Enumerative Geometry of Hitchin Systems and Topological Quantum Field Theory

该奖项支持数学物理方面的研究。多年来，各种空间的拓扑和几何结构一直吸引着数学家和物理学家。目前的建议存在于代数、几何、组合学和分析之间的界面上，并应用于二维拓扑量子场论。近年来，国际和平研究所组织了几个讲习班，专门促进不同研究界之间的互动。PI将继续在本项目强调的方向上促进计数几何、代数和量子化理论之间的互动。拟议的项目旨在利用带状图的边收缩操作来理解新的观点。这些运算最初被用来给出任意亏格的广义加泰罗尼亚数的递推关系。这个递推蕴含了稳定的点曲线的模空间上重言余切类的交数的DVV公式。PI和一位合作者通过带状图的边收缩操作发现了2D拓扑量子场论的另一种公理形式。基于边收缩的规则集也代表了拓扑递归的关键结构。边收缩公理既反映了Frobenius代数的结构，又反映了拓扑曲面的裤子分解对。由Mirzakhani首先利用穿孔黎曼曲面的裤子分解，给出了具有固定长度测地边界分量的双曲曲面模空间的Weil-Petersson体积的递推公式。利用Frobenius代数的乘法和余法，我们的目的是以同样的方式给出上同调场理论的另一种公理定义。这个项目的目标是研究拓扑递归、Weil-Petersson卷的Mirzakhani递归、半单Frobenius代数的CohFT分类定理和特征标变量之间的相互作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

This award supports research in mathematical physics. The topological and geometric structures of various kinds of spaces have captivated mathematicians and physicists for many years. The current proposal lies at the interface between algebra, geometry, combinatorics and analysis with applications to 2D-topological quantum field theory. In recent years the PI has organized several workshops that have specifically facilitated interactions between different research communities. The PI will continue to promote interactions between enumerative geometry, algebra and the theory of quantization in the direction emphasized in this project.The proposed project is aimed at understanding a new point of view utilizing edge contraction operations of ribbon graphs. These operations were originally used to give a recursion relation of the generalized Catalan numbers of arbitrary genus. This recursion implies the DVV formula for the intersection numbers of tautological cotangent classes on the moduli space of stable pointed curves. The PI and a collaborator have discovered, an alternative axiomatic formulation for 2D-topological quantum field theory by edge contraction operations of ribbon graphs. The set of rules based on edge contractions also represent the key structure of topological recursion. The edge contraction axioms reflect both the structure of a Frobenius algebra and the pair of pants decomposition of a topological surface. The pair of pants decomposition of a punctured Riemann surface was first used by Mirzakhani to give a recursion of Weil-Petersson volumes of the moduli space of hyperbolic surfaces with geodesic boundary components of fixed lengths. Using the multiplication and comultiplication of the Frobenius algebra we aim at giving alternative axiomatic definition of cohomological field theories in the same way. The goal of this project is to study the interplay between topological recursion, Mirzakhani recursion of Weil-Petersson volumes, the classification theorem of CohFT for semi-simple Frobenius algebra and character varieties.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.