出现在模运算中）的一种变体，在拓扑空间范畴中将TR运算转化为TR余运算，它配备了一个自然测度。TR协运算的体积是用来确定作为原TR运算基础的Eynard-Orantin相关函数的。该项目的成功完成将大大有助于理解拓扑递归的作用和性质，并将在枚举几何和Gromov-Witten理论中提供一些有用的应用。研究的领域是黎曼曲面的模空间及相关空间。这样的结构出现在高能物理中，对我们理解镜像对称性和更一般的弦理论至关重要。然而，潜在的应用并不局限于理论物理。由于表面在日常生活中无处不在，对于这样一个抽象的主题，有一个令人惊讶的具体应用。例如，带状图，用于探索拓扑递归的基本工具之一，可以用于面部识别算法。此外，它们出现在费曼图展开中，用于模拟粒子碰撞。

Topological recursion is a recent development in geometry that assigns an infinite family of quantities (called correlation functions) to a spectral curve. Although it plays an important role in many branches of mathematics, such as enumerative geometry, Gromov-Witten theory, and random matrix theory, it is still a poorly understood phenomenon. The goal of the proposed project is to establish a general mathematical framework underlying topological recursion. This structure is expressed through the newly introduced concepts of a topological recursion (TR) operad and TR co-operad, and allows one to understand the types of enumerative and geometric problems where Eynard-Orantin theory appears, a precise mechanism to calculate the spectral curve of such a problem, and, conversely, a concrete geometric or enumerative problem associated to an arbitrary spectral curve. To be specific, the combinatorial structure of Eynard-Orantin recursion is easily seen to be modeled on a TR operad, while in many specific examples where Eynard-Orantin recursion appears, such as intersection numbers of moduli spaces of stable curves, Hurwitz theory, Kontsevich's matrix integral, etc., there is a naturally appearing TR co-operad. In general, the Laplace transform of a TR operad is conjectured to determine the spectral curve. On the other hand, a variation of the Feynman transform (which appears in modular operads), takes a TR operad to a TR co-operad in the category of topological spaces, which comes equipped with a natural measure. The volume of the TR co-operad is conjectured to determine the Eynard-Orantin correlation functions which underlie the original TRoperad.Successful completion of the project would go a long way to understanding the role and nature of topological recursion in general, and would provide several useful applications in enumerative geometry and Gromov-Witten theory. The area being investigated is the moduli space of Riemann surfaces, and related spaces. Such constructions arise in high energy physics and are crucial in our understanding of mirror symmetry, and more generally string theory. However, potential applications are not limited to theoretical physics. Because of the ubiquity of surfaces in everyday life, there are a surprising number of concrete applications for such an abstract subject. For example, ribbon graphs, one of the fundamental tools used to explore topological recursion, can be utilized in facial recognition algorithms. In addition, they appear in Feynman graph expansions, which are used to model particle collisions.