Algebraic Geometry and Moduli Spaces in String Theory

弦论中的代数几何和模空间

基本信息

  • 批准号:
    1501612
  • 负责人:
  • 金额:
    $ 16万
  • 依托单位:
  • 依托单位国家:
    美国
  • 项目类别:
    Standard Grant
  • 财政年份:
    2015
  • 资助国家:
    美国
  • 起止时间:
    2015-08-01 至 2019-07-31
  • 项目状态:
    已结题

项目摘要

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.
这一研究项目的目的是在抽象数学和理论粒子物理的两个领域--量子引力和弦理论之间建立一座桥梁。这个项目的主要目标有两个。一方面,它的目的是发现新的结构,并利用自然物理现象作为概念灵感的来源来解释长期存在的数学问题。另一方面,严谨的物理问题的数学表述有望为黑洞熵和量子粒子动力学等重要理论物理问题提供有价值的见解。一个具体的说明性例子是三维节点的分类,这是一个在抽象数学和理论物理中都有深刻影响的问题。三维结很容易想象:一个结只是一根绳子,它的两端连接在一起,以一种复杂的方式缠绕在一起。纽结分类问题的主要目标是将具体的数学不变量分配给这样的对象,以区分两个不同的缠结。事实证明,这个问题出人意料地难。这项研究的一个重要部分集中在通过计算弦理论抽象模型中的量子粒子来构造多项式纽结不变量。虽然这样的理论模型与我们的世界没有直接关系,但它们确实带来了新颖而迷人的数学结构,以及理论物理中重要的概念进步。这项提案中包含的其他项目也同样围绕弦理论中的量子粒子计数和计数代数几何中几何对象的计数之间的关系展开。这一研究项目为所有层次的学生--从本科生到高级研究生--以及博士后研究员提供了多种机会,让他们在职业生涯的早期参与研究。从更技术的角度来看,目前的建议旨在在几个长期存在的数学问题上取得重大进展,如特征簇的上同调、纽结不变量的构造以及与枚举几何中的模相关的问题。这项工作的中心思想是,这样的问题自然与弦理论中的BPS态计数问题有关。然后,弦对偶最终导致新的和精确的数学猜想,表明许多这样的问题自然地发生在Calabi-Yau三重不变量的动机Donaldson-Thomas不变量的背景下。这提供了新的证明策略,开辟了新的研究方向,进一步解开了意想不到的关系。更具体地说,目的是应用这一策略来研究广泛分枝特征簇的上同调,非代数纽结的Khovanov-Rozansky不变量,以及K3-fibered Calabi-Yau三重不变量的模形式与Donaldson-Thomas不变量之间的关系。

项目成果

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Duiliu Diaconescu其他文献

Duiliu Diaconescu的其他文献

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{{ truncateString('Duiliu Diaconescu', 18)}}的其他基金

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
  • 财政年份:
    2018
  • 资助金额:
    $ 16万
  • 项目类别:
    Continuing Grant
D-BRANE MODULI SPACES IN MATHEMATICS AND PHYSICS
数学和物理中的 D 膜模空间
  • 批准号:
    0854757
  • 财政年份:
    2009
  • 资助金额:
    $ 16万
  • 项目类别:
    Continuing Grant
Geometry and Vacuum Structure in String Theory
弦理论中的几何和真空结构
  • 批准号:
    0555374
  • 财政年份:
    2006
  • 资助金额:
    $ 16万
  • 项目类别:
    Continuing Grant

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