Algebraic Geometry and Moduli Spaces in String Theory

本研究项目旨在建立抽象数学与理论粒子物理的两个领域，量子引力和弦理论之间的桥梁。这个项目的主要目标是双重的。一方面，它旨在发现新的结构，并利用自然物理现象作为概念灵感的来源，为长期存在的数学问题提供新的思路。另一方面，物理问题的严格数学公式有望为重要的理论物理问题（如黑洞熵和量子粒子动力学）提供有价值的见解。一个具体的说明性例子是三维结的分类，这是一个在抽象数学和理论物理中都具有深刻分支的问题。三维结可以很容易地可视化：一个结只是一根绳子以一种复杂的方式缠绕在一起，它的两端连接在一起。结分类问题的主要目标是为这些物体分配具体的数学不变量，以区分两种不同的缠结。这个问题出人意料地难。提出的研究的一个重要部分是通过计算弦理论中出现的抽象模型中的量子粒子来构造多项式结不变量。虽然这些理论模型与我们的世界没有直接关系，但它们确实导致了新颖而迷人的数学结构，以及理论物理学中重要的概念进步。本提案中包含的其他项目同样集中于弦理论中的量子粒子计数与枚举代数几何中几何物体计数之间的关系。这个研究项目为各个层次的学生提供了多种机会，从本科生到高级研究生，以及博士后，在他们职业生涯的早期阶段参与研究。从更技术性的角度来看，当前的提案旨在在几个长期存在的数学问题上取得重大进展，例如字符变体的上同调，结不变量的构造以及与枚举几何中的模块化相关的问题。这项工作的中心思想是，这些问题与弦理论中的BPS状态计数问题自然相关。然后，弦对偶最终导致了新的和精确的数学猜想，表明许多这样的问题在Calabi-Yau三倍的动机Donaldson-Thomas不变量的背景下自然发生。这提供了新的证明策略，开辟了新的研究方向，揭示了进一步意想不到的关系。更具体地说，目的是将这一策略应用于广泛分支特征变体的上同调、非代数结的Khovanov-Rozansky不变量以及k3纤维Calabi-Yau三倍的模形式与Donaldson-Thomas不变量之间的关系的研究。

This research project aims to construct a bridge between abstract mathematics and two areas in theoretical particle physics, quantum gravity and string theory. The main goal of this project is twofold. On the one hand, it aims to discover new constructions and shed new light on longstanding mathematical problems using natural physical phenomena as a source of conceptual inspiration. On the other, a rigorous mathematical formulation of physical problems is expected to provide valuable insight into important theoretical physics problems such as black hole entropy and quantum particle dynamics. A concrete illustrative example is the classification of knots in three dimensions, a problem with deep ramifications both in abstract mathematics and theoretical physics. Three-dimensional knots can be easily visualized: a knot is simply a piece of rope tangled up in a complicated way with its ends joined together. The main goal of the knot classification problem is to assign concrete mathematical invariants to such objects that can distinguish between two different tangles. This question turns out to be surprisingly difficult. A significant part of the proposed research is focused on constructing polynomial knot invariants by counting quantum particles in abstract models emerging from string theory. While such theoretical models are not directly related to our world, they do lead to novel and fascinating mathematical constructions, as well as important conceptual advances in theoretical physics. The other projects contained in this proposal are similarly centered on the relation between quantum particle counting in string theory and counting of geometric objects in enumerative algebraic geometry. This research program offers multiple opportunities for students of all levels -- ranging from undergraduate to advanced graduate -- as well as postdoctoral fellows to get involved in research at early stages in their careers. From a more technical point of view, the current proposal aims to make significant advances in several long standing mathematical problems such as the cohomology of character varieties, the construction of knot invariants, and questions related to modularity in enumerative geometry. The central idea of this work is that such problems are naturally related to BPS states counting problems in string theory. Then string duality ultimately leads to new and precise mathematical conjectures showing that many such problems occur naturally in the context of motivic Donaldson-Thomas invariants of Calabi-Yau threefolds. This provides new proof strategies and opens new directions of research, unraveling further unexpected relations. More concretely, the aim is to apply this strategy to the study of the cohomology of wildly ramified character varieties, Khovanov-Rozansky invariants for non-algebraic knots, and the relation between modular forms and Donaldson-Thomas invariants of K3-fibered Calabi-Yau threefolds.