Operadic Structures in Topological Recursion

拓扑递归中的操作结构

• 批准号: 1308604
• 负责人: Brad Safnuk
• 金额: $ 10.85万
• 依托单位: Central Michigan University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-09-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1308604&HistoricalAwards=false
• 关键词: Operadic; Structures; Topological; Recursion

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.

拓扑递归是几何学中的一项最新发展，它将无穷量族(称为相关函数)赋给一条谱曲线。虽然它在许多数学分支中扮演着重要的角色，如计数几何、Gromov-Witten理论和随机矩阵理论，但它仍然是一个鲜为人知的现象。拟议项目的目标是在拓扑递归基础上建立一个通用的数学框架。这种结构通过新引入的拓扑递归(Tr)算符和tr合作算符的概念来表示，并允许人们理解出现Eynard-Orantin理论的计数和几何问题的类型，计算这种问题的谱曲线的精确机制，反之，与任意谱曲线相关的具体几何或计数问题。具体地说，Eynard-Orantin递推的组合结构很容易被模拟在一个TR算子上，而在许多出现Eynard-Orantin递归的具体例子中，如稳定曲线的模空间的交数、Hurwitz理论、Kontsevich矩阵积分等，自然地出现了一个TrCoopad算子。通常情况下，为了确定谱曲线，需要猜想一个tr算子的拉普拉斯变换。另一方面，Feynman变换的一个变体(出现在模算子中)，将一个tr算子转化为拓扑空间范畴中的一个tr合作算子，它配备了一种自然度量。成功地完成这个项目将对理解拓扑递归的作用和本质有很大帮助，并将在列举几何和Gromov-Witten理论中提供一些有用的应用。所研究的区域是黎曼曲面的模空间及其相关空间。这样的结构出现在高能物理中，对我们理解镜像对称性以及更广泛的弦理论是至关重要的。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，有数量惊人的具体应用。例如，带状图是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，它们还出现在用于模拟粒子碰撞的费曼图展开中。