Enumerative Geometry, Algebra, and Combinatorics in String Theory

弦理论中的枚举几何、代数和组合学

• 批准号: 1802410
• 负责人: Duiliu Diaconescu
• 金额: $ 18.5万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-08-01 至 2022-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1802410&HistoricalAwards=false
• 关键词: Enumerative; Geometry; Algebra; Combinatorics; String

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.

该奖项支持旨在进一步在抽象数学和理论物理(特别是弦理论)之间架起桥梁的研究。这种关系自然源于超对称性的概念，该概念假定在非常小的长度尺度下，在自然界中玻色子和费米子粒子之间存在精确的对等。由此产生的代数结构一方面导致弦理论和黑洞熵之间新的深层次的联系，另一方面导致抽象的代数几何和拓扑之间的关系。这种独特的互动反过来又导致了这两个学科的重要进步和新颖的理想。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奥比诺德理论方面取得重大进展，并发展一种新的方法来研究Calabi-Yau三重层上的层的模空间的上同调不变量。第一个方向是利用局部化和跨越墙的方法证明新的组合公式。这是当前至少两个主要公开问题的重要组成部分，涉及驯服分支特征簇的上同调以及纽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发展一种新的方法来证明Calabi-Yau三重上的二维层轮模空间的上同调。其主要思想是在Laumon空间上这种模空间、滑轮和手征代数的上同调之间建立具体的关系。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。