Enumerative Geometry, Algebra, and Combinatorics in String Theory

该奖项支持旨在进一步建立抽象数学与理论物理（特别是弦理论）之间桥梁的研究。这种关系自然来自超对称的概念，它假设在非常小的长度尺度上，玻色子和费米子粒子在自然界中存在精确的宇称。由此产生的代数结构一方面在弦理论和黑洞熵之间建立了新的、深刻的关系，另一方面在抽象的代数几何和拓扑之间建立了关系。这种独特的互动反过来又导致了两个学科的重要进步和新的理想。它还促进了一种新的思维方式，将数学的严谨性与弦理论的物理洞察力和灵活性相结合。本项目的目标是在改进的Donaldson-Thomas轨道理论的基础上取得重大进展，并发展一种新的关于Calabi-Yau三倍轴模空间的上同不变量的方法。第一个方向是利用局部化和壁交证明新的轨道精细化稳定对不变量组合公式。这是目前至少两个主要开放问题的一个重要组成部分，这两个问题是关于纯分支特征变体的上同调，以及结不变量的代数几何构造。该项目的第二部分旨在利用弦对偶性开发一种新的方法来研究Calabi-Yau三折上二维轮轴模空间的上同调。主要思想是建立这类模空间的上同调与劳门空间上的手性代数之间的具体关系。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识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 that aims to further build the bridge between abstract mathematics and theoretical physics (in particular string theory). This relationship follows naturally from the concept of supersymmetry, which postulates that at very small length scales there is an exact parity between bosonic and fermionic particles in nature. The resulting algebraic structure leads to new and deep relations between string theory and black hole entropy on one side, and abstract algebraic geometry and topology on the other. This unique interaction leads in turn to important advances and novel ideals in both disciplines. It also promotes a new way of thinking combining mathematical rigor with the physical insight and flexibility of string theory. The goal of this project is to make significant advances in the refined Donaldson-Thomas theory of orbifolds as well as develop a new approach to cohomological invariants of moduli spaces of sheaves on Calabi-Yau threefolds. The first direction aims to prove new combinatorial formulas for orbifold refined stable pairs invariants using localization and wall-crossing. This is an important component of at least two current major open problems, concerning the cohomology of tamely ramified character varieties, as well as an algebraic geometric construction of knot invariants. The second part of the project aims to develop a new approach to the cohomology of moduli spaces of two dimensional sheaves on Calabi-Yau threefolds using string duality. The main idea is to build a concrete relation between the cohomology of such moduli spaces sheaves and chiral algebras on Laumon spaces.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.