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学，量子引力和弦理论两个领域之间的桥梁。该项目的主要目标是双重的。一方面，它旨在发现新的结构，并使用自然物理现象作为概念灵感的来源，对长期存在的数学问题进行新的阐述。另一方面，物理问题的严格数学表述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中的结的分类，这是一个在抽象数学和理论物理中都有深刻分支的问题。三维的结可以很容易地可视化：一个结只是一段绳子以复杂的方式缠绕在一起，其两端连接在一起。纽结分类问题的主要目标是将具体的数学不变量分配给可以区分两种不同缠结的对象。这个问题原来是出奇地难。拟议的研究的一个重要部分是集中在构建多项式结不变量通过计数量子粒子在抽象模型出现弦理论。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。这个提议中包含的其他项目同样集中在弦理论中的量子粒子计数和枚举代数几何中的几何对象计数之间的关系上。该研究计划为所有级别的学生提供多种机会-从本科到高级研究生-以及博士后研究员在职业生涯的早期阶段参与研究。从一个更技术的角度来看，目前的建议，旨在取得重大进展，在几个长期存在的数学问题，如上同调的字符品种，建设结不变量，并在枚举几何模块相关的问题。这项工作的中心思想是，这样的问题是自然相关的BPS状态计数问题在弦理论。然后，弦对偶性最终导致了新的和精确的数学表达式，表明许多这样的问题自然发生在动机唐纳森-托马斯不变量的卡-丘三倍。这提供了新的证明策略，并开辟了新的研究方向，进一步揭示了意想不到的关系。更具体地说，我们的目的是将这种策略应用于研究广泛分歧的字符簇的上同调，Khovanov-Rozansky不变量的非代数结，和之间的关系的模形式和Donaldson-Thomas不变量的K3纤维卡-丘三倍。

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.